THE THEORY OF


FUNCTION S


OF A
REAL VARIABLE
AND


THE THEORY OF


FOURIER'S SERIES




CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
C. F. CLAY, MANAGER.
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[All Rights reserved]




THE THEORY OF FUNCTIONS
OF A
REAL VARIABLE
AND


THETHEHORY OF FOURIER'S SERIES
by
E. W. HOBSON, Sc.D., F.R.S.
Fellow of Christ's College, and Stokes Lecturer in Mathematics
in the University of Cambridge
Cambridge
at the University Press


I907




rCamibtitrg.
PRINTED BY JOHN CLAY, M.A.
AT THE UNIVERSITY PRESS.




PREFACE.


THE theory of functions of a real variable, as developed during the last
few decades, is a body of doctrine resting, first upon a definite conception
of the arithmetic continuum which forms the field of the variable, and which
includes a precise arithmetic theory of the nature of a limit, and secondly,
upon a definite conception of the nature of the functional relation. The
procedure of the theory consists largely in the development, based upon
precise definitions, of a classification of functions, according as they possess, or
do not possess, certain peculiarities, such as continuity, differentiability, &c.,
throughout the domain of the variable, or at points forming a selected set
contained in that domain. The detailed consequences of the presence, or of
the absence, of such peculiarities are then traced out, and are applied for the
purpose of obtaining conditions for the validity of the processes of Mathematical Analysis. These processes, which have been long employed in the
so-called Infinitesimal Calculus, consist essentially in the ascertainment of
the existence, and in the evaluation, of limits, and are subject, in every case,
to restrictive assumptions which are necessary conditions of their validity.
The object to be attained by the theory of functions of a real variable consists
then largely in the precise formulation of necessary and sufficient conditions
for the validity of the limiting processes of Analysis. A necessary requisite
in such formulation is a language descriptive of particular aggregates of
values of the variable, in relation to which functions possess definite peculiarities. This language is provided by the Theory of Sets of Points, also
known, in its more general aspect, as the Theory of Aggregates, which
contains an analysis of the peculiarities of structure and of distribution in
the field of the variable which such sets of points may possess. This theory,
which had its origin in the exigencies of a critical theory of functions, and
has since received wide applications, not only in Pure Analysis, but also in
Geometry, must be regarded as an integral part of the subject. A most
important part of the theory of functions is the theory of the representation
of functions in a prescribed manner, especially by means of series or sequences
of functions of prescribed types. Much progress has recently been made in




vi                             Preface
this part of the subject, results having been obtained which have led to
a classification of functions in accordance with the modes of representation
of which they are capable. The special case of the conditions of representability of functions by means of trigonometrical series was historically
the starting-point in which a great part of the modern development of the
theory of functions of a real variable had its origin.
The course of study, of which the present treatise is the outcome, followed
an order very similar to the historical order in which the subject was
developed. Commencing with the study of Fourier's series, in their application to the problems of Mathematical Physics, and provided with a
knowledge of the Differential and Integral Calculus, of the traditional kind
in which notions of the nature of continuity and of limits founded on an
uncritical use of intuitions of space and time are the stock in trade, I was
led, by the difficulties connected with the theory of these series, and through
an attempt to understand the literature which deals with them, to a study of
the theories of real number, due to Cantor and Dedekind, and to that of the
theory of sets of points. A study of the foundations of the Integral Calculus,
and of the general theory of functions of a real variable formed the natural
continuation of the course. The present work has been written with the
object of presenting in a connected form, and of thus rendering more easily
accessible than hitherto, the chief results which are to be found scattered
through a very large number of memoirs, periodicals, and treatises. I have
endeavoured, as far as possible, to fill up gaps in the various theories which
occur in different parts of the subject. The proofs of theorems have in many
cases been simplified, often in accordance with developments of the theory
later in date than the original proofs; other theorems have been given in a
form more general than that in which they were first discovered. In the
literature of the subject, errors are not infrequent, largely owing to the fact
that spatial intuition affords an inadequate corrective of the theories involved,
and is indeed in some cases almost misleading. Although I have made every
endeavour to attain to accuracy both in form and in substance, it is practically
certain that the present work will form no exception to the rule of fallibility.
Where I have called attention to what I regard as inadequate statements or
errors on the part of other writers, I have done so solely for the purpose of
directing the attention of students to the points in question, and with full
consciousness that, at least in some cases, close examination might shew that
what appeared to me to be erroneous was rather due to some misapprehension
on my part of the meaning of the writers to whom reference is made. On
some points connected with the theory of aggregates, which are at present
matters of controversy, I have expressed definite opinions, although I fully
recognize that, on such matters, a dogmatic attitude of mind is at the present
time wholly out of place, and not unlikely to be avenged when the points
concerned are finally settled to the general satisfaction of mathematicians.




Preface


vii


Chapter I contains a discussion of Number, and includes a full account
of the theories of Real Number, due to Cantor and Dedekind. Whilst an
indication has been given of the fundamental notions upon which the
conceptions of cardinal and ordinal numbers rest, I have not attempted to
reduce these fundamental notions to a minimum of indefinables from which
the whole theory might be deduced by means of formal logic. A slight
perusal of the extremely extensive literature of the Philosophy of Arithmetic
will shew that any such attempt could only have been made by entering upon
a prolonged discussion of a philosophical character, wholly unsuited to a
treatise of professedly mathematical complexion, and that any views expressed
would have had but little prospect of giving general satisfaction to logicians
and philosophers. The modern theory of Real Numbers has been the object
of much criticism by philosophers and others. It has been represented that
the modern extension of the notion of number to the case of irrational
numbers is a sophistical attempt to obliterate the fundamental distinction
between the discrete and the continuous. I venture to think that such
objections consist, in large part at least, of criticisms of the current terminology of the mathematical theories, especially in respect of the extensions
of the use of the word " number," and I think it probable that many of these
criticisms would not survive a fair examination of the theories themselves
apart from the language in which they are expressed. An appropriate
terminology, although a matter of convention, is no doubt a very important
matter in relation to such fundamental matters, as it is conducive to clearness of thought; but the substance of the theories is of incomparably
greater importance than the forms in which they are expressed, and those
theories may be found on examination to be essentially sound, even if their
terminology be regarded as in some respects defective.
Chapter II contains an exposition of the theory of sets of points, and
includes an account of transfinite cardinal and ordinal Arithmetic, of a
somewhat simpler and less general character than will be met with in the
treatment of the general theory of aggregates, in Chapter iII. Students
who do not care to embark upon the discussions in Chapter III will find a
study of Chapter II amply sufficient to enable them to apply the ideas there
developed in the general theory of functions. A slight account only has been
given of the properties of plane sets of points. An account of the important
recent investigations which had their origin in Jordan's theorem, that a
closed curve divides plane space into two regions, would have occupied more
space than was at my disposal. This omission will be less felt than might
have been the case, were not an excellent account of this subject to be found
in Dr W. H. Young's treatise on the theory of sets of points, which has
appeared since this portion of the present work was printed.
In Chapter IV, there will be found a discussion of the main properties of
functions, in relation to continuity, discontinuity, &c., and investigations of




viii
Villl


Preface


the properties of important classes of functions. Although the treatise is
mainly one on functions of a single variable, a considerable amount of space
has been devoted to the consideration of functions of two variables, not only
on account of the intrinsic importance of that subject, but because no
adequate consideration of the properties of functions of a single variable is
possible without the use of functions of two variables, as is seen, for example,
from the consideration that a function defined by means of a sequence of
functions of a single variable is virtually defined as a limit of a function of
two variables.
The foundations of the Integral Calculus, as based upon Riemann's
definition of a definite integral, and its extensions, are discussed in
Chapter v, where an account of the development of the subject from the
point of view of Lebesgue's new definition of the definite Integral is also
given. In later parts of the book I have introduced extensions of Lebesgue's
definition, to the cases of improper integrals, taken over finite or infinite
domains, regarded as the limits of sequences of Lebesgue integrals.
Chapter vi is concerned with functions defined as the limits of sequences
of functions, and contains an account of the principal properties of functions
represented by series, and a discussion of important matters connected with
the modes of convergence of series through whole intervals, or in the neighbourhood of particular points. Various matters relating to the processes of
the Integral Calculus, which had not been considered in Chapter v, are here
dealt with, because their adequate treatment presupposes a knowledge of the
theorems relating to the convergence of sequences of functions. An account
of the very general results recently obtained by Baire, relating to the representability of functions by means of series, will be found in this Chapter.
Chapter vII is devoted to the theory of Fourier's series. No apology is
needed for the selection of this particular mode of representation of functions
for full discussion in a treatise on the theory of functions of a real variable,
in view of the historical relation of Fourier's series to the development of the
general theory. The history of the theory of Fourier's series is exceedingly
instructive, not merely from the point of view of the mathematician, but
also from that of the epistemologist. I have therefore endeavoured, in my
treatment of the subject, to preserve as much of the historical element as
was possible in an account which should contain, in a moderate compass, not
only indications of the various stages of development of the subject, but also
the most recent results that have been obtained. I have made full use of the
greater generality which can be introduced into many of the known results
by means of the employment of the theory of integration developed by
Lebesgue.
In the preparation of the work, the treatises from which I have most
largely drawn information are the German edition of Dini's treatise on the
subject, Stolz's Grundziige der Differential- und Integral-Rechnung, Schonflies'




Preface                             ix
Bericht entitled "Die Entwickelung der Lehre von den Punktmannigfaltigkeiten," and the various treatises on different parts of the subject by Borel
and Lebesgue. I have consulted a very large number of memoirs, articles,
notes, and books, far too numerous to be here particularized. In respect to
the references given throughout the book, I wish it to be understood that
I have made no attempt to settle questions of priority of discovery. The
references given are to be regarded solely as indicating sources of information
from which I have drawn, or where more detailed information on the various
topics is to be found.
I owe a debt of gratitude to my friend Mr J. W. Sharpe, formerly Fellow
of Gonville and Caius College, who has read with the greatest care the proofs
of about two-thirds of the book. Many points of difficulty I have fully
discussed with him; many obscurities of expression have been removed, and
many improvements in substance have been made, owing to the care he has
bestowed in reading the proofs. I felt it as a great loss when, owing to a
temporary failure of health, he was unable to continue his laborious work.
To Dr H. F. Baker, F.R.S., Fellow of St John's College, and Cayley Lecturer
in Mathematics, who has kindly read some of the earlier proofs, I owe several
valuable suggestions. On several points connected with the treatment of
Number in Chapter I, I have had the advantage of consulting Dr James
Ward, F.B.A., Fellow of Trinity College, and Professor of Mental Philosophy
and Logic in the University.
My thanks are due to the officials of the University Press for the
readiness with which they have met my views, and for the care which
they have bestowed upon the work connected with the printing. I desire
especially to express my sense of the value of the excellent work done by
the readers of the Press; to their care is due the elimination of many
typographical and other blemishes which would otherwise have remained
undetected.
E, W. HOBSON.
CHRIST'S COLLEGE, CAMBRIDGE.
Mcay 15, 1907.








SECTIONS
1.
2-4.
5, 6.
7-9.
10-12.
13, 14.
15, 16.
17.
18 —21.
22-27.
28.
29-31.
32, 33.
34.
35-37.
38, 39.
40, 41.
42.
43, 44.


CONTENTS.
CHAPTER I.
NUMBER.
PAGES
Introduction.......  1-3
Ordinal numbers.....                3-7
Cardinal numbers.......        7-10
The operations on integral numbers. 10-12
Fractional numbers.                        12-16
Negative numbers, and the number zero.. 16-18
Irrational numbers...........  19-21
Kronecker's scheme of arithmetization.      21-22
The Dedekind theory of irrational numbers......  22-26
The Cantor theory of irrational numbers.. 26-33
Convergent sequences of real numbers.       33-35
The arithmetical theory of limits....       35-38
Equivalence of the definitions of Dedekind and Cantor... 38-40
The non-existence of infinitesimals.......  40-41
The theory of indices........    41-45
The representation of real numbers... 45-49
The continuum of real numbers.... 49-51
The continuum  given by intuition.......  52
The straight line as a continuum........    53 —56


45.
46, 47.
48.
49-51.
52, 53.
54-56.
57, 58.
59.
60-62.
63.
64-71.
72-75.
76.


CHAPTER II.
THEORY OF SETS OF POINTS.
Introduction
The upper and lower boundaries of a set of points
Limiting point of a set of intervals..
The limiting points and the derivatives of a linear set
The distribution of points of a set in the interval
Enumerable aggregates.
The power of an aggregate 
The arithmetic continuum.
Transfinite ordinal numbers.
The transfinite derivatives of a set of points
Sets of intervals...
Non-dense closed and perfect sets.
Properties of the derivatives of sets..  57-58.  58-60
60.  61-64.  64-66.  66-70.  70-72.  72-74. 74-79.  79-81.  81-90. 90-97
98




xii


Contents


SECTIONS
77-84.
85, 86.
87, 88.
89.
90, 91.
92, 93.
94, 95.
96-99.
100-108.
109.


The content and the measure of sets of points
The content of closed sets 
Ascending sequences of closed sets
Sets of the first and of the second category.
Diminishing sequences of closed sets
The common points of a system of open sets
The analysis of sets in general
Inner limiting sets...
Non-linear sets of points.
Sets of sequences of integers.


PAGES
98-108
i...  108-110. 110-113..  114-117. 117-119. 119-122. 123-127. 127-135
I-..  135-146. 146-147


CHAPTER III.
TRANSFINITE NUMBERS AND ORDER-TYPES.


110.    Introduction....
111.    The cardinal number of an aggregate.
112, 113. The relative order of cardinal numbers.
114, 115. The addition and multiplication of cardinal numbers 
116.    Cardinal numbers as exponents.
117, 118. The smallest transfinite cardinal number
119, 120. The equivalence theorem..
121.    Division of cardinal numbers by finite numbers 
122.    The order-type of simply ordered aggregates
123, 124. The addition and multiplication of order-types
125, 126. The structure of simply ordered aggregates 
127-129. The order-types r7, 0, r...
130-133. Normally ordered aggregates..
134-136. The theory of ordinal numbers.
137-139. The ordinal numbers of the second class
140.    The cardinal number of the second class of ordinals 
141, 142. The general theory of Aleph-numbers.
143-145. The arithmetic of ordinal numbers of the second class
146.    The theory of order-functions..
147-151. The cardinal number of the continuum.
152-163. General discussion of the theory.


148-149
149-151
151-152
152 -153
153-154
154-156
156-159
159-162.162-163
163-164
164-167
167-170
170-175
175-177
177-180
180-181
181-183
183-185
185-186
187-194
194-212


CHAPTER IV.
FUNCTIONS OF A REAL VARIABLE.


164.
165, 166.
167, 168.
169-171.
172.
173, 174.
175.
176-179.
180-182.


Introduction 
The functional relation..
The upper and lower limits of functions
The continuity of functions..
Continuous functions defined for a continuous interval
Continuous functions defined at points of a set 
Uniform continuity.
The limits of a function at a point.
The discontinuities of functions...213-214.214-219.219-221.221-225.225-226.226-228.229-230.230-233.233-237




Contents


xiii
* * 
Xlll


SECTIONS
183-185.
186, 187.
188-190.
191, 192.
193-196.
197-199.
200, 201.
202-208.
209-213.
214, 215.
216-223.
224.
225-228.
229.
230-235.
236, 237.
238-240.
241, 242.


Semi-continuous functions...
The classification of discontinuous functions.
Point-wise discontinuous functions....
Definition of point-wise discontinuous functions by extension
Functions with limited total fluctuation.
The maxima, minima, and lines of invariability of continuous
functions......
The derivatives of functions.
The differential coefficients of continuous functions
The successive differential coefficients of a continuous function 


PAGES
237-240
240-243
243-248
249-252
252-259
259-263
263-266
266-275
275-281


Oscillating continuous functions.                281-284
General properties of derivatives...285-294
Functions with one derivative assigned..       294-295
The construction of continuous functions.       296-299
Functions of two or more variables......299-301
Functions of two variables.....                         301-311
Partial differential coefficients....                311-316
Higher partial differential coefficients...                   316-321
Maxima and minima of a function of two variables.. 322 —324


243-245. Properties of a function continuous with respect to each variable. 324-328
246-249. The representation of a square on a linear interval... 329-336
CHAPTER V.
INTEGRATION.


250.
251-254.
255.
256.
257.
258-264.
265.
266-269.
270-272.
273, 274.
275-280.
281 —284.
285.
286.
287-291.
292-300.
301, 302.
303-309.
310-312.
313-316.
317.
318- 322.
323, 324.
325-328.


Introduction 
The definite integrals of limited functions..
Particular cases of integrable functions...
Properties of the definite integral....
Integrable null-functions and equivalent integrals
The fundamental theorem of the Integral Calculus
Functions which are linear in each interval of a set 
Mean value theorems....
Improper integrals.
Absolutely and conditionally convergent integrals.
Existence and properties of absolutely convergent improper integrals
Non-absolutely convergent improper integrals.
The fundamental theorem of the Integral Calculus for the case


337
338-342
343-344
344-347
347-349
349-357
357
358-364
364-369
369-371
371-377
377-386




of improper integrals..
Geometrical interpretation of integration
Lebesgue's theory of integration
Integrals with infinite limits.
Integration by parts.
Change of the variable in an integral
Double integration...
Repeated integrals.....
Properties of the double integral 
Improper double integrals
Double integral,; over infinite domains 
The transfo:umation of double integrals.


387-388
388-390
390-397
398-407
407-409
410-415
415-421
421-430
430-431
432-442
442-445
445 —452




xiv                                Contents
CHAPTER VI.
FUNCTIONS DEFINED         BY SEQUENCES.
SECTIONS                                                                 PAGES
329, 330. Introduction.....                                453-454
331.    Non-convergent arithmetic series...                     455-457
332-334. Absolutely convergent, and conditionally convergent series.    457-461
335.    Series of transfinite type.....               461-463
336-338. Double sequences and double series.                         464-469
339.    Functions represented by series.....469-470
340, 341. Uniform convergence of series.....          470-474
342.    Non-uniform convergence...                         474-475
343, 344. The continuity of the sum-function..                       475-477
345-348. Tests of uniform convergence.....477-482
349.    Relation of the theory with that of functions of two variables.483-485
350.    The distribution of points of non-uniform convergence..485-487
351, 352. The limits of a sum-function at a point.....487-489
353, 354. The necessary and sufficient conditions for the continuity of the
sum-function......... 489-495
355-359. The convergence of power-series......495-499
360, 361. The convergence of the product of two series....500-501
362-367. Taylor's series........501-512
368.    Maxima and minima of a function of one variable...512-513
369.    Taylor's theorem for functions of two variables....514-515
370-372. Maxima and minima of functions of two variables...515-522
373-377. Functions representable by series of continuous functions..522-532
378.    Baire's classification of functions....... 532-534
379-388. The integration of series.........534-550
389-393. The fundamental theorem of the Integral Calculus for Lebesgue
integrals...........550-559
394.    Integration by parts for Lebesgue integrals...      559-560
395-398. The differentiation of series....                        561-565
399-403. Repeated improper integrals....                        566-576
404-407. Repeated Lebesgue integrals...576-582
408.    Repeated integrals of unlimited functions..      582-586
409 —411. Repeated integrals over an infinite domain..... 586-594
412-414. The limit of an integral containing a parameter.... 594-599
415-420. Differentiation of an integral with respect to a parameter.. 599-607
421-423. The condensation of singularities....607-618
424.    Cantor's method of condensation of singularities...      618-620
425, 426. The construction of continuous non-differentiable functions..620-625
427-429. The construction of a differentiable everywhere-oscillating function  626-634
CHAPTER VII.
TRIGONOMETRICAL SERIES.
430.    Introduction......635
431, 432. The problem of vibrating strings....... 636-638
433.    Special cases of trigonometrical series.....638-639
434, 435. Later history of the theory........639-641




Contents


XV


SECTIONS                                                                      PAGES
436-440. The formal expression of Fourier's series.           641-648
441-443. Particular cases of Fourier's series.......648-656
444.     Dirichlet's integral.......       656-658
445-448. Dirichlet's investigation of Fourier's series.           658 —666
449, 450. Application of the second mean value theorem.            666-669
451.     Uniform  convergence of Fourier's series..            669-670
452 —455. The limiting values of the coefficients in Fourier's series..671-678
456-457. Sufficient conditions of convergence of Fourier's series at a point  678-683
458-461. Conditions of uniform convergence of Fourier's series..      683 —695
462-464. Further investigations of Dirichlet's integral..695-701
465 —467. The non-convergence of Fourier's series.           701-707
468-471. The series of arithmetic means related to Fourier's series..707-714
472.     Properties of the coefficients of Fourier's series...       715-717
473.     The integration of Fourier's series..                          718-719
474.     Properties of Poisson's integral...                         719-722
475, 476. Approximate representation of functions by finite trigonometrical
series........722-725
477.     The differentiation of Fourier's series.....       725-728
General Examples.......728- 730
480-484. Riemann's theory of trigonometrical series.         730-742
485.     Investigations subsequent to those of Riemann..       743-746
486.     The limits of the coefficients of a trigonometrical series.     746-748
487.     Proof of the uniqueness of the trigonometrical series representing
a function.......748 —749
488-490.   The representation of integrable functions.           750-756
491.     The convergence of a trigonometrical series at a point.       756-757
492, 493. Fourier's integral representation of a function.           758-762
APPENDIX.
On transfinite numbers and order-types.                  763-766
LIST OF AUTHORS QUOTED.......767-768


GENERAL INDEX


GEEA. 769-772




CORRIGENDA AND ADDENDA.


Page 96. In example 2, line 7, for "Of each rational number, there is a double representation" read " Of each rational number, not represented by a recurring radix-fraction,
there is a double representation."
Page 267. In the statement of the theorem in ~ 203, for " where 0 is some proper fraction, and
is neither 0 nor 1" read "where 0 is such that 0<0<1."  The number 0 is not
necessarily rational.
Page 268. In line 9, for "for some value of 0 which is a proper fraction, and is neither 1 nor 0 "
read "for some value of 0 which is such that 0< <1."
Page 317. The statement which commences on line 10 from the foot of the page is erroneous.
The repeated limit may have a definite value when lim f (x' 0 + k) -f (xo, y) has
k=o         nk
no definite value. This is illustrated by the example f (x, y) = i (x) + X (y), where
p (x), X (y) are non-differentiable functions. In this case the above single limit does
not exist, but the repeated limit exists, and is zero. The existence of the repeated
limit as a definite number does not therefore necessarily imply the existence of
7f nd     f f
and of     y, the latter of these having been so defined that it only exists
fo                      d 
when     exists. The point is more fully discussed in a paper by the Author in the
ayo
Proc. Lond. Math. Soc., ser. 2, vol. v, " On repeated limits."
Page 318. In the statement of the theorem, for "(3)f (x, y) be continuous with respect to x at
(X0, yc)," read "(3) f (x, y) be such that af(aX0 yo, f (X0 Yboth exist."
aXo   '   Yo
Page 319. Line 5 from the foot of the page. Delete the sentence commencing " The existence
af   af,
ox0 dyo
Page 357. In the Example, line 8, for " let F (x) = - 0 (x, y)" read "let F (x) = - b (x, /)."
Page 394. In line 8 from the foot of the page, for "in accordance with Riemann's definition"
read "in accordance with Lebesgue's definition, the integrand having in each case
only a finite number of values."




CHAPTER I.


NUMBER.
1.  THE   operation  of counting, in which     the integral numbers are
employed, can be carried out by a mind to which discrete objects, which
may be either physical or ideal*, are presented, and which possesses certain
fundamental notions which we proceed to specify.
(1) The notion of unity, a form under which an object is conceived when
it is regarded as a single one. An object so regarded may be either of
a material or of a purely abstract or ideal nature, and may be recognized, for
all other purposes than that of counting, as possessing any degree of complexity. It is sufficient, in order that the object may be regarded under the
form of unity, that it be so far distinct from other objects, as to be recognized
at the time when it is counted, as discrete and identifiable. What external
marks are necessary that an object may be so recognized as discrete, is a
matter for the judgment of the mind at the time when the object is counted.
The unity under which the object is apprehended is a formal or logical, rather
than a natural unity; it is more or less arbitrarily attributed to the object by
the mind.
(2) The notion of a collection or aggregate of objects which is conceived
of as containing more or fewer objects, or as possessing a greater or less
degree of plurality. A group of objects regarded as an aggregate is conceived
of not merely as a plurality of objects to each of which unity is ascribed
as in (1), but also as itself an object to which unity is ascribed when it is
regarded as a single whole. The single objects of which the aggregate is
composed may be spoken of as the elements of the aggregate; such elements
need not possess any parity as regards size or any other special quality, but
may be of the most diverse characters: a certain logical parity is however
* It is held by some authors that the operation of counting is primarily applicable to physical
objects only. Thus, J. S. Mill writes:-" The fact asserted in the definition of a number is a
physical fact. Each of the numbers, two, three, four, etc., denotes physical phenomena, and
connotes a physical property of these phenomena." See Logic, 9th edition, vol. IT, p. 150. That
objects which are not physical, can be counted, was maintained by Leibnitz and by Locke. See
also Frege's Grundlagen der Arithmetik, Breslau, 1884, where an account is given of various
views as to the nature and origin of the idea of Number.


H.


I




2


Number


[CH. I


ascribed to them in the process of counting, in virtue of the fact that each,if
them is regarded as a single object. A sensibly continuous presentation cannot
be regarded as an aggregate containing a plurality of elements, until the mird
has recognized in it sufficiently distinct lines of division- to serve the purpose
of marking off distinct objects within it, the totality of which makesm~p —*ife
whole presentation; for instance, the history of a country could be regarded
as an aggregate of distinct periods, only when sufficiently salient features had
been recognized in that history to warrant a judgment that periods were to
be found in it, each of which had a sufficient degree of discreteness to be
subsumed under the form of unity. In actual counting, the aggregate is
not necessarily determinate before the counting is commenced, but becomes
so when the process is completed; the notion of an aggregate is thus still
necessary to the process of counting, if the process is ever to come to an end,
or to be conceived of as having come to an end.
It has been held* that when an aggregate is counted, the elements must
remain distinct from one another, not disappearing or combining with each
other during the process. That this condition is unnecessary may be seen,
for example, by considering the case of counting breakers on the sea-shore,
or that of counting the vibrations of a pendulum; thus no physical permanence, but only an ideal one, is necessary.
A discussion of the characteristics which an aggregate (not necessarily
finite) must possess, in order that it may be an object of mathematical thought,
will be given in Chapter iI.
(3) The notion of order, in virtue of which relative rank is given to each
object in a collection, so that the collection becomes an ordered aggregate. In
actual counting, the order is assigned to the objects during the process itself,
as an order in time, and this may be done in an arbitrary manner; the order
of the elements in an aggregate may, however, be assigned in a manner
dependent upon their sizes, weights, or other qualities, or in accordance with
their positions in space. Order may, however, be regarded as an abstract
conception, independent of a particular mode of ordering; for an aggregate
to be an ordered one, it is necessary that in some manner or other, each
element be recognized as possessing a certain rank, in virtue of which it is
known as regards any two elements which may be chosen, which of them
has the lower, and which the higher rank. An element is said to precede
any other element of higher rank than itself.
(4) The notion of correspondence, which underlies the process of tallying.
The elements of one aggregate may be made to stand in some logical relation with those of another one, so that a definite element of one aggregate
is regarded as correspondent to a definite element of another aggregate.


I See Helmholtz's Zdhlen und Messen, Leipzig, 1887; Wissens. Abhandl. vol. II, p. 372.




1, 2]                    Ordinal Numbers                           3
The correspondence may be complete, in the sense that to every element
of either aggregate there corresponds one element, and one only, of the other
aggregate; or the correspondence may be incomplete, in which case one of the
aggregates has one or more elements to which no elements in the other
aggregate correspond. In the latter case we say that the aggregate with the
superfluous element or elements contains more elements than the other
aggregate, and that the latter contains fewer elements than the former.
A correspondence between two aggregates is defined when specifications
or rules are laid down which suffice to decide which element of one aggregate
corresponds to each element of the other; so that, in the case of complete
correspondence, no element of either aggregate is without a correspondent one
in the other.
Whether, or how far, these fundamental notions of unity, aggregate, order,
and correspondence should be regarded as derived empirically from experience,
by a process of abstraction, or whether it must be held that they are original
forms which the mind possesses prior to, and as the necessary conditions of
the possibility of such experience, are questions into which it is beyond our
province to enter. It is certain that civilized man possesses these fundamental
notions, and it is highly probable that primitive man possessed them long
before the notion of abstract number had appeared in an explicit and
developed form. The investigation of the origin of these notions, and their
further analysis is a matter for the Psychologist and for the Philosopher.
Mathematical Science, as any other special science, must take its fundamental
notions as data; it is concerned with the analysis of them, only so far
as suffices to establish that they possess the degree of definiteness which
such data must have, if they are to lie at the base of a logically ordered
system.
ORDINAL NUMBERS.
2. If from an ordered aggregate some of the elements are removed, the
aggregate which remains is said to be a part of the original aggregate. It
will be observed that the relative order of any two elements in the part is the
same as the relative order of those elements in the original aggregate.
An ordered aggregate is said to be finite when it satisfies the following
conditions:(1) There is one element which has lower rank than any of the others.
(2) There is one element which has higher rank than any of the others..(3) Every part of the aggregate has an element which has higher rank
than every other element in the part, and also it has an element which has
lower rank than any other element in the part.
1-2




4


Number


[CH. I


These conditions are equivalent to the statement that a finite aggregate,
and also each part of it, has a first and a last element.
Every part of a finite ordered aggregate is also a finite ordered aggregate.
If M be the aggregate, and M1 a part of it, then 1M has a highest and a
lowest element; also every part of M1 being also a part of M, has a lowest
and a highest element; therefore M1 is itself finite.
3. Two finite ordered aggregates are said to be similar when they can be
made to completely correspond, so that to each element of either of them
there corresponds a single element of the other, and so that to any two
elements P, Q of the one there correspond two elements P', Q' of the other,
which have the same relation as regards rank; viz. that if P is of lower rank
than Q, then P' is of lower rank than Q', an: if P is of higher rank than Q,
then P' is of higher rank than Q'.
Two finite ordered aggregates which are similar are said to have the same
ordinal number.
If each of two ordered aggregates is similar to a third, they are similar to
one another. For if an element P of the first corresponds to an element R of
the third, and the element Q of the second corresponds to R, it is clear that
if we make P correspond to Q, the first two aggregates are made to correspond
in such a way that the relative order is preserved.
It thus appears that an ordinal number is characteristic of a class of
similar ordered aggregates.
An aggregate which consists of a single element A, is said to have the
ordinal number one, denoted by the symbol 1. The ordinal number 1 is
characteristic of every aggregate which consists of a single element.
If to the aggregate which consists of an element A, we adjoin a new
element B, and assign to B a higher rank than A, we obtain an aggregate
(A, B) which has an ordinal number 2, characteristic of all aggregates which
are formed in this manner; A is said to be the first element, B the second.
If to an ordered aggregate (A, B), of which the ordinal number is 2, we adjoin
another element C, and regard this as having higher rank than A and B, we
obtain an ordered aggregate (A, B, C), of which the ordinal number is called
3, and is characteristic of all ordered aggregates formed in this manner
Proceeding in this way, if we have formed an ordered aggregate (A, B, C......
H), of which the ordinal number is n, and adjoin to this aggregate a new
element K, we obtain a new aggregate (A, B, C,......H, K), of which the
ordinal number n' is different from n.
Every ordered aggregate which can be formed in the manner described is
finite.
This can be proved by induction. Let us assume that M is a finite ordered




2, 3]                      OrdCical Numbers                             5
aggregate: it will then be proved that (Ml, e), the ordered aggregate obtained
by adjoining an element e of higher rank than the elements of 1M, is also
finite.  Since 21I has a lowest element, (1, e) has the same lowest clement,
also (111, e) has a highest element e. Again if Al, is a part of (Ml, e) which
does not contain e, then A1il is a part of lM, and therefore has a highest and
a lowest element.  If 1ll, is a part of (if, e) which contains e, let it be (1112, e),
where 1/12 is a part of 11, and therefore contains a lowest element which is
also the lowest element of (M2, e); also (1/A, e) contains a highest element e.
It has thus been shewn that (M1, e) satisfies the requisite conditions that
it should be finite, provided 1lI does so. The aggregates A, (A, B) are clearly
finite: hence the method of induction proves that every ordered aggregate
which can be formed by continually adjoining new elements to an aggregate
which originally contained one element is a finite one.
Conversely, it can be shewn that every finite ordered aggiregate can be
formed in the manner above described.
Let 1  be a finite ordered aggregate, and let e' be its highest element,
thus M= (M1, e'). Nowv M2I being a part of ll, has a highest element e", thus
M   1= (M2, e"), or M1 =(l2, e", e'). Proceeding in this manner, if we do not
reach an aggregate Mif. which contains a single element only, we shall have
found a part (...... e"', e", e', e) of 1Ml which has no element of lowest rank.
But this is impossible, since Ml is by hypothesis finite, and therefore contains
no part without a lowest element.  It has thus been shewn that ill can be
reduced, in the manner indicated, to an aggregate with a single element: and
conversely, starting with this latter aggregate, Ml is obtained by adjoining to
it successively new elements.
A finite ordered aggregate is not similar to anzy part of itself.
This theorem may also be proved by induction. For if we assume that the
finite ordered aggregate M is not similar to any part of itself, it can be shewn
that the same holds for (M, e). If possible let 312 be a part of (M, e) which is
similar to (IV, e); then if Mf contains e, it must be of the form (JL12, e), and if
(M2, e) is similar to (il, e), Mil1 must be similar to M, which is contrary to
the hypothesis that Ml contains no part similar to itself.  If lfli does not
contain e, it must be of the form (iM2,f), wheref is the element which corresponds to e in (ll, e); in this case again i111 is similar to M1, and is a part of
it; thus we have again a contradiction.  The theorem holds for (A, B), an(l
therefore generally.
It. follows from this theorem that the o'rdinal nlumbers 1, 2, 3....... which
have been defined as the ordinal numbers of aggregates (A), (A, B),
(A, B, C)...... 'Cae all different from one another, for each of these agglegates
being,a part of each of those which follow it, cannot be similar to any of the
aggregates which follow it.
Each of the ordinal numbers is to be regarded as a unique ideal object




6


Number


[CH. I


in that it is a permanent object for thought. The relation of an ordinal
number to an ordered aggregate of objects which is characterised by that
number, may be illustrated by the analogy of the relation between the colour
red, and a particular red object.
4. A simply infinite ascending aggregate, or simple sequence, is an ordered
aggregate which has no element of higher rank than all the others, and is
such that every part which has an element of higher rank than all the other
elements in that part, is a finite ordered aggregate.
It follows from this definition, that in a simple sequence there is one
element of lower rank than all the others; and further, that every part
of the simple sequence has an element of lower rank than all the other
elements in that part.
A simply infinite ascending aggregate differs from a finite ordered
aggregate in having no element which is of higher rank than all the other
elements.
The totality of ordinal numbers forms a simply infinite ascending
aggregate; these objects may be represented by a set of signs
a, /, y, 7,...
or                          1, 2, 3, 4,...
where it is assumed that some adequate scheme of such signs has been
devised.
The order of the elements is assigned by the successive formation, as
above, of aggregates having the various elements for their ordinal numbers,
and it has been shewn that if an aggregate has the ordinal number n, another
aggregate having a different ordinal number n', taken to be of next higher
rank than n, can be formed. There exists therefore no highest ordinal number.
Instead of using the expressions "of higher rank" and "of lower rank," it
is usual to say that a number m is less than a number n, when m is of lower
rank than n in the ordered aggregate of ordinal numbers, and that n is greater
than m. The terms " greater" and "less" are borrowed from the language
primarily applicable to the description of magnitudes: but in pure arithmetic
and pure analysis generally, they are used only in the sense in which they
indicate higher or lower rank, and this rank has no necessary reference i'
relations of magnitude or of measurable quantity.
The operation of counting a finite aggregate of objects of any kind i:ay
be conceived of as the process of putting the objects into correspondence Ait
the elements of the aggregate of ordinal numbers, in such a way, that:'heA
any ordinal number has an element of the aggregate which corresponds. to'it;
each of the preceding ordinal numbers also has an element which corresponds
to it. The finite aggregate is usually ordered by the process itself, the ranks




3-5]


Cardinal Numbers


7


of the various elements being successively assigned to them as the counting
proceeds. Those ordinal numbers which are employed in counting such an
aggregate may be regarded as forming an aggregate which is similar to the
given aggregate, as ordered by the process of counting. The last of the
ordinal numbers employed in counting a finite aggregate, is the ordinal
number, or simply the number (Anzahl) of the ordered aggregate.
The theorem that an ordered aggregate is not similar to any of its parts,
holds only as regards finite aggregates. It will appear in the course of the
discussion in Chapter III. that every aggregate which is not finite has parts
which are similar to the whole; and this property is sometimes taken as the
basis of the definition of an infinite, or transfinite, aggregate. For example,
the aggregate of ordinal numbers 1, 2, 3,... is similar to the part 2, 4, 6,...
which contains the even numbers only.
CARDINAL NUMBERS.
5. If any finite ordered aggregate be re-ordered in any manner, the
new ordered aggregate is finite, and has the same ordinal number as the
original one.
In order to prove this theorem, the following particular case will be first
established:-If Q is a finite ordered aggregate, the aggregate (Q, e) obtained
by adjoining to Q a new element e of higher rank than all the elements of Q,
is similar to (e, Q), in which e has a lower rank than all the elements of Q.
For let Q  (Q,,f), and let us assume that the theorem holds for Q1, i.e. that
(Q1, e) is similar to (e, Q1); it follows, since a complete correspondence can be
established between the elements of (Q,, e) and (e, Q1), that the same is true
of the two aggregates (Q1, e, f) and (e, Q1, f). Now (Q1, e, f) is similar to
(Q1, f, e), since Q1 can be made to correspond to itself, e to f, and f to e, therefore (Q1, f, e) is similar to (e, Q1, f), or (Q, e) to (e, Q), and thus the theorem
holds for Q-(Q,, f), provided it holds for Q1. Now it clearly holds if Q1
consists of a single element; hence by induction it holds for any finite ordered
aggregate Q. To prove the theorem in the general case, let us assume that
it is true for an aggregate M; it will then be shewn to be true for (M, e). For
let an aggregate obtained by re-ordering (M, e) be (R, e, S), where either R
or S may be absent; (R, e, S) is similar to (R, 5, e), for R corresponds with
itself, and it has been shewn above that (e, S) is similar to (S, e). Since
(R, S) is by hypothesis similar to M, it follows that (R, S, e) is similar to
(M, e), and therefore (R, e, S) is similar to (M, e). The theorem clearly holds
for an aggregate (A, B) which contains two elements, hence by induction it
holds for every finite ordered aggregate.
It follows from the theorem which has been established above, that, for
any aggregate which can be ordered as a finite ordered aggregate, the ordinal
number is independent of the mode in which the aggregate is ordered.




8


Number


[OH. I


It will be found, when the generalization of ordinal numbers for non-finite
aggregates is considered in Chapter III., that this property, that the ordinal
number of an aggregate is independent of the mode of ordering, is peculiar to
finite aggregates.
6. Two aggregates are said to be equivalent, when their elements can be
placed into correspondence so that to each element of either aggregate there
corresponds one and only one element of the other aggregate.
It will be observed that the relation of equivalence differs from that of
similarity in that it contains no reference to order. It is clear that two
aggregates which are each equivalent to a third, are equivalent to one
another.
An unordered aggregate is said to be finite, when it can be so ordered that
the ordered aggregate is finite in accordance with the definition given in ~ 2.
Two (finite) aggregates which are equivalent are said to have the same
cardinal number.
It thus appears that a cardinal number is characteristic of a class of
equivalent aggregates.
Each of the cardinal numbers is to be regarded as a unique ideal object;
the relation of a cardinal number to a member of the class of equivalent
aggregates of objects, of which it is characteristic, may be illustrated in the
same manner as in ~ 3, in the case of the ordinal numbers.
Since all similar aggregates are also equivalent, and since, in the case of a
finite aggregate, the ordinal number is independent of the mode in which the
aggregate is ordered, it follows that for every finite ordinal number there is a
corresponding cardinal number.
The cardinal numbers of finite aggregates are denoted by the same
symbols 1, 2, 3,... as the corresponding ordinal numbers. The two kinds
of numbers are not symbolically distinguished from each other, although
logically they are not identical.
It will be seen in Chapter iI. that this practical identity of ordinal and
cardinal numbers is confined to the case of the numbers corresponding to
finite aggregates, and therefore called finite numbers. The finite cardinal
numbers form a simple sequence 1, 2, 3,... similar to the sequence of finite
ordinal numbers; the expressions "greater" and "less " are used in relation
to two cardinal numbers in the same purely ordinal sense, denoting higher
and lower rank, as in the case of ordinal numbers.
It is impossible, in a purely mathematical work, to enter into a discussion
of the nature and proper definition of number from a philosophical point of
view. One view of number which is widely held, is embodied in the definition




5, 6]                         Cardinal Numbers                                    9
by abstraction, in which the cardinal number* is regarded as the concept of
an aggregate which remains when we make abstraction of the nature of the
objects forming the aggregate, and of the order in which they are given;
the ordinal number is then regarded as the concept obtained             by making
abstraction of the nature of the objects only, retaining the ordert in which
they are given in the aggregate. The view has also been maintained+ that
a cardinal number is simply the class of all equivalent aggregates. A tendency has been exhibited amongst mathematicians~ to regard numbers, at
least for the purposes of analysis, as identical with the symbols which
represent them. In accordance with this view, abstract arithmetic is cut
entirely adrift from the fundamental notions related to experience in which it
had its origin, and it is thus reduced to a species of mechanical game played
in accordance with a set of rules which, when divorced from their origin, have
the appearance of being perfectly arbitrary; though it may, of course, be
said that it is possible at the end of any arithmetical process to reconnect
the symbols employed, with the ideas which originally suggested them, and
thus to interpret the results of the purely symbolical processes. Whatever
view ll be adopted as to the real nature of number and its place in a general
* This view is that of G. Cantor; see Math. Annalen, vol. XLVI, p. 481, where the following
definition is given:-" ' Machtigkeit,' oder 'Cardinalzahl' von M nennen wir den Allgemeinbegriff
welcher mit Hiilfe unseres activen Denkvermogens aus der Menge M hervorgeht, dass von der
Beschaffenheit ihrer verschiedenen Elemente m, und von der Ordnung ihres Gegebenseins abstrahirt wird."  See also Peano, Forniulaires de Mathematiques, 1901, ~ 32, '0 Note.
t Ordinal numbers are frequently regarded as logically prior to cardinal numbers, but this
order of procedure is not a necessary one. In Dedekind's tract "Was sind und was sollen die
Zahlen," Brunswick, 1887 and 1893, which has been translated into English by Prof. W. W.
Beran, under the title "Essays on the Theory of Numbers," 1901, a detailed treatment of the
subject is given, in which the notion of order is regarded as fundamental.
+ See B. Russell, The Principles of Mathematics, vol. I, chap. xi.
~ For example see Heine, Crelle's Journal, vol. LxxIV (1872), p. 173, where the matter is stated
in the following plain form: " Ich nenne gewisse greifbare Zeichen Zahlen, sodass die Existenz
dieser Zahlen also nicht in Frage steht." Again, Helmholtz appears to hold a view closely
approaching the notion that Arithmetic is the art of manipulating certain -signs according to
certain rules of operation; he writes in Ges. Abh. vol. III, p. 359, "Ich betrachte die Arithmetik
oder die Lehre von den reinen Zahlen als eine auf rein psychologische Thatsachenb aufgebaute
Methode, durch die die folgerichtige Anwendung eines Zeichensystems (namlich der Zahlen) von
unbegrenzter Ausdehnung und unbegrenzter Moglichkeit der Verfeinerung gelehrt. wird." Reference may be made to an essay by A. Pringsheim in the Jahresberichte der d. math. Vereinigung,
vol. vi, 1899, " Ueber den Zahl- und Grenzbegriff im Unterricht." In an article entitled "Die
Du Bois Reymond'sche Convergenz-Grenze," Sitzungsberichte d. bayer. Akad. vol. xxvii, 1897,
Pringsheim speaks of numbers as " Zeichen, denen lediglich eine bestimmte Succession zukommt."
See p. 326. This article contains various remarks on arithmetization, and especially a criticism
of the views of P. Du Bois Reymond. A searching criticism of the tendency to reduce Arithmetic
to the formal manipulation of symbols is given in L. Couturat's work De l'infini mathematique,
Paris, 1896, which contains a valuable account and discussion of theories of the philosophy of
arithmetic.
EI References to the literature relating to the Philosophy of Number will be found in the
Article i. A. 1, " Grundlagen der Arithmetik," by H. Schubert, in the Encyclopddie der mathe



10


Number


[OH. I


scheme of thought, the assumption of the right to hypostatize numbers
would appear to be an essential condition of the possibility of developing an
abstract arithmetic, and consequently of the establishment of mathematical
analysis in general.
THE OPERATIONS ON INTEGRAL NUMBERS.
7. If two finite ordered aggregates A and B, of which the ordinal
numbers are a and b respectively, are combined into a single ordered
aggregate in which the elements of A have all lower rank than those of
B, and in which any two elements of A, and any two elements of B, have
the same relative orders as in the original aggregates, then the ordinal
number of the combined aggregate is said to be the sum of the ordinal numbers
a and b, and is denoted by a + b.
It can be shewn that the new aggregate is a finite one, and that its
ordinal number is unaltered if for A and B there be substituted aggregates
which are similar to them; it thus appears that the sum a+ b is a finite
number which depends only upon a and b.
The aggregate (A, B) has as lowest element the lowest element of A,
and as highest element the highest element of B; moreover any part of
(A, B) is of the form (A', B'), where A' is a part of A, and B' is a part of B;
or else it has one of the forms A', B', and since A', B' have each a. lowest
and a highest element, any such part of (A, B) has a lowest and a highest
element. Thus (A, B) is finite.
Again, if A1, B1 are aggregates which are similar to A and B respectively,
the elements of A may be placed in correspondence with those of Al, and the
elements of B with those of B1; we have then a (1, 1) correspondence between
the elements of (A, B) and those of (A., B1); thus the ordinal number of
(A, B) is the same as that of (A1, B1).
Since (A, B) has the same ordinal number as (B, A) it follows that
a + b=b+a, which is known as the commutative law of addition.
If a, b are the cardinal numbers of two finite aggregates A, B, then the
cardinal number of the aggregate formed by combining the two aggregates
into one is said to be the sum of a and b, and is denoted by a + b. That a + b
is a definite finite number dependent only on a and b, follows at once from
the corresponding theorem which has been proved for cardinal numbers.
matischen Wissenschaften, vol. I; also in E. G. Husserl's Philosophie der Arithmetic, vol. I, chaps.
5 and 6, Halle, 1891. The view that Number is fundamentally dependent on the notion of
Time was developed by Sir W. R. Hamilton; see the Dublin Transactions, vol. xvII (II), 1835,
t" Theory of Conjugate Functions or Algebraic Couples with a Preliminary and Elementary Essay
on Algebra as the Science of Pure Time"; see also Helmholtz's essay " Zihlen und Messen"
(1887), where the view is adopted that the axioms of Arithmetic have a relation to the intuitional
form of Time, similar to that which the axioms of Geometry have to the intuitional form of space.




6-8]


Operations on Integral Numbers


11


The operation of finding the sum of two numbers a and b, is known as the
operation of addition, and it has been shewn that this operation is commutative. It should be observed that the sum of two numbers a and b cannot be
determined merely by contemplating those numbers themselves as abstract
concepts, but can only be defined as above, by referring to aggregates of which
a and b are the numbers, and then combining those aggregates. The number
of the combined aggregate is then conceived of as the result of a symbolical
operation upon the numbers a and b. For example, the equation 5 + 3= 8
does not imply that the concept 8 is obtainable by placing the concepts
5, 3 as it were in juxtaposition, but can only be regarded as a symbolical
expression of the fact that an aggregate of 5 objects together with one of
3 objects make up an aggregate of 8 objects. Bearing this observation in
mind, the numbers 1, 2, 3,... are represented symbolically as the results of
successive operations of addition, 1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, etc.; but
these equations do not express definitions of the numbers 2, 3, 4,..., since
from the concept unity taken by itself, no other concept is directly derivable.
The operation of addition can be extended by continued repetition. Thus
the sum of a, b, c,... k is a finite number represented by a + b + c +... + k,
and, in particular, any number n is represented by n = 1 + 1 + 1 +... + 1. An
immediate induction shews that the result of the operation of addition
repeated any definite number of times is a finite number dependent only
on the. constituents of the summation.
The associative law of addition, a + (b + c) = (a + b) + c, follows from the
irrelevancy of the order in which the operations are performed. This is seen
from the contemplation of aggregates of which a, b, c are either the ordinal
or the cardinal numbers.
8. If in a finite aggregate of which the number is b, each element be
replaced by a finite aggregate of which the number is a, the number of the new
aggregate so formed is said to be the product of b by a, and is denoted by ab.
This operation is said to be that of multiplying b by a. By taking the
aggregates to be ordered, it is seen at once that the new aggregate satisfies
the conditions that it is finite, and that its number is unaltered by the
substitution of similar aggregates of other objects for those originally employed. Thus ab is a definite number dependent only on i::: n,-d,
It is clear that ab may be regarded as the sum a + a  +.... wh ere a
occurs b times in the operation.
If the ordered aggregate of which the number is ab, be re-ordered in the
following manner:-take the first element of each of the aggregates of which
a is the number, then the second elements of these aggregates, and so on,
with lastly the ath elements of these aggregates, then we have as the result
of the process an aggregate of which the number is a, and each element of




12


Number


[CH. I


which consists of an aggregate of which the number is b; the re-ordered
aggregate has the number ba. It has thus been shewn that ab = ba, which
is expressed by saying that the operation of multiplication of finite integers
is commutative.
The distributive law for multiplication, a (b +c)= ab + ac, follows from
the definition of the operation, by considering the aggregates of which
a, b, c are the numbers.
An immediate induction shews that the repetition of the operation of
multiplication any definite number of times gives a finite number dependent
only on the numbers multiplied, and independent of the order in which the
operations are performed.
The result of the operation of multiplying the number a by itself is
denoted by a'l, where n is the number of times a occurs in the product
a. a. a... a. From this definition the law am. a = am'n, is directly deducible.
9. If the sum of two numbers a, b be denoted by c, the number a is
uniquely determined when b, c are fixed; and it is then regarded as the result
of the operation of subtracting b from c. The operation of subtraction is
thus defined as inverse to that of addition. If c = a + b, a is obtained as the
result of the operation denoted by c -b, which is such that (c - b) + b = c.
It is obvious that the operation of subtraction of b from c is only possible in
case c > b.
If the product of two numbers a, b be the number c, then the number a
is uniquely determined when b and c are given; and a is regarded as the result
of the operation of division of c by b. The operation of division so defined
is inverse to that of multiplication; it is clear that the operation is only
possible in case c is one of the class of numbers b, 2b, 3b,.......
FRACTIONAL NUMBERS.
10. The operation of multiplying two integers a, b together, is one
which is always a possible operation, in accordance with the definition of the
operation of multiplication which has been given above; the inverse operation of division is however, as we have seen, not always a possible one. This
restriction upon the possibility of the operation of division suggests the
introduction into Arithmetic of a new class of numbers, the rational fractions,
which, when defined, shall be such that the operation of division, within the
whole aggregate of integers and fractions, may be a possible one without
restriction. Stated in algebraical form, the demand arises for a scheme of
numbers such that the equation ax=b, shall always have a solution in x,
where a, b are any two numbers which belong to the contemplated aggregate
of numbers.




8-11]


Fractional uumbers


13


The actual use of fractional numbers arose historically from the necessities of the process of measurement of extensive magnitude, and the
conception of a fraction which arises in this connection is the one which is
used in ordinary life, and is made the basis of the treatment of the theory
of fractions, even in recent scientific text-books.  In accordance with this
view, a unit of magnitude of some kind is divided into b equal parts, and a
of these parts are taken; the resulting magnitude is then denoted by the
fraction a/b.
This notion of the essential nature of a fraction, dependent as it is upon
the notions of a unit, and of the divisibility of such unit into equal parts, is
incompatible with the modern view that Mathematical Analysis should be
developed upon the basis of a Pure Arithmetic, quite independently of all
notions connected with the measurement of extensive magnitude. The
modern tendency known as Arithmetization manifests itself in the construction of theories of Number and of the operations involving numbers, which
depend entirely upon the conceptions connected with the process of counting,
measurement being regarded as a process foreign to Pure Arithmetic. The
process of counting is an exact one: whereas measurement can in practice
only be carried out with a greater or less degree of approximation, and can
only ideally be made an exact process. Pure Arithmetic is made the basis of
Analysis, not only in accordance with the general principle that the fundamental conceptions of a branch of science should be irreducible to simpler
conceptions, but also because the theory of ideally exact measurement has
peculiar difficulties of its own. Our essentially inexact intuitions of spatial,
temporal, or other magnitudes, necessitate a process of idealization in which
the objects of perception are replaced by ideal objects subject to an exact
scheme of definitions and postulates, in order that an exact science of
measurement may be possible. The view is at present held by the majority
of mathematicians that the nature of the abstract continuum, and that of a
limit, are capable of exact formulation only in the language of a Pure
Arithmetic; and that this science must therefore be developed upon an
independent basis before it can be applied to the elucidation of the conceptions requisite for an abstract theory of continuous magnitude. The
theory of measurement is, in accordance with this view, regarded as an
application, and not as part of the basis, of Mathematical Analysis.
11. By those writers who are under the influence of the modern
arithmetizing tendency, the traditional non-arithmetical definition of a
fraction has been abandoned, and in its place a formal definition has been
substituted, in which the fraction is regarded as an association of a pair of
integers. The associated integers are regarded as making a single object, and
laws of combination of these objects are then postulated.
If a, b are two integers, a new number (a, b), or in ordinary notation




14


Number


[CH. I


-, is formed by the association of a and b, the new number being defined to
be such as to satisfy the following conditions:
(1) (a, b) is regarded as ordinally greater, equal to, or less than (c, d),
according as ad is greater, equal to, or less than be. The expressions greater,
equal to, or less than, are here used, not in their primitive sense as referring
to magnitude, but in the sense in which we have used them in the case
of integers, as assigning relative order to the numbers.
(2) (a, 1) is defined as equal to a; thus if b=l1, the association is
regarded as equivalent to the integer a. Taking (1) in conjunction with
this postulate, the new numbers have their orders assigned, not only relatively to one another, but relatively also to the integral numbers; so that
the whole aggregate of integers and fractions is ordered, in the sense that,
of two given numbers, it can always be said which has the higher rank.
(3) The addition of two fractional numbers is defined by
(a, b) + (c, d) = (ad + be, bd).
(4) The multiplication of fractional numbers is defined by
(a, b) x (c, d)= (ac, bd).
(5) The use of a fraction as an index, is defined by the postulate
X (a, b) X  (c, d) =  X(a, b) + (c,d)
where x is any number, either integral or fractional. The symbol x(ab) is
to be interpreted subject to this postulate, in case such interpretation
is possible.
It will be observed that, in the- case b = 1, d = 1, the above definitions are
consistent with those which have been adopted in the case of integral
numbers; and thus the new numbers, together with the integers, form an
aggregate with uniform laws of operations. It is easily seen that the operations with new numbers satisfy the commutative, associative, and distributive
laws. The inverse operation of division is now one which is always possible
within the domain of the numbers; thus (a, b) - (c, d) =(ad, be).  The
inverse operation of subtraction, (a, b)- (c, d)= (ad - be, bd), is only possible
if (a, b)> (c, d).
The association of a pair of integers is a "number" in quite a different
sense from that in which the cardinal and ordinal numbers, hitherto discussed,
are numbers. The justification of the extension of the term "number" to
the fractions, lies in the fact that a consistent scheme of operations can be
imposed upon them, of which the laws are in agreement with those which
hold for operations which involve integers only.




11, 121]              Fractional Numbers                        15
12. The scheme which has been above indicated suffices for a formal
definition and logical development of the properties of fractions, but it is
subject to the objection that it is of an arbitrary character; indeed it is not
easy to see why the particular laws of operations have been postulated,
except as suggested by the traditional non-arithmetical conception of a
fraction.
To remedy this defect, a view of the nature of a fraction will be here
given which relates the fraction with the process of counting, in such a
manner that fractional and integral numbers have similar relations to that
process. It will appear that the laws of combination given above naturally
follow from this mode of regarding the fraction, with the exception of (5),
which is however immediately suggested by the rule for integral indices.
Consider an aggregate of b objects, and out of these b objects pick out
any a (- b) of them. If we regard these a objects not only as single objects
of number a, but also as belonging to an aggregate whose number is b, we
may denote the a objects by (a, b), where their number a is associated with
the cardinal number b of the aggregate to which they belong. This process
being independent of the particular aggregate used, the abstract fraction
(a, b) is related to this process in an analogous manner to that in which the
number b is related to the process of counting an aggregate whose cardinal
number is b. Thus the fraction (a, b), or a/b, is characteristic of an aggregate
of a objects each of which belongs to an aggregate of b objects. The
extension of the definition to the case a > b, is clear when we observe that
it is unessential that the a objects taken should all belong to one and the
same aggregate of b objects; it is sufficient that each of them be regarded
as essentially belonging to some aggregate of cardinal number b.  In
accordance with this view, a fraction, say 3/5, is characteristic of any three
things each of which belongs to an aggregate of five things, i.e. 3/5 means
3 out of 5. That the three things taken out of five should necessarily
all be equal in respect of size, or some other kind of magnitude, is as
irrelevant to the true nature of a fraction as the assumption of five things
necessarily meaning five equal things, is to the true nature of the number
five.
Since (a, 1) is characteristic of an aggregate of a things each of which is
also regarded as a single object, it is clear that (a, 1) is identical with a.
If we suppose each of the b elements in an aggregate, of which the
cardinal number is b, to be replaced by an aggregate of n elements, we
have now an aggregate with nb for its cardinal number; and instead of a
elements chosen out of this aggregate we now have na of the new elements,
each of which is to be regarded as associated with the cardinal number nb.
We represent these na elements by (na, nb), which is equivalent to (a, b),
since the two forms represent two different aspects of the same process.




16


Number


[OH. I


Therefore we have (a, b)= (na, nb), or in the ordinary notation a/b = na/nb.
This relation is in complete accordance with the law of logical (not arithmetical) addition, that a mere repetition of a term yields only the term
itself.
Since (a, b)=(ad, bd), and (c, d) = (bc, bd), we regard (a, b) as greater,
equal to, or less than (c, d), in the purely ordinal sense of the terms,
according as ad is - be. For the two numbers (ad, bd), (be, bd) are characteristic of the process of taking ad, be elements respectively from an aggregate
of the same cardinal number bd; and thus the relative order of the two
numbers (a, b), (c, d) will naturally be fixed in accordance with the relative
order of the two numbers ad, be.
The addition of the two numbers (a, b) and (c, d) is equivalent to that of
(ad, bd) and (be, bd), and is consequently naturally defined as given by
(ad+ be, bd), which characterises the amalgamation of two aggregates of
which the numbers are ad, be, the elements of each of which all belong to
an aggregate of number bd, or to one of several such aggregates.
To interpret the operation of multiplication, let us consider an object
represented by (c, d); this consists of c things each belonging to an aggregate
of d things. To multiply it by (a, b), is to take a such objects each of which
belongs to an aggregate of b such objects; we have on the whole one or
more aggregates of bd elements, and out of these, ac elements are to be
taken. Thus the multiplication of the number (c, d) by the number (a, b)
may be understood to characterise the result of taking a objects each of
which is characterised by (c, d), out of one or more collections of b objects
each of which objects is characterised by (c, d). This is the same thing as
the process of taking ac objects out of one or more aggregates of bd objects,
and is characterised by the number (ac, bd); we are thus led to the law of
multiplication
b o  c  ac
(a, b) x (c, d)= (ac, bd), or  x d = b
NEGATIVE NUMBERS, AND THE NUMBER ZERO.
13. Although the operation of addition is always possible within the
aggregate of integral and fractional numbers, yet the inverse operation of
subtraction is not always possible; thus a number x cannot be found such
that x + (c, d) = (a, b), unless (a, b)> (c, d). As the limitation of the possibility of division suggests the introduction of fractional numbers, so this
limitation of the possibility of subtraction suggests the introduction of a
further set of new numbers, which shall be such that within the so completed
aggregate, subtraction may always be a possible operation.




12-14]


Negative Numbers


17


If a, /3, 7, 8 denote integral or fractional numbers such that a >,, y > 8;
we may put a =     + x, =  + y; then x =a-, y =7-. We have
a+,7=3+     + +  y,
hence                   x + y = (a + 7) - (3 + ),
or                 (a-3) + (7 - 8)= (a + 7/)- (/ + 8)................(1).
Again, if a-/,= y- 8, i.e. x=y, we have a+8=38+8+z =3+ y;
or                   a+8=13+7, if a-3=7y-8.....................(2).
Lastly, we have
a = (3+ x) (8+y) =   (8 + y) + x ( + y)
= 18 + 3y + x + xy;
hence              a' + 38 =, (y + + )+ 8 ( x + /3) + xy
= 13, + a8 + xy;
hence             (a - 3) (7- 8) = (ar +3,)-(a + 37,y)..................(3).
The rules (1), (2), (3), with regard to the numbers a -/, y - 8, which so
far exist only when a > 3, y > 8, suggest the mode of the extension referred
to above.
14. Let a, 38 be any two numbers integral or fractional, and conceive a
new number D (a, 18), formed by the association of a and 13, to be defined as
subject to the laws
(4) D(a, )= D (, 8), if a+ 8=    +7,
(5) D (a, /3)+ D (, 8)= D (a +, 3+ ),
(6) D (a, 3) x D     (y, 8)= D (a + /8, aS +/3y):
it will be observed that when a > /, and 7 > 8, D (a, /) may denote a -- 3,
the three laws becoming (2), (1), (3). It will now be shewn that the symbol
D (a, /3) defines a number of an aggregate within which the operation of
subtraction is always possible.  For, to find a number x, such that
x + D (a, fi) = D (y, ), we see that x = D (/3 +, a + 8),
since       D (a +/3 + 7, a + 3 + 8) =D (y, 8), in virtue of (4).
Since D(a, a) = D (, y), we see that D (a, a) is independent of a; and
thus D (a, a) defines a new number which is called the number zero, and
is denoted by the symbol 0.
The number zero is regarded as characteristic of the absence of all
elements from an aggregate of which the existence has been contemplated;
it is the number of such a hypothetical aggregate, in a sense similar to that
in which a positive integer is the number of an actual aggregate.
The number D (a + k, k) depends only on a, and we shall postulate that:it is identical in meaning with a itself.


H.


2




18


Number


[OH. I


The numbers D (a, 3), or a-/3, for which a > /, are called positive
numbers, and form the aggregate of integral and fractional numbers we
have previously considered.
Those numbers for which a < /3, are called negative numbers.
Since by (5), D (a, /) + D (/, a) = D (a + /, a + /?) = 0, the number D (/3, a)
may be denoted by - D (a, /), or in ordinary notation - (a - /). Thus to every
positive number x there corresponds a single negative number - x, which is
such that x + (- x) = O.
We may now use the notation a - / in every case for D (a, 3), and thus
a -/3 = - (/ - a).
From (6), it is seen that the operation of division is always possible for
two members of the complete aggregate of positive and negative numbers, and
zero, except when the divisor is the number zero, in which case the operation
is meaningless.
From (6), we see by putting y = 8, N = D (a, /), that N. 0 = 0. From (5),
we have N+0 = N.
Any number D (a, /) is said to be greater in the ordinal sense than
D (y, 8), when D (a, /3)- D (y, 8) is positive; thus the complete aggregate of
positive and negative integral and fractional numbers together with the
number zero, is one in which all the numbers are arranged in a definite
order. This aggregate is known as the aggregate of rational numbers.
In the aggregate of rational numbers so ordered, the number zero has
lower rank than any of the positive numbers, and higher rank than any of
the negative numbers. Further, if x, y are two positive numbers of which x
has higher rank than y, the negative number -x has lower rank than - y.
If x, y are any two rational numbers, such that x < y, there exist an unlimited number of rational numbers each of which is > x, and < y.
Such numbers are said to be between x and y. For it can be seen at once,
from the definition of order given above, that - (x + y) is one such number;
between ~ (x + y) and either x or y, another rational number can, in a similar
manner, be found. This process can be carried on without end; and it is
clear that in accordance with the mode of ordering of the aggregate, defined
above, all the numbers thus determined are between x and y.
If x, y are any two positive rational numbers such that x < y, an integer n
can be found which is such that nx > y.
For Y is a rational number such that Y. x = x; again if be any positive
x                            x                q
rational number, there exist integers which are > -, for p + 1 is itself such
an integer. If n is an integer which is >, we have nx >. x > y; thus the
theorem is established.




14-16]


Irrational Numbers


19


IRRATIONAL NUMBERS.
15. The only numbers of which the existence was recognized by the
Greek geometers were the rational numbers, although the fact that the ratio
of two geometrical magnitudes is not necessarily exactly representable by
such numbers appears to have been discovered at a very early period. Euclid
gave, in the fifth book of his treatise, a discussion of the theory of ratios, and
in the tenth book, a theory of those incommensurable magnitudes which are
ideally constructible by means of straight lines and circles. In later times*,
the idea was current that, to the ratio of any two magnitudes of the same kind,
there corresponds a definite number; and in fact Newton in his Arithmetica
Universalis expressly defines a number as the ratio of any two quantities.
Before the recent development of the arithmetical theories of irrational
number, and to a considerable extent even later, a number has been regarded
as the ratio of a segment of a straight line to a unit segment, and the
conception of irrational number as the ratio of incommensurable segments
has been accepted as a sufficient basis for the use of such numbers in
Analysis.
In accordance with the doctrine that Mathematical Analysis must rest
upon a purely arithmetical basis, the introduction of irrational numbers into
Analysis must be made without an appeal to our intuition of extensive
magnitude, but rather by an extension of the conception of Number, resting
on a further development of the ideas which have been here discussed in
connection with the theory of rational numbers. The necessity for this
extension of the domain of Number arises not only on account of the
inadequacy of rational numbers for application to ideally exact measurement, but also, as will be explained later in detail, because the theory of
limits, which is an essential element in Analysis, is incapable of any rigorous
formulation apart from a complete arithmetical theory of irrational numbers.
Before the recent establishment of the theory of irrational numbers, no
completely adequate theory of Magnitude was in existence. This is not
surprising, if we recognize the fact that the language requisite for a complete
description of relations of magnitudes must be provided by a developed
Arithmetic.
16. The successive extensions of the domain of Number, by the introduction of fractional and of negative numbers, were suggested by the
desirability of so completing the domain that the operations of division and
subtraction, which are not always possible in the more limited domain, might
* A good short account of the history of this subject will be found in the Article I. A 3,
"Irrationalzahlen und Konvergenz unendlicher Prozesse," by A. Pringsheim, in the Encyclopddie der Math. Wissenschaften, vol. I. See also M, Cantor, Geschichte der Math., vol. I.
2-2




20


Number


[CH. I


always be so in the more extended one. In the aggregate of rational
numbers, the operations of addition, subtraction, multiplication, and division
are always possible operations; but it can be readily shewn that the inverse
operation involved in determining a fractional power of a rational number is
not, in general, a possible one.
As the simplest case of this impossibility of such operation, we may take
the problem of finding the square root of a positive integer m which is not
a square number. It can be shewn that such a number has no square root
within the aggregate of rational numbers.
If possible* let m be the square of a rational fraction p/q in its lowest
terms; thus p2 - rmq2 = 0. There always exists a positive integer X such that
X2 < m < (X + 1)2; we then have Xq < p < (X + 1)q.
Now let us consider the identity
(mq - Xp)2 - m (p - Xq)2 = (2 - m) (p2 - mq2) = 0.
From this identity it follows that m is the square of the rational number
(mq -Xp)/(p- Xq), of which the denominator is less than q, and this is
contrary to the hypothesis that m is the square of the fraction p/q which
is in its lowest terms. It thus appears that there exists no rational number
of which the square is m.
On the formal side of Arithmetic, a demand for the extension of the domain
of number arises from the impossibility of carrying out, with the requisite
generality, certain operations, as in the example given above. Such extensions
of the domain of number as are made when fractional, negative, irrational,
and complex numbers are successively adjoined to the original integral
numbers, are made in accordance with a principle known as that of the
permanence of forms, which was first indicated by Peacockt, and further
developed by Hankel+. This principle may be stated in the form that, in
order to generalize the conception of number, the following four requisites
must be satisfied:
(1) Every operation which is represented by a formal expression involving the unextended class of numbers, and which does not result in the
representation of a number of the unextended class, must have a meaning
assigned to it of such a character that the formal expression may be dealt
with according to the same rules as would be applicable if the expression
represented one of the unextended class of numbers.
(2) An extended definition of number must be given, such that a formal
* This proof is given by Dedekind in his tract Stetigkeit und irrationale Zahlen. An extension
of Dedekind's method to the case of nth roots has been given by S. M. Jacob. See Proc. Lond.
Math. Soc. Ser. 2, vol. i, p. 166.
+ British Association Report for 1834; also Symbolical Algebra, Cambridge, 1845.
+ See his Theorie der komplexen Zahlsysteme, Leipzig, 1867.




16, 17]


Arithmetization


21


expression, as in (1), may represent a number in the extended sense of the
term.
(3) A proof must be given, that for numbers of the extended class the
same formal laws of operation hold as for the unextended class.
(4) Definitions must be given of the meaning of greater, equal, and less,
in the extended domain of number, these terms being taken in the ordinal
sense.
The arithmetical theory of irrational numbers has been developed in
three main forms, of which the first * was given by Weierstrass in his lectures
on Analytical Functions; the secondt is that of G. Cantor, which was developed in further detail by Heine +, and was also developed independently
by Ch. Me'ray~; the third, that of R. DedekindlJ, appeared about the same
time as that of Cantor. We shall give an account of the theories of
Dedekind and of Cantor, and shall shew that they are fundamentally
identical.
KRONECKER'S SCHEME OF ARITHMETIZATION.
17.  As it is now    generally understood, the term    "arithmetization" is
used to denote the movement which has resulted in placing analysis on a
basis free from all notions derived from the idea of measurable quantity, the
fractional, negative, and irrational numbers being so defined that they
depend ultimately upon the conception of integral number. An extreme
theory of arithmetization has however been advocated by Kronecker~, who
proposed the abolition of all modifications and extensions of the conception
of number, the integral numbers being alone retained. His ideal** is that
every theorem in analysis shall be stated as a relation between integral
numbers only, the terminology involved in the use of negative, fractional, and
* For an account of this Theory see S. Pincherle, Giorn. di mat., vol. xvIII (1880), p. 185;
also O. Lurmann, Theorie der analytischen Funktionen, Leipzig, 1887, p. 19.
t Math. Annalen, vol. v (1872); see also Math. Annalen, vol. xxI, where Cantor discusses all
the three theories.
+ Crelle's Journal, vol. LXXIV (1872).
~ Nouveau Precis d'Analyse infinitesinmale, Paris, 1872.
II Stetigkeit und irrationale Zahlen, Brunswick, 1872.
'[ See Crelle's Journal, vol. ci, "Ueber den Zahlbegriff."
** He writes (loc. cit. p. 338) "Und ich glaube auch, dass es dereinst gelingen wird, den
gesammten Inhalt aller dieser mathematischen Disciplinen zu ' arithmetisiren,' d. h. einzig und
allein auf den im engsten Sinne genommenen Zahlbegriff zu griinden, also die Modificationen
und Erweiterungen dieses Begriffs (ich meine hier namentlich die Hinzunahme der irrationalen
sowie der continuirlichen Grbssen) wieder abzustreifen, welche zumeist durch die Anwendungen
auf die Geometric und Mechanik veranlasst worden sind." He proceeds to shew in detail, how
the notions of negative, fractional, and algebraical numbers can be avoided by substituting for
equalities in which these numbers occur, congruences relative to certain moduli or systems of
moduli. A similar suggestion had been made by Cauchy with reference to imaginary numbers.




22                             Number                             [CH. 
irrational numbers, being entirely removed. This ideal, if it were possible
to attain it, would amount to a reversal of the actual historical course
which the science has pursued; for all actual progress has depended upon
successive generalizations of the notion of number, although these generalizations are now regarded as ultimately dependent on the whole number for
their foundation. The abandonment of the inestimable advantages of the
formal use in Analysis of the extensions of the notion of number could only
be characterised as a species of Mathematical Nihilism.
THE DEDEKIND THEORY OF IRRATIONAL NUMBERS.
18. Let us consider the aggregate of all the rational numbers ordered in
the manner which has been previously discussed, and let us take any one
such number NV. We may conceive all the rational numbers to be divided
into two classes R1, and R2, such that every number of R, is, in the ordinal
sense of the term, less than every number belonging to the second class R2,
the two classes being separated by the number N, which may itself be
assigned at choice either to the first or to the second class. If N belongs
to the first class, it is the greatest number in that class, and the numbers of
the second class have no number which is less than all the others of that
class; if N be taken to belong to the second class, it is the least number in
that class, and there exists no number in the first class which is greater than
all the others; for if any rational number less than N be taken, it is always
possible to find another greater one which is less than N.. Such a division
of the rational numbers into two classes is called a section (Schnitt), and we
therefore say that corresponding to any given rational number there exists a
section which divides the aggregate of rational numbers into two classes, such
that all the numbers of the first class are less than all those of the second class;
and such that either in the first class there is no greatest number, or else in the
second class there is no least number.
It can be shewn by means of examples, that sections of the as!Tregate
of rational numbers exist which are different in character from those just
described. If m is a positive integer which is not a square number, we may
conceive the rational numbers to be divided into two classes, the first of
which contains all the negative numbers and also those positive numbers
of which the square is less than m, including zero; the second class contains
all the positive numbers of which the square is greater than m. The first
class contains no greatest number, and the second class contains no least
number; this section is said to be related to an irrational number V/m, in the
same way as a section such as has been considered above is related to a
rational number. This example shews that sections of the rational numbers
R exist, such that R is divided into two classes R,, R2, where every number
of R, is less than every number of R2, and such that R, contains no number




17, 18]                   Dedekind's Theory                            23
greater than all the others, and also R2 contains no number less than all the
others.
A new aggregate of objects, the real numcbers, may now be defined as
follows:
To every section (R1, R2) of the aggregate R of ratiolnal numbers, such that
every,nmber of R belongs tv one or other of the two classes R1, R,, and every
nzmber in R, is ordiinallj less than every znuber in1 R2, there corresponds a
real number.
In case neither R, contains a number which is ordinlally greater than all
the others in R1, nor -R2 contains a nvumber which is ordiuctally less than all the
others in R2, the real number corresponding to the section is said to be an
irrational numtber.
In case either R, has a greatest nmzber x, or R2 has a least inumber x, the
section is said to define a real number corresponding to the rational numeber x.
The real number which corresponds to a rational number x, though
conceptually distinct fiom x, has no properties distinct from those of x, and
is usually denoted by the same symbol.
The definition of a real number can be put into a different and somewhat
less abstract form, by employing the notion of a lower segment of the
aggregate of rational numbers.   A  lower segment of the aggregate 11 of
rational natibers is any class of rational numbers which contains no num1ber
greater than all the others, and such that if any number whatever of the
class be ta'len, the class contains all those tlnumbes of' I which are less than
that numbler.  A lower segment of R is identical with one of Dedekind's
classes R,, in case R, contains no greatest number.
A real nuember may be defined* to be a lower segment of the aggregate R of
rational,rnumbers  and thu s every real number, whether irrational or not, is a
definite class of rational numbers..in accordance with this definition, the real number 3, for example, is
defined o,.be the aggregate of all rational numbers which are less than the
ratioalnflia-mber 3; the irrational number \/3 is defined as the aggregate of
all rntional- numbers which are either negative, or if positive have their
squ;    less than 3, the number zero being also included in the aggregate.
use, here adopted, of the term real number, is sanctioned by general
us.     The employment of the term real, has originated from the contrasti'rg OI Uiese numbers, not with rational numbers, but with complex numbers.
'The extension of the term Number to the real numbers, is justified by the
fact that it is possible to define the operations of addition, multiplication, &c.,
* This form of the definition is that given by B. Russell, see The Principles of Mathematics,
vol. I, chaps. xxiii and xxiv; it was suggested by Peano, see Rivista di lMatenmatica, vol. vi,
pp. 126-140.




24


Number


[OH. I


for real numbers, so that the foruial laws of these operations are in agreement
with those which hold for operations within the domain of the rational
numbers.
19. It will now be shewn that the aggr-egate of real numbers, defined in
Dedekind's manner, can be so ordered, that every real number has a definite
rank in the aggregate, i.e. of any two real num bers it is determinate which
has the higher and which the lower rank.
The basis of the scheme of order being taken to be the ordered aggregate
of rational numbers, let us denote by n, n' any two jeal numbers, and let
the sections by which they are defined be denoted bJy (RI, R2), (R1, R2')
respectively.
The following cases may arise:
(1) If (-R, R2) and (RI', R2') are identical, that is, if every number in R1
is also in R1', and every number in R2 is also in R2', the tw o numbers n, n'
are identical; thus n - n'.
(2) Let us next suppose that there is one rational number r1 - r2', which
is contained in RI, but not in R,'; it is consequently contained in R2'. All
the numbers in R,' are less than r2', and hence all the numlers in R,' are
in RI. Since r, is the only number in R, which is contained in R2', it
follows that r, is greater than all the other numbers in R1; and thus the
number n defined by (R1, R2) is a number corresponding to the rational number
riT or^r2'. All the elements in R' are contained in -R, and are less than r';
^4she numbers in R2' except r', are greater than r2', for if not they would
f: contained in R1: hence the section (R1', R') defines the real number
Ti n, corresponding to the rational number r2' ri. The two se ctiohs are
essentially identical, the only difference being that the rational number
r,- r', is regarded as belonging to the first class in one section aLr to the
second class in the other section.
(3) If there are two different numbers belonging to R, which also belong
to R2', there are an indefinite number of other numbers which have th' same
property, since an unlimited number of rational numbers can be found ivhich
lie between two given rational numbers. In this case we define: the nutnbeir
n or (Ri, R2), to be greater, in the ordinal sense of the term, thr  or
(R', R2'), agreeably with the definition already down for the rational r  trs.
The cases in which one, or more than one number which belong;"
also belongs to R2, may be treated in a similar manner; thus we de..,^o?Lmeaning of the relation n < n'. It is easily seen that if n > n', and n' > >nl
then the relation n > n" is also satisfied. Thus the system of real numbersl
is arranged in a regular order, such that those of them which correspond Ito,
rational numbers have the same relative rank as the corresponding rational
numbers have in the aggregate of rational numbers.




18-21]


Dedekind's Theory


25


20. The aggregate of real numbers has the following properties:
(1) If a > 3, and 3 > 7, then a > y.
(2) Between any two real numbers a, y there are an unlimited number
of real numbers. This is easily proved from the corresponding property of
rational numbers, by considering the sections which define the numbers.
(3) If a is a fixed real number, then all real numbers may be divided
into two classes RI, R2, such that RI contains all the real numbers which are
less than a, and R2 contains all real numbers which are greater than a. The
number a may be regarded either as belonging to RJ, in which case it is the
greatest number in R1, or else as belonging to R2, in which case it is the least
number in R2. This also follows from the definition above.
(4) If the aggregate of real numbers falls into two classes R1, R2, such
that every number of R1 is less than every number of R2, then there exists
one and only one number by which this section is produced.
To prove this, we observe that the section (R1, R2) of the aggregate of
real numbers also defines a section (R1, R,) of the aggregate of rational
numbers, such that all rational numbers belonging to R, correspond to real
numbers which belong to RI, and all numbers belonging to R2 correspond to
real numbers which belong to R2.
Let N be the real number defined by the section (R1, R2), and let N' be
any real number different from N, defined by the section (RI', R2'). There
are an indefinite number of rational numbers n which belong to only one of
the aggregates Ri, R1'; let n be the real number corresponding to n. If
N'< N, then n belongs to R1, and therefore n belongs to Rl; and since
N' < n, it follows that N' belongs to R,. Similarly, if N' > N, we can shew
that N' belongs to Ra. It has thus been shewn that every number different
from N belongs to R1 or to /2, according as it is less or greater than N.
Thus N is either the greatest number in RI or the least in R2, and therefore N is the only number by which the section (R1, R,) can be made.
21. The operations between two real numbers may, in accordance with
the above definition of real numbers by means of sections, be so defined that
the result of each operation corresponds to a section of the rational numbers;
thus the arithmetical operations are reduced to operations with rational
numbers.
A complete theory of the operations involving real numbers can be
established; and the formal laws of the operations can be shewn to be the
same as in the case of the rational numbers, the range of possibility of
operations being greater in the case of real than in that of rational numbers.
This theory has been worked out to some extent by Dedekind: but as the
Cantor theory of real numbers lends itself to a simpler detailed treatment of




26


Number


[CH. I


the operations than that of Dedekind, and as it will appear that the two
theories are fundamentally equivalent to one another, it will be sufficient, as
an example of the general method of treating operations in accordance with
Dedekind's theory, to take only the case of the addition of two real numbers.
Let a, b be two real numbers defined by means of the sections (R1, R,),
(RI', R2') respectively; then the sum a + b, of a and b, is defined by means
of a section (R1", R2") which satisfies the following conditions:-If cl is any
rational number, it is put into the class R"/, provided there are two rational
numbers a, in R/, and b1 in R1', such that a, + b1 > cl; all rational numbers C2
for which this is not the case fall into the class R2". It is clear that every
number cl is less than every number c2, hence the section (R,", R2") is
defined by means of this condition.
It can be shewn that, when a, b both correspond to rational numbers, this
definition is in agreement with the ordinary definition of the sum of two
rational numbers, so that the sum of the numbers corresponds to the sum of
the corresponding rational numbers. Every number cl in R1", is - a + b,
because a, - a, b, - b, and therefore a, + b, - a + b. Further, if there were
contained in R2" a number c < a + b, so that a + b = c, +p, where p is a
positive rational number, we should have 02 = (a - Ip) + (b - Ip), and this is
contrary to the definition of c2, because a - p belongs to R1, and b -p
to R,'; thus every number 02 in R2", is - a + b, and it has consequently been
shewn that (R1", R2") defines the number a + b. As is usual, we have denoted
the rational numbers a, b and the conceptually distinct real numbers a, b by
the same symbols.
THE CANTOR THEORY OF IRRATIONAL NUMBERS.
22. The Cantor theory of irrational numbers essentially depends upon
the use of convergent simply infinite ascending aggregates, or convergent
sequences (Fundamentalreihen) in which the elements are rational numbers;
we therefore proceed to define and discuss these aggregates.
A simply infinite ascending aggregate (a,, a, a3,... a,...) in which each
element is a rational number, is said to be convergent, if it is such that
corresponding to any fixed arbitrarily chosen positive rational number e, as
small, in the ordinal sense, as we please, a number n can be found such that
I an-an+m <e, for m =, 2, 3,....
The symbol Ix is here used to denote that one of the two numbers
x, - x, which is positive; Ix t is said to be the absolute value of x.
This definition is equivalent to the statement that, in a simply infinite
convergent aggregate, an element can always be found whose absolute
difference from any element whatever which comes after it is as small as
we please.




21-23]


Cantor's Theory


27


It should be observed that the terms "as small as we please," or
"arbitrarily small," as applied to a positive number which is at choice, have
reference to the conception of order only, and not to the non-arithmetical
notion of magnitude. These expressions denote only that the number can be
so chosen as to be of lower rank than any other arbitrarily chosen positive
number.
To each value of e there corresponds a value of n, which will in general
have to be increased when e is made smaller.
We may denote the aggregate by the symbol {an}, and shall speak of it
shortly as a convergent sequence; that it is simply infinite will in future
be understood.
In a convergent sequence, corresponding to any arbitrarily chosen positive
number e, a number n can be found such that from and after that value of n
the absolute difference of any two elements is less than e.
For choose n so that | an-an+m  <   e, for all positive integral values
of m; then I an+m -an+m' I  a Ctn-an+m [ + | an-an+m' < e.
In the convergent sequence {an}, if we choose n such that a- an+m I < e,
then for m = 1, 2, 3,..., the value of an+m for all values of m, lies between an + e,
and an- e; that is to say, from and after some value of n all the elements
lie between two rational numbers whose difference is arbitrarily small. There
exist therefore two positive numbers a, a', of which the smaller a' may be zero,
such that, from and after some fixed value of n, all the elements lie in absolute
value between a and a'.
23. If the aggregate (a1, a,,... a,...) is such that, from and after some
fixed element, each element is less than the following one, and if all the
elements are less than some fixed number N, then the aggregate is a convergent sequence.
For if the aggregate is not convergent, there must exist some positive
number 3, such that an indefinite number of increasing values n, n,, 2,,...
of n can be found, for which i a,-,n i, an - an, (, I a, - an2... are all _ 3.
Since a,, - an, a2, - a,,,,... are all positive,.we have an, = an + r3, where r can
always be taken so large that an + r3 > N, or a,,n > N, which is contrary to the
hypothesis. Hence the aggregate is convergent.
It may in a similar manner be shewn that the aggregate is convergent
if, from and after some fixed element, each element is greater than the
following one, and if all the elements are greater than some fixed number.
If {an},, {bn, are two convergent sequences of rational numbers, a value of n
can be found corresponding to any arbitrarily assigned number e, such that both
an+m -an+m' I and b bn+m  bn+m,  | are less than e, m and m' having all positive
values.
For we have only to choose for n the greater of the two values corresponding to e, for each aggregate separately.




28


Number


[CH. I


24. It will now be shewn that the aggregates
{a + bn, {an-b n, {la } b.,
in which the elements are the sum, difference, product, and quotient, respectively
of the corresponding elements of the two convergent sequences [an}, {bn}, are also
convergent sequences, with a certain restriction in the last case.
We have I (an  bn) - (an+m + bn+m) I   n a- an+m | +  bn - bn+m; now n
can be so chosen for a given e, that for all values of m, an - an+m <  e,
and I b  bn - b+m  < I e, hence so that (an + bn) - (aq+m + bn+m) I < e; therefore
the aggregates {an + b,}, {an - bn} are convergent.
Again,
bnb - an+mbn+m I = an (bn - bn+m) + bn+m (an - an+m)
< a I bn - bn+m I +   an - an+m 
where a, /3 are the two positive numbers which are such that [ a i < a,
I bn+m I < 3, for all values of n and m.
We can take n so large that I b -bn+,n |     < 8, a n-an+m I < 8, where 8
is at our choice, and may be taken to be. Hence for this value of n,
a+13'
anbn - an+mbn+m < E, for every value of m; and thus {anbn has been shewn
to be a convergent sequence.
Lastly, in the case of ibn,' we shall suppose that all the elements of {bn}
are numerically greater than some fixed positive number /3'.
We have then
an  an+ _ an (bn+m - b) + b(an - an+m)
bn  bn+mn            bbn+m
hence
a,,  an+m   a I bn - bn+m 1 + p | an - a,,nm
b,  bn+m                "/2
If now n be chosen so that bn - bn+m I, Ia, - a,+,  I are both less than
1312                                                    an  anm+
+3 e, for every value of n, then, for such a value of n, bn   n+m;
therefore  an- is a convergent sequence, provided  bn is, for all values of n,
greater than some fixed positive number 3', which may be as small as we
please, but must not be zero.
25. The essence of Cantor's theory consists in the postulating of the
existence of an aggregate of objects for thought, the real numbers, ordered in
a definite manner, which manner is assigned by means of certain prescribed




24, 25]


Cantor's Theory


29


rules. Any element of the aggregate of real numbers is regarded as capable
of symbolical representation by means of a convergent sequence of which the
elements are rational numbers; and the mode in which the aggregate of real
numbers is ordered is specified by means of formal rules relating to these
convergent sequences. The aggregate of real numbers contains within itself
an aggregate of objects which is similar to the ordered aggregate of rational
numbers which has already been considered, in the sense that to each rational
number there corresponds a certain real number; and the relative order of any
two rational numbers, in the ordered aggregate of rational numbers, is the
same as the relative order of the two corresponding real numbers in the new
aggregate of real numbers. The rational numbers are frequently regarded as
identical with the real numbers to which they correspond, and are denoted
by the same symbols. In the development of Analysis, this identity leads
to no difficulties; but in the fundamental theory of the aggregate of real
numbers, a conceptual distinction between rational numbers and the real
numbers to which they correspond must be made, in order to obviate logical
difficulties, and especially with a view to coordinating Cantor's theory with
that of Dedekind. Those real numbers which do not correspond to rational
numbers are called irrational numbers; and those real numbers which
correspond to rational numbers are usually spoken of as themselves rational
numbers.
The rules by which the order of the real numbers in their aggregate is
assigned are the following:
(1) Any convergent sequence {an}, of which the elements are rational
numbers, is taken to represent a real number, which we may denote by a.
Two such aggregates {al}, {bn} are taken to represent the same real number
provided they satisfy the condition that, for any arbitrarily chosen positive
rational number e, a value of n can be found such that a+m - bn+m < e, for
this value of n, and for all values 0, 1, 2, 3... of mn. Symbolically*, we have
tan =- bn,, under the condition stated.
(2) The real number represented by {an} is regarded as of higher rank,
or in the ordinal sense greater, than the real number represented by {ba},
if, corresponding to any arbitrarily chosen positive rational number e, a value
of n can be found such that an+m - b,+m is positive for this value of n, and for
all values 0, 1, 2, 3,... of in, and greater than some fixed positive rational number
8 which may be dependent upon e. If, under similar conditions, a,4+m - bn+m
is negative and numerically greater than some fixed positive number 8, the
number represented by {an} is taken to be less than that represented
by {b,,.
* Those who hold the view, advocated by Heine and others (see ~ 6, note), that a real
number is identical with the set of symbols by which it is represented, can attach no direct
meaning to this equality. It can only be taken to indicate that the two expressions may be used
indifferently in any operation which involves the number.




30


Number


[cH. I


The aggregate (x, x, x,...) or {x}, in which all the elements are identical
with one rational number x, represents, since it is a convergent sequence,
a real number which corresponds to the rational number x. It is clear, from
the definition of order in (2), that the relative order of any two rational
numbers, in the aggregate of rational numbers, is the same as that of the real
numbers which correspond to them, in the aggregate of real numbers. The
aggregate of rational numbers, and that of the real numbers which correspond
to them, are similar aggregates.
Cantor's theory of irrational numbers, in the form in which it was presented
by himself and by Heine, has been criticized* on the ground that an assumption
is made that the sequence {x}, in which all the elements are the same
rational number x, represents the rational number x itself, and that this
amounts to an assumption that x is the limit of the sequence {x}; whereas
the theory of arithmetical limits is represented by Cantort as deducible from
his theory of irrational numbers, and as not assumed in the construction
of the theory itself. The theory in the form presented above is not open
to this objection.
It can be shewn that any two convergent sequences {an}, {bn}, satisfy one
or other of the conditions laid down in the above definitions of equality and
inequality, i.e. symbolically {an} {[bn}.
For, as has been shewn in ~ 22, corresponding to any arbitrarily chosen
positive rational number 8, a value of n can be found such that a,+m lies
between a, + 8, and an - 8, and such that, for the same value of n, bn+m lies
between bn +8, and b - 8; from this it follows that, for such value of n,
an+m - bn+,, lies between an - bn + 28 and a - b - 28; or a,+m - bn+m differs
from an -bn by not more than 28. If corresponding values of 8 and n can
be found, for which an-b b + 28, an-b - 28 have the same sign, then
an+m - bn+m has the same sign as a - b,, and is numerically greater than
a fixed number; the condition of inequality of {an}, {bn} is then satisfied.
If no such values of 8 and n can be found, then an+m - bn+m is numerically less
than 48; and since 8 is arbitrarily small, the condition of equality of {an}, {bn}
is then satisfied.
Although Cantor's form of the theory of irrational numbers, or rather
of real numbers, is more convenient for detailed development than is
Dedekind's form, yet it lies under the disadvantage that the nature of
any single real number is veiled by the fact that, although it is a unique
object, it is capable of representation by an unlimited number of convergent
sequences, and therefore that the formal character of the theory does not
make it clear what such a number really is. The comparison between the
* See B. Russell, The Principles of Mathematics, vol. I, p. 285.
+ See Math. Annalen, vol. xxi, p. 568.




2&jiSB]


Cantor's Theory


31


two theories which * will be given later on will throw light upon this point:
for it will be shewn that a convergent sequence of the rational numbers is
sufficient to define a section, of the kind fundamental in Dedekind's theory;
and this, as we have seen, is equivalent to the definition of a lower segment,
which is itself a certain definite class of rational numbers.
26. The sum   a + b, of two real numbers represented by the sequences
{an}, {bnj, is defined to be the real number represented  by the sequence
{an + bn; and the difference a - b is defined as the number represented by
{an-bn}.
It has been shewn in ~ 24 that the two sequences {an + bn}, {a,-bn} are
convergent.
If {an}, {bn} represent the same number, the sequence {an-bn} defines
the real number zero; for the condition that I an+, - bn+m  < e, where e is
arbitrarily small, for a sufficiently great value of n, and for m = 0, 1, 2, 3,...,
is in this case satisfied.
The product ab, of two real numbers, is defined to be the number represented
by the sequence {anbn}, which has been shewn in ~ 24 to be convergent.
The quotient a/b is defined to be the number represented by the convergent
(an\
sequence bI '.
The only restriction on this definition is that b is not to be zero; for, when
this condition is satisfied, the elements of the sequence [bn] which represents b,
can be so chosen as to satisfy the restrictive condition given in ~ 24, that
n} may be convergent.
It is necessary to shew that the sum a+ b, the difference a - b, the
product ab, and the quotient b, of two numbers a, b, as they have been
defined above, are definite numbers independent of the particular convergent
sequences used to represent the numbers a and b. Thus it must be shewn
that if {an} = {an', {bn} = {bn', then
{an + bn} = {an 4+ b'}, {a - b = {an' - bn'}, {aubi = {ann b},
an)   (an/ 
and                            bn    l
We have
(an+m ~ bn+m)- (a' +m T b'n+m)  an+ - a'n+m + | bn+m- bn+m
Now n can be so chosen, corresponding to a fixed number e, that
n+m - a'n+, I < 1 e, | bn+m - b'n+m <  6e, for m = 0, 1, 2, 3,...;
with this value of n, we now have the condition
| (an+m ~ bn+m) - (a' n+m + b'n+m) I < e,
* In Tannery's work Introduction a la theorie des fonctions d'une variable, chap. I, the theory
of irrationals is treated by a combination of the two methods of Cantor and Dedekind.




32


Number


[OH. I


satisfied; and this is the condition that {an ~ b,,} represents the same number
as fa,' + bn'.
Again,
I an+mbn+,, - an'+mb n+'+, I< I an+n (bn+m - b'n+m) I + b'n+ (n+m- a'n+m) I
< A I b|n+n - b'n+~, i B I a,,+m - a n+m
where A, B are fixed positive numbers. It is now clear that n may be so
chosen that [ anbn+,- a'n+mnb'n+m I < <, where 7 is an arbitrarily chosen
positive number; thus {anbn}, {an'bn'} represent the same number.
Again,
an+    a n    n+m a                   I b    bn+m 
— c              n          Yn+m) + -          (an^in-a a'+m7
bn+n   b'n+m    b,+mb+n+m(     bn+p+m n+m
whence it can easily be seen that the condition is satisfied that
{an+m  and fa
bn+m)     Ib n+m)
represent the same number.
It is readily seen- that the same commutative, associative, and distributive
laws hold for the operations between real numbers, as for those involving
rational numbers.
27. If, from  and after some fixed element an, all the elements of {an}
are positive and greater than some fixed positive rational number 8, then the
real number represented by {an, is positive, i.e. it is ordinally greater than zero.
For, if we take any convergent sequence {bn} which defines the number
zero, we have {an} > {bn}; because, for some fixed value of n, an+m - b+m is
certainly positive for all values of m, and is greater than a fixed positive
number; since n can be taken so large that an+m> 8, and bn+m < 8', where
8' is a positive rational number chosen less than 8.
Similarly it may be shewn that the number defined by {an} is negative,
if, from and after some fixed value of n, all the an are negative and numerically
greater than some fixed rational number 8.
The term "numerically greater" denotes that an > } 8, and thus refers
to the absolute values of the numbers concerned.
It is easily seen, that unless {a}, is such that, from and after some fixed
value of n, all the elements have the same sign, then {an} must represent the
number zero.
If {an}, {bn} define two different real numbers a, b, then there lie between
a, b an unlimited number of those real numbers which correspond to rational
numbers.
Suppose a > b, then there exist a rational positive number 8, and an
integer n, such that for all positive integral values of m including zero,




26-28]                Convergent Sequences                       33.(74b~n':> >'i a.-:n,,+m < e, b- bn+  < e, where e is a rational number
Chose:b Q toab <, & 'If we take any rational number x, which is < 8, and > e,
the number -{an-  }, in which all the elements are identical, lies between
{an} and {b,, since, for every value of m, we have a,,+, - (an-x) > x-;
therefore a is greater than the real number which corresponds to a, - x.
Again, (a - x) - bm,+ = (acc - b,,) + (b, - bn+m) - X > 3 - e - x, therefore provided x is chosen to be < 8- e, the real number which corresponds to an -x
is greater than b, and thus lies between a and b. The rational number
an -  may be chosen in an unlimited number of ways, since x is any rational
number whatever which lies between  - e and e.
CONVERGENT SEQUENCES OF REAL NUMBERS.
28. Convergent sequences will now be considered, of which the elements
are real numbers. It might at first sight be imagined that we should be led,
by the employment of such sequences, to a further extension of the domain
of number; it will however be seen that this is not the case.
The definition of a convergent sequence of real numbers is precisely
similar to the definition which has been given in the case of sequences of
rational numbers; thus (al, ao,... a,...) is a convergent sequence of real
numbers, provided that, corresponding to each arbitrarily chosen positive real
number V7, a value of n can be found such that an - On+m i < V, for m = 1, 2, 3,....
If we conceive that each such convergent sequence of real numbers represents
a single ideal object, and if we give definitions of equality and inequality, and
of the fundamental operations, precisely analogous to those given in ~ 25 and
~ 26, and assume as before that a convergent sequence in which all the elements
are identical with the real number a is taken to represent that one of the new
aggregate of objects which corresponds to a, it will be shewn that the new
aggregate of objects is similar to the aggregate of real numbers, i.e. to each of
the new objects there corresponds one of the real numbers, and also that the
relation of order between corresponding pairs of elements in the two aggregates is the same. It thus appears that the aggregate of new objects is
practically identical with the aggregate of real numbers, since the two are
ordinally similar. We saw however that there is no such relation between the
aggregate of real numbers and that of rational numbers. Therefore the
passage from rational numbers to real numbers involves a real extension of
the domain of number; but the passage from real numbers to an aggregate
of objects represented, in accordance with the rules referred to above, by
convergent sequences of real numbers, does not lead to any essential extension
of the domain of number...L     e any convergent sequence of rational numbers, and let {1a}
denqo    iequence of those real numbers which correspond to the rational
numnf~ich form the elements of {ta}. It can easily be shewn thtt ad,,n
3




34                            Number                          [CH. I
is a convergent sequence: for if e is an arbitrarily chosen positive rational
number, and c the corresponding real number, the condition of convergence
of {an} is that, for every E, a value of n can be found such that an+m lies
between an +, an - e, for m= 1, 2, 3,.... It follows from this that an+,,, lies
between C a+ c,,, - e; and this ensures the convergence of the sequence {a,,}.
Conversely, we see that if {an} is convergent, so also is {aC}.
Next, let {an} be a convergent sequence of real numbers; then, between
an and an+i, a real number an can be found which corresponds to a rational
number an. Let this be done for every pair of consecutive elements in {ca},
and let us consider the sequences {an}, an}j.
Since,-n- an+, = (an  - a,) + (an - an+m) + (an+ - G,+-,), we have
I     a n- n+mI     I I  an-an  + |I. +   - n ~n+?.
Now, corresponding to any real positive number 8, n may be so chosen that
for every value of in, I cn-a,, I, a-a,+,      -n+m-n+m|, ( are each less
than 3 8; hence for such a value of n
n n- an+m < 8, for n, ==1, 2, 3,...,
thus the sequence {Cn] is convergent.
Again,    {on} - {n} = {an - n}, and I,, - on I < a, - an+,
and, since {an} is convergent, n may be chosen so great that, for that and all
higher values of n, all the differences j a - a,,+, are less than an arbitrarily
fixed number, hence [ a - cn I satisfies the same condition; and therefore the
two convergent sequences {a,}, {-,,} satisfy the condition of equality, or they
represent the same one of the new objects. It has been shewn above that
since {a,,} is convergent, so also is {an}. Now {a,} corresponds to a single real
number a; therefore to any convergent sequence {an,, of which the elements
are real numbers, there corresponds a real number a.
We have further to shew that, if {ca}, {/3n} are two convergent sequences
of real numbers, and a, b the corresponding real numbers as just determined,
then a > b, according as {a,} {/3n}.
We know that a ' b, according as {an}  {bn}l, where {bn} denotes the
sequence of rational numbers which defines b, in the same way as {an} defines a.
Now {an }-{/3n} = {an- /3}, and an- fi =(aln - a) + (bn -in) + (,n-,b); and we
can choose n so large that I an- dI and I bn- fn are each less than -i, where
v} is an arbitrarily chosen real positive number: therefore we see that an - ~i lies
between an- bn+r and an- bn-. It follows easily that {an;} {/f}, according
as {a,} _ {bn,} or according as {ac,} {bn}, and hence as shewn above, according
as a  b. It has now been shewn that the objects which are represented by
convergent sequences of real numbers have the same ordinal relattenl to one
another as the real numbers to which those sequences have been shewn to
correspond.




28, 29]


Theory of Limits


35


It appears, from what has now been proved, that, to every convergent
sequence of real numbers there corresponds a real number which may be
taken to be defined by means of that sequence.
There does not necessarily exist any rational number which corresponds in
the same sense to a convergent sequence of rational numbers. The property
of the aggregate of real numbers here stated embodies the characteristic
difference between that aggregate and the aggregate of rational numbers;
for the latter does not possess the corresponding property. It is this property
of the aggregate of real numbers which makes it suitable to be the field of
the real variable in the Theory of Functions.
THE ARITHMETICAL THEORY OF LIMITS.
29. If x1, x2, x,... xn,... is a sequence of real numbers such that a number
x exists which has the property that, corresponding to any arbitrarily chosen
positive number e, a value of n can be found such that  - x - x, x - xn,
x - x,1+o... are all less than e, then the number x is said to be the limit of the
sequence x,, x2,... x,.... This fact may be denoted by the equation x = L x.
n= co
This definition is known as the arithmetical definition of a limit, and was
first given *, in a form substantially identical with the above, by John Wallis.
It will be observed that the above definition contains no assertion as to
the necessary existence of a limit of a sequence of numbers, but contains
only a statement as to the relation of the limit to the numbers of the sequence,
in case that limit exists.
There cannot be two numbers which both satisfy the condition of being
a limit of the same sequence. For, if possible, let x, x be two such numbers
and let I x - x' I =. Choose a value of e, less than  8; then numbers n, n' can
be found such that I x -     x+,,  x' - Cx,,+m\ for all values 0, 1, 2, 3,... of m,
are less than e. Suppose n > n', then x - xn I and x' - xn are both less than
e, hence x - x' < 2 < 8, which is contrary to the condition x - x' =.
It will now be shewn that, if the numbers of the sequence {xa,} are real
numbers, and if the sequence is a convergent one, then the real number x
defined in the manner explained in ~ 28, by the sequence {xn, is the limit
of the sequence.
For the two sequences {x,}, {x} both define the same number x, and
therefore satisfy the condition of equality, which is that x - Xn+m I < e, for any
arbitrarily chosen e, provided n be sufficiently great, and this is the condition that x should be the limit of the sequence t[xn. A sequence of real
numbers which has a limit must be convergent. For if x is the limit of {x,},
* Arithmetica Infinitorum (1655), Prop. 43, Lemma. See M. Cantor's Geschichte der Mathematik, vol. II, p. 823.
3-2




36


Number


[CH. I


then for a sufficiently large value of n, x - x, |, x - x,+, I,... x- x,+, l,.. 
are all less than 4e, where e is arbitrarily chosen; now I x - xn+, I < I - x-,l
+ x - xn+,; hence I xn - Xn+m I < e, which is the condition of convergence
of {xn}.
As the complete result we have now the theorem known as the General
Principle of Convergence:
The necessary and sufficient condition that a sequence xi,, x... xn,... of
real numbers may have a limit, is that, corresponding to every arbitrarily
chosen positive number e, a value of n can be found such that x, - x+,,
X, - Xt+2,  l -xn+3)... shall be all numerically less than e.
This theorem, which contains the criterion for the existence of a limit
as defined in accordance with the arithmetical definition of a limit, is a
deduction from Cantor's theory of real numbers.
30. If the numbers of a sequence {xn} are rational numbers, instead
of real numbers, the definition of the limit is applicable, and it is a necessary
but not a sufficient condition for the existence of the limit, that the sequence
should be convergent. Strictly speaking, if a convergent sequence of rational
numbers has a limit, that limit is also a rational number; but from the
existence of convergent sequences of rational numbers which have no limit
there arises the necessity for the extension of the domain of number, so that
in the extended domain every convergent sequence may have a limit; this
extension has been carried out by substituting Real Number for Rational
Number. However, although a convergent sequence of rational numbers
which has no rational limit, has in this strict sense no limit at all, by reason
of the convergent sequence of those real numbers which correspond to the
rational numbers having an irrational number as limit, and since, as has been
seen above, these real numbers are for practical purposes not distinguished
from the rational numbers to which they correspond, it is usual to consider
this irrational number to be the limit of the sequence of rational numbers.
We may thus assert that any convergent sequence of rational numbers
which has not a rational number as limit, has an irrational number as its
limit. This assertion is a correct one for the practical purposes of Mathematical Analysis.
31. The method of limits, which is essential both to pure Analysis
and to the applications of Analysis in Geometry and in Kinetics, had a
geometrical origin in the Method of Exhaustions, which was applied by the
Greek geometers to determine lengths, areas, and volumes, in simple cases.
This method, supplemented by the notion of the numerically infinite, was
developed in later times, in various forms, into a general method which formed
* This term "das allgemeine Convergenzprinzip" is due to P. Du Bois-Reymond; see his
Allgyemeine Functionentheorie.




29-31]


Theory of Limits


37


the basis of the Infinitesimal Calculus. The traditional geometrical conception of a limit may be exemplified by the case of the determination of the
length of a curve as the limit of a sequence of properly chosen inscribed
polygons. The lengths of the perimeters of the polygons are regarded as
continually approaching the required length of the curve whilst the number
of sides of the polygons is continually increased. The limit, the length of the
curve, is then regarded as actually reached at the end of a process described
as making the number of sides of the polygon infinite, this mode of
attainment of the limit being however inaccessible to the sensuous imagination, and disguising an actual qualitative change of a geometrical figure
which possesses corners and is bounded by segments of straight lines, into
one which has no corners and has a curvilinear boundary. No doubt was
felt as to the existence of the limit, which was regarded as obvious from
geometrical intuition. That a curve possesses a length, or an area, was
considered to require no proof. The first mathematician who recognized
the, necessity for a proof of the existence of a limit was Cauchy, who gave
a proof of the existence of the integral of a continuous function. That the
logical basis of the traditional method of limits is defective has in recent
times received a posteriori confirmation by the exhibition of continuous
functions which possess no differential coefficient, and by many other cases of
exception to what were regarded as ordinary results of analysis resting on the
method of limits, which have been brought to light by those mathematicians
who have been engaged in examining the foundations of analysis.
The arithmetical theory of limits, which is summed up in the general
principle of Convergence, provides a definite criterion for the existence of the
limit of a sequence of numbers; and a considerable part of modern analysis
is concerned with obtaining special forms of the general criterion adapted
for use in special classes of cases. The theory is essentially dependent upon
the theory of irrational numbers; for, in default of an arithmetical theory of
irrational numbers, all attempts to prove the existence of a limit of a cbnvergent sequence are doomed to inevitable failure; and this for the simple
reason that a convergent sequence of rational numbers does not necessarily
possess a limit which is within the domain of such numbers. The definition
of real numbers by means of convergent sequences of rational numbers is
not a mere postulation of the existence of limits to such sequences; it
involves rather the introduction of an enlarged conception of number, of such
a character that the scheme of ordered real numbers should form a consistent
whole, and such that every convergent sequence of numbers in the domain of
real number necessarily has a limit within that domain. The postulation of
the existence of the aggregate of real numbers is justified by shewing that


* An interesting discussion of various methods which have been suggested of proving the
existence of a limit will be found in Du Bois-Reymond's Allgeneine Fuictionentheorie.




38


Number


[CH. I


a complete scheme of definitions and postulates can be set up for the elements
of this aggregate, and that such a scheme does not lead to contradiction*. As
regards the existence of limits in the case of lengths, areas, volumes, &c.,
referred to above, the order of procedure is a reversal of the traditional one,
the existence of the limit being no longer inferred from geometrical intuition.
For example, in the case of the determination of the length of a curve, that
length is not assumed to be independently known to exist, but is defined as
the arithmetical limit of the sequence of numbers which represent the
perimeters of a suitable sequence of inscribed polygons. When this sequence
is convergent, and its limit is independent of the particular choice of the
polygons, subject to suitable restrictions, then the limit so obtained determines
the length of the curve. In case no such limit exists, the curve is regarded
as not having a length.
EQUIVALENCE OF THE DEFINITIONS OF DEDEKIND AND CANTOR.
32. In order to establish the equivalence of the definitions of irrational
numbers, as given by Dedekind and by Cantor, it must be shewn that every
convergent sequence of rational numbers defines uniquely a section of all
the rational numbers, and that this section is the same for all convergent
sequences which represent the same real number in accordance with rule (1)
in ~ 25. Conversely, it must be shewn that any number defined by a section
can also be represented by a convergent sequence of rational numbers.
To shew that, corresponding to the convergent sequence {x,, which, in
accordance with the Cantor theory, defines the real number x, a section can
be found: Let r be any rational number, and let r be the corresponding real
number represented by {r}. The number x-r is represented by {x - r};
and if this number is not zero, then (see ~ 27), from and after some fixed
value of n, x,, - r has a fixed sign, positive or negative according to the value
of r. A section of the rational numbers may now be defined as follows:Let every number r such that x - r is negative, from and after some fixed
value of n, be placed in the class R,; and let every number for which x, - r
is positive, from and after some fixed value of n, be placed in the class R1.
If there exists a rational number r, such that neither of these cases arises,
then  _ r, and r may be put into either of the classes R,, R,. It has thus
been shewn that a section of the rational numbers can be determined, corresponding to the convergent sequence {xn}.
Next, let {xn'} be any other convergent sequence which represents the
same real number x, as {xn} does. We have to shew that the section of the
rational numbers which corresponds to {xn'}, is identical with that which
* On this mode of regarding the aggregate of real numbers as dependent upon a complete
consistent scheme of definitions and axioms, see Hilbert, " Ueber den Zahlbegriff," Jahresber. d.
deutsch. math. Vereinigung, vol. vII (1900).




01 03]
d lOo


Equivalence of Definitions


39


corresponds to {xn. If, as before, r denote any rational number, we have
{x - r} = [x, - r. Now a value of n can be found, from and after which,
xn-r and x,' - r both have fixed signs independent of n, and they must have
the same sign. It follows that a number r which belongs to the class R1,
must also belong to the class R1/, by which the section corresponding to [xn'}
is defined; and also a number r which belongs to the class R2, necessarily
belongs to R,', except in the case xn} = {x,'} = r. It has thus been shewn
that the section (R,, R,) which corresponds to {xc}, is identical with the
section (R/,, R,') which corresponds to {x,,'}.
33. To shew that a convergent sequence can always be found such as
to define the number corresponding to a given section (R1, R,), we observe
that two rational numbers can always be found, one of which is in R1 and the
other in Ri, and such that their difference is numerically less than a given
arbitrarily small rational number e. Let A be any rational number in RB, and
let ' be a rational number < e. Then of the numbers A +e', A + 2e',... A +re',...
there must be a last one A + re' which falls in R1, for A + ne' may be made as
large as we please by taking n large enough; the next number A+(r+1)e' is
then in R2; and these numbers A + re', A 1 (r + l)e', whose difference is e < e
are the two numbers required. Moreover, if B is a rational number in R2,
the two numbers may be so chosen that both lie between A and B; for we need
only take e' to be of the form - (B - A), where s is a positive integer so
chosen that - (B - A) < e.
Now let {e,} be any convergent aggregate of rational numbers, which has
zero for its limit. Choose x, in Ri, and x2 in R2, so that x2 - x2 < 61; next take
*x3 in Rl, and x4 in R2, so that Ix - X1 < 62; and that x,, x4 both lie between
x, and x2. Proceeding in this way, we can take x2,,-, X22, rational numbers
of different classes, so that x2n-i - x2,, < e,,; then either of the sequences
{X', x3, 2, Ix.., }  x4,... }, defines the number which is represented by the
section (RI, R2).
To prove this, we observe that {x2n-1} is a convergent sequence, since all
the elements are < x2, and x c< x 3< x....
Again, suppose a is a rational number belonging to R2, we can shew that,
provided a rational number b exists in R2 which is less than a, then a is
greater than all the numbers xl, x3,... by more than a - b. For
a - x2,-_ = (a - b) + (b - xn-1) > a - b,
however small b - x2,,~ may become. Hence, unless a is the smallest rational
number in R2, the real number {a} which corresponds to a, is greater than
the number (x,, x3,...).
Again, the sequences {x2,n-,, {x24} represent the same number, since their
difference is the aggregate {e,,} which defines zero. It now appears, by




40


XNumber


[CH. I


reasoning similar to the above, that any number a in jR is such that the
real number {a} is less than the number {x2}j, unless a is the greatest
rational number in Ri.
If either R, has a greatest rational number, or R, has a least one, the
real number {a} which corresponds to this rational number a, is itself defined
by (RI, R2), and is the number represented by either of the sequences {Xn-i},
{X2n}. In any case, either of these two sequences defines the number given
by the section (Ri, R2).
The complete equivalence of the two theories of Dedekind and of Cantor
has now been established. The first theory operates with the whole aggregate
of rational numbers, the second with sequences selected out of that aggregate.
THE NON-EXISTENCE OF INFINITESIMALS.
34. It should be remarked that, in assuming that every section of the
aggregate of real numbers defines a single real number, it has been implicitly
assumed that if a, b are any two positive real numbers, such that a < b, then
a positive integer n can be found such that na > b.
This is the arithmetical analogue of the so-called principle of Archimedes.
If any real numbers existed which are ordinally greater than all the
numbers a, 2a, 3a,..., then a section of the aggregate of real numbers would
be defined by considering all numbers greater than all the numbers
a, 2a, 3a,... to be in one class, and all the remaining real numbers to
be in the other class; and this section would define a real number N. If
now e be an arbitrarily chosen positive number less than a, then N - e is
a number which is less than some of the numbers a, 2a, 3a,...; and there
must be a first of this set of numbers such that N- e is less than it. Let
this be pa; thus N- e < pa, hence N < pa + e < (p + l) a; which is contrary
to the hypothesis that no number na is in the class of numbers which
are > N.
The property of the aggregate of real numbers which has been established
may be denoted by the statement that the aggregate of real numbers forms
an Archimedean system; and this property of the aggregate is essentially
equivalent to the property that every section of the aggregate defines a single
number of the aggregate.
A consequence of the fact that the aggregate of real numbers forms an
Archimedean system, is that so-called infinitesimal numbers do not exist
within the aggregate. Every positive number e, being such that an integer n
can be found such that ne > 1, is a finite number, in the sense in which
finite numbers were distinguished from infinitesimals, in the older forms of
the Infinitesimal Calculus. In Arithmetical Analysis, the conception of the




33-35]


Theory of Indices


41


actually infinitesimal has no place. When the expression "infinitesimal" is
used at all, it is to describe the process by which a variable to which the
numbers of a sequence converging to zero are successively ascribed, as values,
approaches the limit zero; thus an infinitesimal is a variable in a state of
flux, never a number. Such a form of expression, appealing as it does to a
mode of thinking which is essentially non-arithmetical, is better avoided.
THE THEORY OF INDICES.
35. When m is a positive integer, and x a rational number, xm was
defined to denote x x x x x... x x (m factors); and this definition may be
extended to the case in which x is any number defined by a convergent
sequence; so that if x is defined by [x,,}, xlm is defined by {x,,l}. It thus
appears that for any real numbers a, we have, provided in and n are positive
integers, Xm x x" = Xv'm++1.
If we assume x~ and x-fi to be defined as having such a meaning that
this law of indices holds when vz or n is zero, or a negative integer, we can
at once interpret x~ and x-mn; for
X0 X xn = xl+~ = xn, thus xa = 1,
1
and                 x-a2 x x = x~ = 1, thus z-J  -1.
When p/q is a rational fraction, we shall define xq to have such a
meaning that the above law of indices holds when either or both of m, n
may be rational fractions. With this assumption
P   P    P      P
xa x xq x aq... X xq (q factors) ==P;
p           P
hence (xaq) = xp; or xq is, if it exists, a number whose qth power is xp. The
p
problem of determining, if possible, a number xs, is that of finding a number
whose qth power is a given number; and it has been already shewn that this
is not always a possible operation within the domain of rational number.
It will now be shewn that, in the domain of real numbers, the' operap
tion of finding xq is always a possible one when x is positive, and also when
x is negative; provided however that in this latter case, q is an odd number, or
if it is even, p is not odd.
The following lemma will be required:-If a is any real positive number
less than unity, a positive integer mn can be found such that a'l < e, where e is
an arbitrarily prescribed positive number, or, in other words, L an = 0.
Sn=cco
Since all > al+1, the sequence (a, a2,... an,...) is convergent.




42


Number


[CH. I


Suppose, if possible, that the sequence represents a positive number ko
different from zero; then in may be so chosen that am", alc+l,... all differ from k
by less than the arbitrarily prescribed number 8, say a =k-F+, where V< 8. We
have therefore an+l1 = (kJ + V) a < (k + 8) a; now 8 can be chosen to be equal to
(1   a), then ax+  < k -; and this is contrary to the condition imposed
14- a
in the choice of m.
It follows that k cannot be different from zero; and thus the lemma is
established.
Suppose now that a is any positive number, rational or not, which lies
between N1q and (AV + 1)q, where LA is a positive integer; we shall first shew
that a number N + h, where h < 1, can always be found such that a - (N + h)q
is positive, and less than a - NK.  We find by division
(N + ) -,)   = {(N+ h) N-  } {(N + -l)r-l + (N~+ h) —2N +... + Nw-};
hence, if h is positive and less than unity, (N + h)q - -Ar lies between
qhNP- and qh (N + 1)2-l.
Since         a - (N + z)q = (a- N2) - {(N + h)~ - N'7,
we must take h not greater than  a, in order that a - (N + h)q may
we tke geae  q (N + 1)q-, i
certainly be positive; and the difference a - (N + h)q is then less than
(c - N') - qhN-.-l
( c - ffq
Let                        h      -
q (N + l)a-i'
then            a -(Nh + ) < (a - N)     - (N+     1} 
Let N, = N + h, then N1 is such that
a      <-N ( < (a- ') -  
and                             Nr > N.
In a similar manner, we can shew that a number N2 exists which is > N1,
and such that
a -N2q < (a -21-    ( N) {1-    }
Proceeding in this manner, we obtain a series of numbers N, IVI, N2,...
Nr... such that N,. > N,.-,, and that a - N,. is positive, and less than
(a - N1)     (N,+ I) '
We shall now shew that (AN, N1,,... N,...) is a convergent sequence
which defines a number whose qth power is a.




35, 36]                 Theory of Indices                          43
The sequence {N,.j is convergent since N,. > N,._, and every N. is less than
N+ 1. The qth power of the number defined by this convergent sequence
is jNj,}, and we shall shew that this defines the number a or {a}.
We have
a                 -.(<(a-  ))j (j)[l-  (N._2 )              N   j)]
<(ca - N) [I   (N    )q ]r
for                       A     <,NV-1 <N           +1'
and hence         >1-)          >       _      )
Now 1 - (T -1      is a proper fraction, hence from the lemrma proved
above, we infer that a power r of the expression can be found which is less than
an arbitrarily chosen positive number, which number we may take to be - -q 
Hence, corresponding to every e, a number r can be found such that
a - NQ+, < e, for s = 0, 1, 2,..., and therefore the sequence {NV}9, defines
the number {a} or a.
If a is a positive proper fraction, we have (a2)Q < a, hence we may take
N to be equal to a2, instead of to a positive integer. Then a < (N + 1)q; thus
this value of N will play the same part as the integral value in the above
proof, and the reasoning is the same as before.
36. It has now been shewn that in every case a real number can be found of
p
which the qth power is a given positive number a. It thus appears that xq has
an interpretation within the domain of real numbers, when x is any positive
number, and - is a positive rational fiaction.
We interpret  to be such that
We interpret x q to be such that
P    P
X    q X Xq =XO= 1,
p      p
or                            X q =1 /q.
If x is a negative number - x', we have (- ')q, defined as a number whose
qth power is (- x')p; and (- x')P is x' or - x', according as p is even or odd.,p.p
If p is even, (- x')q can be interpreted as the value of x'q. If p is odd,




44


N  imber


[CH. I


p                         p
and q is odd, (- ')q may be interpreted as - x'.  When p is odd, and q
is even, we have obtained no interpretation of (- x').
2r+l
To complete the theory of indices in such a way that (- x') 2S may have
an interpretation, we should require a further extension of the conception of
number. This further extension takes place by the introduction of complex
number, which is however outside the limits imposed upon this work as a
treatise dealing only with real number.
37. The only case in which xL, for a positive x, has not been defined, is
that in which n is not a rational number. To extend the definition to this
case, we suppose n to be defined by a convergent sequence [n,., in which all the
numbers n,. are rational. We shall shew that the aggregate {[xt' is convergent,
and the number which it defines we shall denote by x*1.
We have xI' - x,'++s =  r l{1 - xr11+S-h'}; now, since {n,.} is a convergent
aggregate, all the numbers n,. are numerically less than some fixed number,
and therefore I xni <A, where A is some fixed number.
First suppose x > 1, then,n - Xn-+s = - -r+s (.Xnl-.s - 1) = XO"r (1 - X1lr+s-2),
hence               I xn. - x,.+s I < A I x1"n-~N+sI - 1.
Now let r be so chosen that, for all values of s,
1
|r - U'.s I < -,
where q is a positive integer; then
1            x-1
XInr-r+sl - 1 < X - 1 <  1   2       q-1
I +-xq+xq+...+- X q
hence                   I x l,.-,. +s - 1 -   <  i
or                     [ x?' - X+<s < A,and if q is chosen so that  <          here e is a fixed number we see
q  A (x - 1)' w
that r' may be so chosen that | xr - x'+ts I < e, for all values of s, therefore
{Sx"'} is a convergent sequence.
If x < 1, then {q  is a convergent sequence, and therefore {x"r} is also
convergent, since it is the quotient of {1} and {xn'}. If x = 1; then {x}- = 1.
Thus in every case {xn,' is a convergent sequence if {n,.} is convergent.




36-38]           Representation of real numbers                   45
Since {xr} x {xkr } = {Xnr.+mr} we see that the definition of xn, when n is
not rational, is such that the relation Xm x x- -= x1f+n is satisfied.
THE REPRESENTATION OF REAL NUMBERS.
38. The ordinary mode of representation of a real number is by means
of a decimal, or more generally by a radix-fraction. When the decimal is
non-terminating, this mode of representation is a case of the representation
by a convergent sequence of rational numbers, in accordance with Cantor's
theory. For example, the number or is represented by the sequence
(3, 3-1, 3-141, 3-1415, 3-14159, 3-141592,...),
where by known processes, any prescribed element can be found as the result
of a definite number of arithmetical operations.
The general theorem will be established that every positive real number N
is uniquely representable by means of a non-terminating series of radixfractions, of which r, the radix, is any integer > 2.
Of the numbers 0, r, 2r, 3r,..., there is (see ~ 34), of all those which are
less than rN, a greatest one cor, which may be zero; thus
rN> cor, and <(co + 1)r;
it follows that              N= co + -
r
where N1- is a positive number less than r.
In a similar manner we obtain
N,=cl +   2, N,=C2N           NN= -  c +Nn
r            r                3"
where N2, N3,... N,,+~ are all < r; therefore
l c2        cn   Nn+l
-c=-Co + -:+ +... +  -4+
where Co, cl, c2,... cn are each of them positive integral or zero, and
0 < NVn+ < r.
Since            N-(    +   -   -... + +  < r
and it has been shewn that - has the limit zero as n is indefinitely increased,
we see that the sequence, of which the nth element is
C1  C\ 2,,
cO +- + 2 +... + -
is convergent, and represents the real number N. This is expressed by
N = Co + l + 2+      n
h= iecs  b  +... +  r a+...,
r   ri      rn
in which N is represented by a non-terminating radix-fraction.




46


Number


[CH. I


Let us now consider the case in which N is a rational number a, in its
b 
lowest term. We have a = a0b + 3o, where,0 < b; and r/3 =.alb + /3, where
/3 < b; r/3 = a2 b + 2;..., r/3-, = anb + /3, where 1/, 3 s,...*  are all less
than b.
If one of the numbers /, say /n, is zero, we have
N --  = a a0+  + ~  +..'  + 
and thus N is expressed in terminating radix-fractions; this case can only
arise when b contains only prime factors of r. The terminating series of
radix-fractions can be replaced by a periodic one which does not terminate.
For if we use a - 1 instead of a,, as the numerator of rn, we have
r/3n_ = (a, - 1)b + b;
thus 3n becomes b instead of zero, and
rb = (r -1) b + b;
thus n, fn+l,... are all equal to b; and  a+ n+2,... are all equal to r — 1.
Thus N is represented by
a      a1  a9       a,-I    r-l r9-i
N -=a     +al +  +    + an-   +      -+     +
b      r   r2        r1     r+     r+2...
It thus appears that a rational number, which in its lowest terms has a
denominator which contains only prime factors of r, is capable of a double
representation; (1) by a terminating series of radix-fractions; (2) by a nonterminating series of radix-fractions, of which the numerators after some fixed
one are all r- 1.
In case none of the numbers /,8, 3,... kt,... vanishes, it is clear that
since all these numbers are either 1, 2, 3,... b - 1, they cannot be all
unequal. Suppose,3n is the first which is repeated, and let,n =/3n+m; it
is then clear that /,,l+ =fn+m+l, /n+2 = /,+m+2,..; and therefore the number
is represented by a recurring series of radix-fractions.
39. When a number is defined by means of a convergent sequence of some
special form, it is in general not immediately obvious whether the number is
rational or irrational.  Many special investigations relating to particular
cases, and various general criteria have been given by well-known mathematicians.
One of the most important modes of such representation of a number is
that by an endless continued fraction. This fraction may be regarded as an
aggregate, each element of which is a finite continued fraction. Legendre
established the fundamental theorem  that a number represented by an
endless continued fraction
al a2 as      an
b, + b 2~ b +'"b +"




38, 39]


Representation of real numbers


47


that is, by an aggregate of which the nth element is
a,  a2  a3    an
bl+ b      + b3 +'" bn'
is irrational*, provided the positive integers a,, b, are such that for every
value of n, bn - acn > 1; except that when b - an = 1, for every value of n > m,
where m is some fixed number, and when at the same time the signs before
all the fractions -, for n > n, are negative, then the continued fraction
converges to unity, or to a rational fraction, according as nz = 1, or mn > 1.
This theorem contains as special cases the theorems previously established
by Lambert, that el, tan x, loge x, tan — x, wr are irrational for rational values
of x. The irrationality of e and e2 was first proved by Eulert.   Legendre +
himself applied the general theorem to prove the irrationality of 7r2, although
his proof was lacking in rigidity.
The following general theorem has been proved~ by Cantor:If b, b', b",... is a set of positive integers such that, q being any
arbitrarily chosen integer, all the numbers 1, b, bb', bb'b",... from  and after
some fixed number of the sequence, are divisible by q; then any number N
can be uniquely represented by
I + -, +...,
~b + bl+ b_7ib/+
where I is an integer, and X, /A, v,... are integers such that
X   b-1, /,     b'- 1, v    b"-1.
Further, in order that the number N    may be rational, it is necessary that,
from  and after some fixed term   of the series, all the numbers X, Lt, v,...
have their highest possible values. If this condition is not satisfied, N is
irrational.
As an example of this theorem, the number e represented by
I    1       1
2+ - +      +    l    +
2   2.3    2.3.4
is seen to be irrational.
A particular case of Cantor's theorem  is that in which the sequence of
numbers b, b', b",... from a particular element onwards, is periodic. In
* A proof of this theorem is given by Pringsheim, "Ueber die Convergenz unendlicher
Kettenbrtiche," Sitzungsberichte d. bayer. Akad. vol. xxvii, 1897, p. 318.
~ On the history of these theorems see Pringsheim's article " Ueber die ersten Beweise der
Irrationalitat von e and 7r," Sitzungsberichte d. bayer. Akad. vol. xxvii.
+ See his 'lements de Geometrie, Note 4; see also Rudio's work, "Archimedes, Huygens,
Lambert, Legendre," 1892, p. 166.
~ Schlomilch's Zeitschrift, vol. xiv (1869), "Ueber die einfachen Zahlensysteme."




48                             Number                            [CH. I
this case, the necessary and sufficient condition that the number represented by
/3  /3'   /"
++ +b+...
should be rational, is that the sequence /, /3', /",... be, from and after some
fixed number of the sequence, periodic. This is a generalization of the
theorem relating to a number represented by radix-fractions.
If b = 2, b' = 3, b" =,... we obtain the theorem* that the number
represented by
C1  C2  C3        Cn
2!  3!  4!        n!
where cn < n - 1, is rational, only if, from and after some particular value of n,
Cn= n-1.
A mode of representation of numbers by sequence of products has been
givent by Cantor. He shews that every number N> 1, can be uniquely
represented in the form
(1+ I) (   ) (1  + ) (+ d)..
aA     bj\   c/\    d...
where a, b, c,... are integers such that
b - a, c  b2, d   c,....
N        Na
The number a is determined as the integral part of N-. If     = -B,
]T-1      a+I
B        Bb
b is the integral part of; if b ---= C c is the integral part of
B - i'      ~1
C
C-; and so on.
As an example, V/2 is represented by
(1     + 3)(   17) (   577  + 66h58 7) '
where    17= 2.32- 1, 577 = 2. 1712, 665857 = 2. 5772-1,....
The criterion for determining whether N is rational or irrational is the
following:The number represented by
(i+    II +   I1 +) -..,
where                      b  a2, c - b2,...,
* See Stdphanos, Bulletin de la soc. math. de France, vol. vii (1879). For further information on the history of this subject see Pringsheim's article I. A. 3, in the Encyclopddie der
Math. Wissenschaften.
t Schlimilch's Zeitschrift, vol. xiv (1869), " Ueber zwei Satze...."




39, 40]


~The Continummz


49


all the numbers a, b, c,... being positive integers, is rational if, from and after
some fixed number of the sequence a, b, c,..., each number is the square of
the preceding number of the sequence; but the number is irrational if
this condition is not satisfied.
THE CONTINUUM OF REAL NUMBERS.
40. If a, b, are any two real numbers such that a,< b,, then two real
numbers a, b2, (a, < b2), can be found both lying between a,, b,, and such that
the difference between a,, b2 is as small as we please, i.e. b2- a < e, where e is
an arbitrarily prescribed number. Between ca, b2, two more numbers a,, b,,
(a3 < b,), can be found whose difference is again as small as we please; and
this process may be carried on indefinitely. This property of the aggregate
of real numbers may be expressed, to use the term introduced by G. Cantor,
by saying that the aggregate of real numbers is connex; it arises from the fact
that an indefinite series of numbers can be found which lie between any two
given numbers. If we anticipate a term which will be introduced when we
come to the general theory of aggregates, the property of connexity may be
expressed by saying that the aggregate of real numbers is everywhere-dense.
It will further be observed that the aggregate of rational numbers is also
connex, or everywhere-dense; so that, so far as this property is concerned,
there is nothing to differentiate the one aggregate from the other.
If the difference of a, and bn is denoted by En, and the sequence,6, 62,... En,... satisfies the condition, that corresponding to any fixed
arbitrarily small positive number q, a value of n can be found such that
En, en+,... are all less than Vi, then there exists a single real number x which is
greater than all the numbers a,, a,..., and less than all the numbers b, b,....
This number x is the limit of either of the sequences (a,, a,... an,...) and
(b,, b2,... bn,...), and is defined by a section of all the real numbers.
If we confine ourselves to the domain of rational numbers, there subsists
in that domain no such property; that is, the above numbers a, b being all
rational, no such rational number as x necessarily exists.
In the domain of Real Number, every convergent sequence has a limit
which is a number belonging to that domain; and, conversely, every number
is the limit of properly chosen convergent sequences of numbers belonging to
the domain: but in the domain of Rational Number a corresponding statement does not hold good, although the converse is still valid.
This property, which the domain of real numbers possesses, we express
by saying that the aggregate of real numbers is perfect. -The aggregate of
rational numbers is not perfect.
From the point of view of Dedekind's theory, the property that the
aggregate of real numbers is perfect expresses the fact that every section of


H.


4




50


Number


[CH. I


the real numbers corresponds to a single real number, and the converse.
A section of the rational numbers does not always correspond to a rational
number; consequently the aggregate of rational numbers is not perfect.
We give the name continuum* to an aggregate which possesses the two
properties of being connex, and of being perfect. This is in the first instance
taken to be the definition of the meaning of the word continuum, as it is
used in Analysis. Thus the aggregate of real numbers forms a continuum;
whereas the aggregate of rational numbers is essentially discrete, and does
not form a continuum, since one of the two essential properties of a continuum
is absent.
The aggregate of real numbers is spoken of as the continuum of real
numbers, or the arithmetic continuum.
The real numbers which lie between two numbers a, b do not form a
continuum, but if the two numbers a, b themselves are considered to be
included in the total aggregate, then this completed aggregate does form
a continuum.
All the real numbers x such that a - x < b, in the ordinal sense of the
symbols <, =, >, are said to form an interval (a, b); and such an interval
is frequently described as a closed interval.
The real numbers x which are such that a < x < b, are frequently said to
form an open interval (a, b).
The closed interval (a, b) is a continuum, since it satisfies the two necessary
conditions for the applicability of the term; but the open interval (a, b) is
not a continuum, as it contains convergent sequences which have no limit
belonging to the open interval. Such an open interval has been termed by
Cantor a semi-continuum.
Of the two essential properties of the arithmetic continuum, that of
connexity, and that denoted by the term perfect, the latter is absolutely
indispensable, in order that the arithmetic continuum may be suitable
to be the field of operations in analysis. It will appear, when we come
to the consideration of the theory of functions of a real variable, that many
of the most important properties of a function may still subsist even if the
domain of the variable lacks the property of connexity; but that such
properties would not belong to functions of a variable which is defined for
a domain such that convergent sequences of numbers in it possess no limit
within that domain, and which therefore lacks the property of being perfect.
This is the more remarkable on account of the fact that, in the older
traditional notion of a continuum, the property of connexity was the one
which was regarded as all important; the more essential property of being
* See Cantor, Math. Annalen, vol. xxI, p. 576.






_~


40, 41]                   The Continuum                          51
perfect has only been explicitly formulated in the course of construction
of the modern arithmetical theory.
41. The term arithmetic continuum is used to denote the aggregate of
real numbers, because it is held that the system of numbers of this aggregate
is adequate for the complete analytical representation of what is known as
continuous magnitude. The theory of the arithmetic continuum has been
criticized on the ground that it is an attempt to find the continuous within
the domain of number, whereas number is essentially discrete. Such an
objection presupposes the existence of some independent conception of the
continuum, with which that of the aggregate of real numbers can be compared. At the time when the theory of the arithmetic continuum was
developed, the only conception of the continuum which was extant was that
of the continuum as given by intuition; but this, as we shall shew, is too
vague a conception to be fitted for an object of exact mathematical thought,
until its character as a pure intuitional datum has been modified by exact
definitions and axioms.  The discussions connected with arithmetization
have led to the construction of abstract theories* of measurable quantity;
and these all involve the use of some system of arithmetic, as providing the
necessary language for the description of the relations of magnitudes and
quantities.  It would thus appear to be highly probable that, whatever
abstract conception of the intuitional continuum of quantity and magnitude
may be developed, a parallel conception of the arithmetic continuum, though
not necessarily identical with the one which we have discussed, will be
required. To any such scheme of numbers, the same objection might be
raised as has been referred to above; but if the objection were a valid one,
the complete representation of continuous magnitudes by numbers would,
under any theory of such magnitudes, be impossible. It is clear that it
is only in connection with an exact abstract theory of magnitude, that
any question as to the adequacy of the continuum of real numbers for the
measurement of magnitudes can arise. For actual measurement of physical,
or of spatial, or temporal magnitudes, the rational numbers are sufficient;
such measurement being essentially of an approximate character only, the
degree of error depending upon the accuracy of the instruments employed.
The purely ordinal nature of the conception of the arithmetic continuum,
including the ordinal character of an interval, has been pointed out in the
course of the development of the theory. This will be further elucidated
in connection with the abstract theory of order-types to be discussed in
Chapter III.
* See 0. H. Holder, Die Axiome der Quantitdt mnd die Lehre vomn Mass, Leipziger Berichte,
vol. LIII (1901); also Veronese's work, Fondamenti di Geometria, 1891; and Bettazzi's work,
Teoria delle grandegze, 1890.
4-2




52


Number


[CH. I


THE CONTINUUM GIVEN BY INTUITION.
42. Before the development of analysis was made to rest upon a purely
arithmetical basis, it was usually considered that the field of operations was
the continuum given by our intuition of extensive magnitude, especially of
spatial or temporal magnitude, and of the motion of bodies through space.
The intuitive idea of continuous motion implies that in order that a body
may pass from one position A to another position B, it must pass through
every intermediate position in its path. An attempt to answer the question,
what is meant by every intermediate position, reveals the essential difficulties
of this conception, and gives rise to a demand for an exact theoretical treatment of continuous magnitude.
The implication contained in the idea of continuous motion, shews that,
between A and B, other positions A', B' exist, which the body must occupy
at definite times; that between A', B', other such positions exist, and so on.
The intuitive notion of the continuum, and that of continuous motion negate
the idea that such a process of subdivision can be conceived of as having a
definite termination. The view is prevalent that the intuitional notions of
continuity and of continuous motion are fundamental and sui generis; and
that they are incapable of being exhaustively described by a scheme of
specification of positions. Nevertheless, the aspect of the continuum as a
field of possible positions is the one which is accessible to Arithmetic Analysis,
and with which alone Mathematical Analysis is directly concerned. That
property of the intuitional continuum, which may be described as unlimited
divisibility, is the only one that is immediately available for use in Mathematical thought; and this property is not sufficient for the purposes in view,
until it has been supplemented by a system of axioms and definitions which
shall suffice to provide a complete and exact description of the possible
positions of points and other geometrical objects which can be determined in
space. Such a scheme constitutes an abstract theory of spatial magnitude.
The exact theory of magnitude was developed to a considerable extent
by Euclid; but not until recently, under the influence of the ideas of the
arithmetical theory, has it been perfected in a form which exhibits the exact
system of axioms and definitions necessary for a characterization of continuity, that is adequate for mathematical analysis. Besides the arithmetic
theory of number, there exists at the present time a theory of magnitude
which runs to a certain extent parallel with the former theory.  Some
mathematicians* still prefer to regard number as primarily representing the
ratio of two magnitudes; but they nevertheless to a large extent employ the
methods of arithmetical analysis.
" P. Du Bois Reymond in his Allgqemeine Functionentheorie strongly advocates the view that
linear magnitude forms the basis of the conception of Number. See also Stolz, Allgemeine
Arithmetik, where both views of Number are developed. See also G. Ascoli, Rend. Ist. Lomb. (2)
28 (1895).




42, 43]


The Continutum


53


THE STRAIGHT LINE AS A CONTINUUM.
43. Although it is no part of the plan of the present work to enter fully
into the general theory of Magnitude, it is necessary briefly to consider the
case of those magnitudes which are segments of a straight line, that straight
line which is the ideal object of geometry, and which is the ideal counterpart
of the physical straight line of perception.
The length of the segment between two points A, B, of a straight line,
is a particular case of a magnitude; and we shall take this conception as a
datum, subject to a set of axioms* relating to the notions of congruency,
and to the notions greater and less as applied to magnitudes.
We assume that any number of congruent segments OA, AB, BC,... can
be constructed on the straight line; and that any segment OA can be divided
into any number of segments which are all equal to one another.
Any segment OA may be taken as the unit of length, so that its
magnitude is represented by the number 1; its multiples OB, OC,... are
denoted by the numbers 2, 3,.... If each one of the segments OA, AB, BC,...
be divided into the same number q of equal parts, then, if P is a point of
division, OP is denoted by a fractional number p/q, where p is the number of
the sub-segments in OP. Thus when p, q are any positive integral numbers,
p/q represents a definite magnitude OP, the unit magnitude OA having been
fixed upon beforehand.
Further, the number p/q may also be regarded as representing the
position of the point P itself. In order to represent points of the straight
line on both sides of 0, the convention is made, that points on one side of
O shall be represented by positive numbers, and those on the other side by
negative numbers; thus if P is on the right of 0, and P' on the left of 0,
and if OP= OP', the point P' is represented by the number -p/q. The
length of any segment of the straight line, whose ends are points to which
rational numbers have been assigned in the manner explained above, is the
difference of the above two numbers. In this mannr, we have a correspondence established between the aggregate of rational numbers and an aggregate
of points on the straight line, the relation of order being conserved in the
correspondence, so that the two aggregates are similar.
The set of points, thus represented by rational numbers, we may speak
of as the rational points of the straight line; but it must be remembered
that a definite origin 0, and a definite unit of length OA, are supposed to
have been fixed upon beforehand; and if these be altered, the set of rational
points will in general be altered also.


* These axioms are discussed by 0. Holder, Leipziger Berichte, vol. LIII, 1901.




54


Number


[CH. I


It has been assumed as an axiom that, if PQ, is any segment of the
straight line, it may be divided into any number n, of equal parts: of these,
if P2Q2 be taken as one, the same axiom     asserts that PQ, may be
similarly divided into any number, n2, of equal parts, P3Q3 being one of the
parts; and that this process may be repeated an unlimited number of times.
The axiom is equivalent to an assumption that the straight line is capable
of unlimited divisibility; and this, being a characteristic property of the
intuitional linear continuum, must also hold for its ideal counterpart, the
straight line which we are here considering.
We proceed to assume as another axiom that, P,1Q, P2Q2, P3Q3... being
the segments constructed as above, there exists in the straight line one
point X, and one only, which separates all the points P1, P2, P3,... from all
the points Q1, Q2, Q3,.... If Y be any point other than X, then points
belonging to the sequence P1, P2, P,... and points belonging to the sequence
Q, Q2, Q3,... can be found which are both on the same side of Y.
The point X may be regarded as the limit of either sequence of points;
and the property corresponds to that property of the arithmetic continuum
which is expressed by saying that it is perfect.
In accordance with this axiom there is one single point on the straight
line which corresponds to any given real number; and this point, or the
magnitude of the corresponding segment, may be represented by the
real number.
This axiom has been stated by Dedekind, in a form' corresponding to his
definition of an irrational number:-that a section of the rational points, in
which they are divided into two classes, is made by a single point.
Another form of the axiom is that known as the Axiom of Archimedes*:that if AB, A'B' are any two segments of the straight line, of which AB is
the smaller one, an integer n can always be found such that n. AB > A'B'.
As in the case of the arithmetic continuum, this is equivalent to the negation
of the existence of infinitesimal segments of the straight line.
This axiom being assumed, there is a complete correspondence between
the points of the straight line and the aggregate of real numbers. Thus the
nature of the linear continuum, that is, so far as its possible parts, and the
possible positions in it, are concerned, is completely represented and described
by means of the arithmetic continuum, the axioms relating to the straight line
having been so chosen that this may be the case. It will be observed that
there is no real disparity between the rational points and the irrational
points of the straight line; a point, which with one origin and one unit
of length, is a rational point, may be an irrational point if another origin, or
another unit of length, be chosen.
* The importance of the Axiom of Archimedes in this connection was pointed out and
discussed by Stolz, Miath. Annalen, vols. xxII and xxxix.




43, 44]


The Con/tinummcnz


55


44. The mode which has been adopted above, of establishing a complete
correspondence between the aggregate of real numbers and the aggregate of
points in a straight line, though the most convenient mode, is not the only
possible one. All that is really necessary for the correspondence is that, in
accordance with some systematic scheme, the points in the straight line shall
be made to correspond with the numbers of the arithmetic continuum in
such a way that the relation of order is conserved in the correspondence.
It is not necessary that the difference of two numbers should represent the
length of the segment of the straight line which is terminated by the points
that correspond to the two numbers. The mode of correspondence given
above is however the simplest one, and will therefore be adopted for the
purpose of enabling us to use the language of geometry in analytical
discussion.
In the case of space of two or of three dimensions, it will be assumed
as axiomatic that one point of the space, and one only, corresponds to
each pair or triplet of real numbers which represent Cartesian coordinates.
This axiom may be considered as fundamental in the Cartesian system of
analytical geometry.
The disputable idea that the theory here explained necessarily implies
that a continuum is to be regarded as made up of points, which are
elements not possessing magnitude, has frequently been a stumbling-block
in the way of the acceptance of the view of the spatial continuum which
has been indicated above. It has been held that, if space is to be. regarded as made up of elements, these elements must themselves possess
spatial character; and this view has given rise to various theories of infinitesiL Als or of indivisibles, as components of spatial magnitude. The most
modern and complete theory of this kind has been developed by Veronese *,
and is based upon a denial of the principle of Archimedes which has been
already referred to. In Veronese's system, when a unit segment of a straight
line has been chosen, there exist segments which are too large, and others
that are too small, to be capable of representation by finite numbers; and these
segments are respectively infinite, and infinitesimal, relatively to the unit
segment chosen. Under this scheme, a section of the rational points, or a
section of the points represented by real numbers, is made, not by a single
point, but by an infinitesimal segment. Veronese has consequently introduced systems of infinite and of infinitesimal numbers, each of an unlimited
number of orders, for the measurement of segments which, relatively to a
given scale, are infinite or infinitesimal. From his point of view, the points
on a straight line which represent the real numbers form only a relative
continuum, i.e. one which is relative to the particular scale of measurement
* See his Fondamenti di Geometria, Pisa, 1894; a German translation by Schepp has been
published in Leipzig.




56


Number


[CH. I


employed; and he contemplates the conception of an absolute continuum, for
the representation of which his series of sets of infinite and infinitesimal
numbers are requisite. A segment, which in a given scale is finite, may be
infinitesimal, or infinite of any order, when measured relatively to another
scale.
The validity of Veronese's system has been criticized by Cantor and
others, on the ground that the definitions contained in it, relating to equality
and inequality, lead to contradiction; it is however unnecessary for our
purpose to enter into the controversy on this point. The straight line of
geometry is an ideal object of which any properties whatever may be
postulated, provided that they satisfy the conditions, (1) that they form a
valid scheme, i.e. one which does not lead to contradiction, and (2) that the
object defined is such that it is not in contradiction with empirical straightness and linearity. There is no a priori objection to the existence of two or
more such adequate conceptual systems, each self-consistent, even if they be
incompatible with one another; but of such rival schemes the simplest will
naturally be chosen for actual use. Assuming then the possibility of setting
up a valid non-Archimedean system for the straight line, still the simpler
system, in which the principle of Archimedes is assumed, is to be preferred,
because it gives a simpler conception of the nature of the straight line, and
is adequate for the purposes for which it was devised.  The case of the
non-Euclidean systems of geometry is an instance of the existence of valid
geometrical schemes divergent from one another, which nevertheless all afford
a sufficient representation of physical space-percepts.
An answer to the difficult question, in what sense the straight line or a
space of two or of three dimensions, admits of being regarded as an aggregate
of points, can only be discussed after a full treatment of the nature and
properties of infinite aggregates has been developed. The discussions in
Chapters II. and III. of infinite aggregates, and especially of the notion of
the power or cardinal number of such an aggregate, will throw light upon
this subject.




CHAPTER II.


THEORY OF SETS OF POINTS.
45. AN aggregate of real numbers, each element of which consists of
a single real number, is defined by any prescribed set of rules or specifications
which are of such a nature that, when any real number whatever is arbitrarily
assigned, they theoretically suffice to determine whether such real number
does or does not belong to the aggregate. The difficulty of regarding
an aggregate, so defined, as a definite object, is bound up with the difficulties
connected with the notion of the linear continuum, i.e. the aggregate of
all real numbers, out of which the defined aggregate is to be obtained by
a process of selection which, except in the case of a finite aggregate, can
never be actually carried out in its entirety, but which is determined by a
rule or set of rules. The precise scope of the definition will be rendered
clearer by the consideration of various classes of actually defined aggregates
which will be considered in the present Chapter; moreover, the theoretical
difficulties of the notion of such an aggregate, in general, will be in some
measure elucidated by the discussions in the present and the following
Chapters, of the notion of the power, or cardinal number, of an aggregate.
In accordance with the principle explained in ~ 43, each number of a
given aggregate may be represented by a single point on a fixed straight
line; thus, to an aggregate of numbers, there corresponds an aggregate of
points on the straight line. An aggregate of single numbers, or of their
equivalent points, we shall speak of as a linear set of points.
The theory of linear sets of points, of which the present Chapter contains
an account, arose historically from the discussion of questions connected with
the theory of Fourier's series and of the functions which can be represented
by such series. A consideration of the properties and peculiarities of the sets
of points at which infinities or other discontinuities of such functions exist,
led to a study of the properties of linear sets in general, and to the
development by G. Cantor, P. Du Bois Reymond, Bendixson, Harnack, and
others, of a general theory which has lately received wide applications both
in Analysis and in Geometry.
Corresponding to the theory of linear sets of points, there exist theories
of plane, solid, or n-dimensional sets of points.  A set of points in n




58


Sets of Points


CCH. II


dimensions is an aggregate each element of which is specified by n real
numbers. The theory of such sets proceeds on lines similar to that of linear
sets; indeed most of the investigations in the latter theory are capable of
extension, with slight modification, to the more general cases. For the
sake of brevity, we shall in general confine our attention to linear sets;
some indications will, however, be given of the mode in which the definitions
and properties which arise in the theory of linear sets, may be extended
to the case of plane or solid sets; and a few properties peculiar to non-linear
sets will be given.
The whole theory is fundamentally arithmetical; the geometrical representation and nomenclature being a matter of convenience, not of necessity.
THE UPPER AND LOWER BOUNDARIES OF A SET OF POINTS.
46. Let a set of points be such that every point of the set lies upon
a straight line, the position of each point being determined by its distance
from a fixed origin upon the straight line, in the manner explained in ~ 43.
If a point /8 exists, such that no number of the set is greater than fi, the
set is said to be bounded on the right. In this case it will be shewn that
there is a definite point b, such that no point of the set is on the right of b,
and such that either b is itself a point of the set, or else points of the set
are within the interval (b -e, b) however small the positive number e may
be taken to be.
When b is a point of the set, there may or may not be other points of the
set in every interval (b - e, b). This point b is said to be the upper limit of
the set, when points of the set lie within every interval (b - e, b). In any
case, when b is a point of the set, it is said to be the upper extreme point
of the set. The term upper boundary may be applied to the point b,
whether it be the upper limit or only the upper extreme point.
In case b is both the upper limit, and the upper extreme point, of the
set, the upper limit is said to be attained; and b is then called the maximum point of the set.
To prove the existence*, under the condition stated, of an upper boundary,
as above defined, it may be observed that all the numbers of the continuum of real numbers can be divided into two classes, one of which
contains every number which is greater than all the numbers of the set, and
the other of which contains every number which either belongs to the set or
is less than some or all of the numbers of the set. The section thus specified
defines a number b which is the upper boundary of the set.
* The existence of upper and lower boundaries was proved by Weierstrass, in his lectures.
See also Bolzano, Abh. d. Bihmischen Gesellsch. d. Wiss., vol. v, Prag, 1817.




45-47]


Bounded Sets


59


In a similar manner, it may be shewn that, if the set is bounded on the
left, i.e. if a point can be found such that all the points of the set are on
the right of such point, then a point a exists, which is such that no points
of the set are on the left of a, and such that either a is a point of the set,
or else points of the set are within every interval (a, a + e), where e is an
arbitrary positive number. Both conditions may be satisfied simultaneously.
In case points of the set lie within every interval (a, a + e), then a is
called tIeR lower limit of the set; and the lower limit is said to be attained
if a be itself a point of the set. In any case in which a is a point of the set,
it is then said to be the lower extrem2e point of the set. The term lower
bouncdary may in all cases be applied to a.
A set of points which has both an upper and a lower boundary is said to
be a bounded set.
47. If no point b exists, which is either the upper limit, or the upper
4-tTemne point, of the set, then the set is said to be unbounded on the right;
dtiit is said that the upper limit of the set is + O; the two statements
Wtfg regarded as tautological. Similarly, if no lower limit nor lower
extOreme point a exists, the set is said to be unbounded on the left; or it
iF)id that the lower limit is - x.;, he symbols + AC, - o do not really represent numbers; they must
be atken to represent what is sometimes spoken of as the improperly infinite,
'.e. rthe mere absence of an upper or a lower boundalry respectively.  In order,
however, to avoid circumlocution in the statement of theorems concerning
sets, it is usually convenient to speak of + c, - o, used in the above sense,
if they were numbers which correspond to upper and lower limits
i+ectively.
In the present Chapter, it will in general be assumed that the sets treated
of are bounded; and the interval (a, b) will be said to be the interval in which
the set exists.  This restriction is not so great a one as mnight at first sight
appear; for an unbounded set can be placed into correspondence with a bounded
one, in such a manner that the relative order of any two points in the one
set is the same as that of' the corresponding points in the other set. If
= i, where the radical is taken to have always the positive sign, then
to a point x, in the unlimited line (- c, + c ), there corresponds a point x',
in the limited line (-1, + 1); and also x' - x.', according as a,  S x.,. The
same object might have been attained by using the transformation
x   2 tan-'x.
7r
There is no real loss of generality in considering only such sets as lie




60


Sets of Points


[OH. II


in a given interval, say (0, 1); for the relation x' -  establishes a complete
correspondence between sets in the interval (a, /3) and sets in the interval
(0, 1), the relative order of points being preserved in the correspondence.
The points of the interval (a, /) may be made to correspond in order with
the points of the interval (0, 1), in such a manner that an arbitrarily chosen
point y within (a, /), corresponds to an arbitrarily chosen point within (0, 1);
for example the point 1. This correspondence can be effected by t1: transformation
ac'     x-a 7y-38
x'-1     x- /3'y-a
LIMITING POINT OF A SET OF INTERVALS.
48. Let (al, bi), (a2, b2),... (an, be)... be an unending sequence of interf:^J]
which are such that any one of them (a,, bn) lies entirely in the precerig:
one (an-,, bn_-), the two having at most one end-point common; gtit
an  a,,_,, bn - bnI; moreover, suppose that the lengths b - a,, b2 -ftV9
bn- an,.. form a sequence which converges to zero, the condition for *WAli
is that, corresponding to any arbitrarily small e, n can be so chosen, hat
bm -am, for all values of m which are _ n, is < e.  It will be seen] that
in accordance with the axioms explained in ~ 43, there exists one poin ~a
one only which is in every interval of the sequence. This point may be caleT
the limiting point of the sequence of intervals.
Each of the aggregates (a, a2, a3,... a,...), (b, b,... b,,,...) being'tot
vergent, defines a number; and in fact, in virtue of the definition of eqv.i
in ~ 25, they define the same number x. This number x is not less thap aand not greater than bn, whatever n may be; the point x therefore:f i
in all the intervals, and is the limiting point whose existence was 'tiio
remarked. If    b    y be any number greater (or less) than x, we can find n so
great that bn - x < y - x, if y > x; or that x - as < x - y, if x > y: thus yfa S
not lie in (a,,, bn). Hence there is only one point which satisfies the prescribed conditions.
If for every n, from and after some fixed value, the inequalities an > an_,,
b,, < b,_ both hold, then the limiting point x is in the interior of all the
intervals of the sequence. If, from and after some fixed value of n, say n,
we have a = an,_, bn < bi,, the limiting point x coincides with the common
end-points an,-i, an, a,+1,....




47-49]         Limiting Points and Derivatives                   61
THE LIMITING POINTS AND THE DERIVATIVES OF A LINEAR SET.
49. If a point x be taken in the interval (a, b), an interval (x-e,,
x + 6e) which lies entirely in (a, b) is called a neighbourhood of the point x;
and this neighbourhood may be made as small as we please by proper choice
of e1 and e2. An interval (x, x + 2) is called a neighbourhood of x on the
right, and (x-e, x) is called a neighbourhood of x on the left.  The
end-points a and b can only have neighbourhoods on the right and the left
respectively.
If a linear set of points not finite in number (denoted by G) is in the
interval (a, b), then a point P, in whose arbitrarily small neighbourhood there
exists at least one point of G not identical with P, is called a limiting point
of the set G, whether P belongs to G or not.
The fundamental theorem will now be proved that every set of an infinite
number of points G, in an interval (a, b), possesses at least one limiting point.
Divide (a, b) into in equal parts; then in one at least of these, say
(a1, b1), there is an infinite number of points of G; and if this is the case in
more than one of the parts, we may take any one of these for (a,, b1).
Divide (a,, b,) into m equal parts; then there must be one of these parts
at least, say (a2, b,), which contains an infinite number of points of G.
Proceeding in this manner, we obtain a sequence of intervals (al, b,),
(a,, b2)... (a,,, b)..., of lengths -(b- a),  (b- a),... 1 (b-a),... in each
of which there is an infinite number of points of G. In accordance with the
theorem of ~ 46, there exists one point x which is in all the intervals (a,,, b,).
Take any arbitrarily small neighbourhood of x, say (x - 1, x +6); then
if n be chosen so large that - (b - a) is less than the smaller of the numbers
61, 62, the interval (a,, bn) lies entirely within (x - 1, x + 2e). Hence in the
arbitrarily small neighbourhood of x, there is an infinite number of points
of G; therefore x is a limiting point of G.,
It has thus been shewn that G has at least one limiting point. It may
have a finite number, or an indefinitely great number, of limiting points.
It should be observed that a limiting point of G may or may not itself
be a point of G. If either boundary of the set be not a point of G, then it
is certainly a limiting point of G; it may however be both.
A limiting point P of a set G is a limiting point on both sides, if an
indefinitely great number of points of G lie in every neighbourhood of P on
the right, and also in every neighbourhood of P on the left.  Otherwise
P is a limiting point of G on one side only.




62


Sets of Points


[CH. I1


50. In the case of a set for which either the upper boundary or the lower
boundary is absent, or both are absent, we may use either method given
in ~ 47, of making the set correspond with a bounded set in the interval
(-1, + 1).  To any definite interval in (- oo, + oo ), there corresponds a
definite interval in (-1, + 1), neither end-point of which is at - 1 or 1.
To a limiting point interior to (- 1, + 1), there corresponds a limiting point
in (- oo, + oo ). For if x' be such a limiting point of the set in (- 1, + 1),
there are in any neighbourhood of x' an infinite number of points of the set;
and to this neighbourhood there corresponds a neighbourhood of the corresponding point x of the unbounded set. Thus the point x is such that, in
any neighbourhood of it, there are an infinite number of points of the set,
so that x is a limiting point.  The only case in which the unbounded set has
no limiting point, is when the corresponding bounded set has for its sole
limiting points the end-points -1, +1, or one only of these; and in this
case we may say that oo, - o, or one of these, is the improper limiting
point of the unbounded set. The properties of an unbounded set in relation to its limiting points are thus not essentially different from those of
a bounded set.
51. Returning to the case of a set G in an interval (a, b), we observe
that the limiting points of G form a set of points which may be finite or
infinite; this set is called the derived set*, or first derivative of G, and may
be denoted by G'. In case the set G' contains an infinite number of points,
it possesses itself a derivative set G", which is called the second derivative
of G. If we proceed in this manner, we may obtain a series
G', G", G"',... G(n)
of derivatives of G. If the nth derivative G(?) contains a finite number
only of points, then these have no limiting point, and we may say that
G('+l) = 0. It may however happen that, however large the integer n may
be, the derivative G() contains an indefinitely great number of points; and
thus a next derivative exists.
A set G which possesses only a finite number of derivatives is said to be
of the first species.
In this case, if G(8) contains only a finite number of points, the set G is
said to be of ordert s. Thus, for example, a set of the first species and order
zero, contains only a finite number of points; and a set of the first species
and order 1, has a first derivative which contains only a finite number of
* The notion of the derivative of a set was introduced by Cantor, Math. Annalen, vol. v (1872),
p. 128. Du Bois Reymond contemplated the existence of limiting points of various orders,
Crelle's Journal, vol. LXXIX (1874), p. 30; in Math. Annalen, vol. xvi, p. 128, Du Bois Reymond
defined a limiting point of infinite order.
t Cantor, Math. Annalen, vol. v, p. 129.




50, 51]


Limiting Points and Derivatives


63


points. It will be observed that the order of each derivative of G is less
by unity than that of the one which precedes it.
A set G which possesses an indefinite number of derivatives is said to be of
the second species.
As an example, we may consider the set of rational numbers in the
interval (0, 1). The first derivative of this set contains every real number
in (0, 1), and all subsequent derivatives are identical with the first.
The theorem, that every non-finite linear set of points possesses a
limiting point, is a particular case of the theorem that every non-finite set
in a finite portion of an n-dimensional continuum has a limiting point.
In this case we may take the neighbourhood of a point to be either a
"sphere" of arbitrarily small radius p, or a "rectangular cell" with sides
parallel to the coordinate axes, and the point at the centre. The space
in which the set of points exists can be divided into a finite number of
overlapping "spherical" or of "square" portions, and the argument then
proceeds as in the case of a linear set.
EXAMPLES.
1. Let      *                           ooo I
2' 3' 4  -   )
We see that G' consists of the single point 0, which does not belong to G; thus G is of the
first species and of order 1.
2. Lett the points of G be given by
1   1        1 1
3s1 582 7s3 1184
where sl, s2, S3, 84 each have all positive integral values. Here G' consists of the four sets
of points given by
1 1  51          1   1    1 1    1        1    1 
-    -   7'      ~         3s1+- +-      +s-S1 5st ~S~) 3s+  5 S2 1 1s4 3+  7'8 IlS* 5's  7sa 1 71'4
and of the six sets of points
I   1    I      1   1    I   1   1 1    1   1
3S1 5S2 3s1 7S3) 31  1s 14 5s2 7S  58 11 7 S 3 1184
and of the four. sets of points
1   1   1    1
3s1)  5S2  7S38  I' S4
together with the single point 0. G" consists of the last ten of these sets, and of the
point 0. The second derivative G'" consists of the last four sets, and of the point O; G""
consists of the point 0 only. The set G is of the first species and of the fourth order.
* Cantor, Math. Annalen, vol. v (1872).
t Ascoli, Ann. di Mat., Series In, vol. vi, p. 56, 1875.




64


Sets of Points


[CH. II


3. Let* the points of G be given by
1   1       1
-+ -    + -...+ --
a1  a2      a,
where n is a fixed number, and each of the numbers a,, ac,... an takes every positive
integral value. In this case G is of order n.
4. The zerost of the function sin- form a set similar to that in Example 1.
The zerost of th      e nction sin    form  a set of the second order, those of
(sin      /
sin  /
5. Let+ the points of G be given by
1      1              1
2m-  2m, +'"?2   2mi +m2+...- +m, )
where ml, m2,... m, have all positive integral values, including zero, and n is a fixed
integer. It can be seen that G(n) consists of the point zero only.
THE DISTRIBUTION OF POINTS OF A SET IN THE INTERVAL.
52.  If G1, G2, G3... Gn denote a number of sets of points, a set which
contains all points which belong to any one or more of the given sets is
called their common measure, and is denoted by M(G&, G2,... GO).        That
set which contains all those points which belong to every one of the given
sets is called their greatest common     divisor, and   may be    denoted   by
D(GI, G2,... Gn)
A set of points is said to be an isolated set, when no point of the set is a
limiting point.  Thus if G be such a set, we have D (G, G')= 0.
If from any set we remove those points which also belong to its derivative,
the remainder forms an isolated set; thus G-D (G, G') forms an isolated set.
Any set may be regarded as the sum of an isolated set and of a set which
is a divisor of the derivative.
A  set, all of whose limiting points belong to the set itself, is said to be
closed. Thus in a closed set G, the derivative G' is a divisor of G.
A set which is such that every point of the set is a limiting point is said to
be dense-in-itself.
For a set G, dense-in-itself, G is a divisor of the derivative G'.     The
rational numbers in (0, 1), form an example of a set which is dense-in-itself.
* H. J. S. Smith, Proc. Lond. Math. Soc., vol. vI, p. 145, 1875.
t P. Du Bois Reymond, Journ. f. Math., vol. LXXIX, p. 36.
+ Mittag-Leffler, Acta Math., vol. iv, p. 58.




51, 52]


Distribution of Points in an Interval


65


A set G which is both closed and dense-in-itself is said to be perfect*.
Thus a perfect set G is identical with its derivative. It follows that every
perfect set is of the second species.
By some writerst the term perfect is applied to sets which, in accordance
with the terminology of Cantor here adopted, are only closed, without
necessarily being dense in themselves; what we call a perfect set is then
spoken of as an absolutely++ perfect set.
If in an interval (a, b), a smaller one (a', b') such that a < a', b - b' is
taken, then the latter may be called a sub-interval of the former interval.
If in the interval (a, b), in which a set of points G is contained, no subinterval whatever, however small, can be found which does not contain points
of G, then the set G is said to be everywhere-dense~, or simply, dense in the
interval (a, b).
By Du Bois Reymond l, the term pantachisch was used with the same
meaning as everywhere-dense.
A set which is everywhere-dense is also dense-in-itself, but the converse
does not hold.
It will be seen that, if G is everywhere-dense in (a, b), every sub-interval
of (a, b) must contain an indefinitely great number of points of G. The
derivative G' of G must contain every point in the interval (a, b), since the
arbitrarily small neighbourhood of any point whatever of (a, b) contains an
indefinitely great number of points of G; and therefore every point must be
a limiting point.  This property, that G' contains every point of (a, b), may
be used as the definition~ of an everywhere-dense set.
If in every sub-interval (a', b') of the interval in which G exists, a part
(a', b") can be found which contains no points of G, then G is said to be
nowhere-dense, or non-dense in (a, b).
An example of a set which is everywhere-dense in its interval is the set of
rational numbers in the interval (0, 1). A set which is everywhere-dense in
its interval, or in any sub-interval, is necessarily of the second species; a set
of the first species is nowhere-dense in its interval.
A set which is in no sub-interval dense-in-itself, is said to be separated.
If in an interval (a, b), an indefinitely great number of sub-intervals, which
may or may not overlap, be taken, and no sub-interval (a, A) of (a, b) can
be found which is wholly external to all the given sutb-intervals, then the set
of sub-intervals is said to be everywhere-dense in (a, b).
* Cantor, Math. Annalen, vol. xxI.
t For example Jordan, see Cours d'Analyse, vol. I, p. 19.
+ Borel, Lemons sur la theorie desfonctions, p. 36.
~ Cantor, Math. Annalen, vol. xv, p. 2.
II Math. Annalen, vol. xv, p. 287.
1 Baire, Annali d. Mat., Series 3, vol. II, p. 29.
H.                                                                5




66


Sets of Points


[CH. II


53. The following fundamental theorem will now be proved:
All the derivatives G', G", G"',... G(),... of a given set G are closed sets,
and each of these derivatives, after the first, consists only of points belonging
to the preceding one, and therefore to G'.
If a point P of G(), where n Z 2, existed, which did not belong to G', then
a neighbourhood of P could be found, so small as to contain only a finite
number of points of G, or no such points; and this neighbourhood would
therefore contain no points of G', and therefore none of G", G"',... G;
which would be contrary to the hypothesis that P      belongs to  Gl.
Therefore every point of G() (n.  2) belongs to G'; and consequently G' is a
closed set. If we take G(n-2) to be the original set, it follows from the
above that every point of G('1), the second derivative, belongs to G('-') the
first derivative. We have thus shewn that G~ is the greatest common
divisor of G', G",... G(; that is
G( = D (G',  ",... G()).
The derivative G', of a set G which is dense-in-itself, is perfect.
For G' is closed, and every point of G belongs to G'; thus G' contains no
point which is not a limiting point of the set G'. Therefore G' is densein-itself; and hence it is perfect.
A set G1, which consists of some, but not all, of the points of G, is said to be
a component of G.
If the component G? of G, be such that every point of G is a limiting point
of GI, then the component G, is said to be everywhere-dense relatively to, or
simply, dense in G.
In the case in which G is the continuum (a, b), this definition agrees with
that which has been given for a set which is everywhere-dense in the interval
in which it is contained.
ENUMERABLE AGGREGATES.
54. An aggregate which contains an indefinitely great number of elements
is said to be enumerable *, or countable (abzalbar, denombrable), when the aggregate is such that a (1, 1) correspondence can be established between the elements
and the set of integral numbers 1, 2, 3,....
An aggregate of objects is therefore enumerable if the objects can be
arranged in a series which has a first term, and in which any assigned object
belonging to the aggregate has a definite place assigned by a definite ordinal
number n. Thus the elements of an enumerable aggregate can be represented by a series of terms
U1, U2,.. U. n....


* Cantor, Crelle's Journal, vol. LXXVII (1873), p. 258,




53, 54]


Enumerable Aggregates


67


It follows from this definition that the elements of two enumerable
aggregates are such that a (1, 1) correspondence can be established between
them.
If a new aggregate be formed by selecting elements from those which
belong to an enumerable aggregate, an indefinitely great number of such
elements being taken, then the new aggregate is also enumerable. For such
an aggregate selected from u1, u2,... Un,... is u,., Us, Ut... (r < s < t...), which
satisfies the conditions of having a first term, and of having each element
of the aggregate in a definite place in the series. It thus appears, that a
(1, 1) correspondence can be established between an enumerable aggregate
and one which is a part of that aggregate, provided this part be not finite.
This is the characteristic property which distinguishes an aggregate containing an indefinitely great number of elements from one containing only
a finite number of elements. For example, a (1, 1) correspondence exists
between all the integral numbers and all the odd numbers, or between all the
integral numbers and all the prime numbers.
If a finite number of enumerable aggregates be taken, or even if the
number of such aggregates be indefinitely great, but enumerable, then the new
aggregate formed by the whole is itself enumerable*.
We may denote such a composite aggregate by the letters
U11, ~ U12,  t13,  ***..  * U
lIt2,   1223,... U12n,...
2131,  32....................................
and we shall shew that the double series so formed represents an enumerable
aggregate.  To see this, it is sufficient to write the series in the form
U13    t221 X  ^31......................
21, -1,)  t2, 2 —2,   U3, —3S)..  n-1,1,,,.,,,o........................,..
where the sum of the indices is the same for all the terms which are written
in one horizontal line. It is now clear that each number 1pq has a definite
place in a series in which u,, has the first place; the double series is therefore
enumerable.


* Cantor, Crelle's Journal, vol. LXXXIv (1875).


5-2




68


Sets of Points


[CH. II


An important particular case of the above theorem is the following
theorem:
The aggregate of all the rational numbers is enumerable.
A rational number p/q may be denoted by up,q: therefore the aggregate is
enumerable. It makes no difference that any particular number p/q occurs
an indefinite number of times as Urp,,.q; since, if all such terms except those
for which r= 1, and p/q is in its lowest terms, be removed, the aggregate left
is still enumerable.
55. A more general theorem has also been established by Cantor*.
An algebraical number is one which is a root of an algebraical equation in
which the coefficients are all rational numbers, so that the coefficients may
without loss of generality be taken to be integers. Cantor's theorem is, that
all the algebraical numbers form an enumerable aggregate.
To prove this theorem, let
pon + p1xn-1 +... + pn = 0
be an equation in which Po, Pi... Pn are all positive or negative integers;
and let
Ipol + |p    +| p + pn + n = N;
then N is a positive integer which may be called the rank of the equation.
It is clear that there are only a finite number of equations of any given
rank, these equations having only a finite number of roots. If then we let
N= 3, 4, 5,... successively, we can arrange all the algebraical numbers in a
simple series; and thus they form an enumerable aggregate. The aggregate
which is formed of all the real algebraical numbers is also itself enumerable.
A number which is not an algebraical number is said to be transcendental. The existence of transcendental numbers was first establishedt by
Liouville, who shewed how examples of such numbers could be formed. No
general criterion is known by which it can be decided whether a number,
defined by a given analytical procedure, is algebraical or transcendental.
The first case in which such a number, well known in Analysis, was shewn to
be transcendental was that of the number e, the base of the natural system of
logarithms; and the first proof that e is transcendental was given by Hermite.
The next case in which a number defined analytically was shewn to be
transcendental was that of the number 7r. The first demonstration of this
important fact is due to Lindemann+,,who proved the more general theorem
that, if ex = y, the two numbers x, y cannot both be algebraical, except in the
case x = 0, y = 1. It follows that the natural logarithms of all algebraical
* Crelle's Journal, vol. LXXVII.
t Liouville's Journal, vol. xvI, 1851.
+ See MIath. Annalen, vol. xx.




54-56]


Enumerable Aggregates


69


numbers are transcendental, as also all numbers of which the natural
logarithms are algebraical.
56. The following fundamental theorem will now be established*:The aggregate which consists of the continuum of numbers in a given
interval is not enumerable.
Suppose 0o, o21, o... denote the numbers in an enumerable aggregate; it
will then be shewn that, between any two numbers a, / as near as we please,
a number occurs which does not belong to the enumerable aggregate. It will
then follow that in the given interval there is an unlimited number of
points which do not belong to the enumerable aggregate, and thus that the
latter cannot contain all the points of the continuum.  It the enumerable set
of points is not everywhere-dense in (a, /3), then smaller sub-intervals inside
(a, /) can be taken which contain no points of the aggregate; and thus we have
only to consider the case in which the given aggregate is everywhere-dense in
(a, /3). Let w)K be the first of the points ox, o,,... which lies within (a, /3), and
Wz2 be the next of these points which lies within (a, /), so that Ke < Kc. Let a'
be the smaller, and /' the greater of the numbers oK, oK,, then a < a' <  ' < /3,
and KI < c2; and if /a < K2, then w, does not lie within the interval (a', /3').
Considering this latter interval, let eO), oK4 be the first two of the numbers
of the enumerable aggregate which lie within (a', /f'), and let a" be the
smaller and /3" the greater of these; then a' < a" < /" < /', and Kc, < Kc, < c4.
Proceeding in this manner we obtain a whole series of sub-intervals each
one of which is entirely within the preceding one; thus (a(), /(v)) lies within
(a(V-1), /(-1)); and if /u Kco,, then.o, does not lie within (a(v), /3)); also
CI < IC2 < ~C3... < /C2v —2 < KC2V-1 < I 2v,
and 2v c< 2v; and thus We lies outside (a(v), /3()). Since the numbers a', a",
a"'... are in ascending order, and all lie within (a, /3), they have a limit A;
similarly /3', /8", f"'... have a limit B; and a(v) < A - B < (v). If A < B,
then since all the numbers ov are outside the interval (A, B), the given
aggregate is not everywhere-dense in (a, /); which is contrary to hypothesis.
Hence we have A = B, and the number A or B, is a number which does not
occur in the aggregate o,, eo2,...; which was what we had to prove.
It will be observed that the point of the foregoing proof consists in the
fact, that an everywhere-dense enumerable aggregate necessarily has limiting
points which do not belong to the aggregate.
A second prooft that the continuum is not enumerable is the following:Without loss of generality, the interval may be taken to be (0, 1). Suppose
it to be possible to arrange all the numbers in this interval in order, so that
there is a first, a second, a third and so on; and so that every number occurs
* Cantor, Crelle's Journal, vol. LXXVII.
t Jahresbericht der deutschen Math. Vereiniy. vol. i, p. 77.




70


Sets of Points


[CH. II


somewhere in the arrangement. Let the numbers, in order, be exhibited as
decimals
'Pll2  2 Pl3......
'P21 p22 p23.....
'P1 PS2 P33.....
where each p stands for a digit 0, 1, 2,... 9, and numbers, in which the digits,
from and after some fixed place, are all 9, are excluded; then if a number can
be defined which does not occur in the above series, a contradiction will
have been shewn to be involved in the supposition that all the numbers- can
be exhibited in the above manner. Now this can be done; for the number
'(pn) (P22) (p33)... where (p) denotes any digit except p, say p + 1 or 0,
according as p < 9, or p = 9, differs in at least one place of the decimal, from
every number in the above set; and the contradiction is thus established.
THE POWER OF AN AGGREGATE.
57. A notion of fundamental importance in the theory of aggregates is
that of the power of an aggregate. This notion will be considered more
generally and fully in the next Chapter, where it will be shewn that the
power of an aggregate is the generalization of the notion contained in the
cardinal number of a finite aggregate. At present, an account of the notion
of the power of an aggregate will be given, so far as it is necessary for the
application to the case of sets of points.
Two aggregates of objects are said to have the same power, or cardinal
number, when a (1, 1) correspondence can be established between them, so that
each element of either of the aggregates corresponds to one single element of
the other.
Finite aggregates have the same power when they consist of the same
number of elements, i.e. when they have the same cardinal number. Of
aggregates which are not finite we consider first enumerable aggregates.
Every enumerable aggregate has the power of the aggregate of integral
numbers; and this we may denote by a. It has been shewn above that, if
from an aggregate of power a any elements be removed, then the remaining
aggregate, provided it contains a non-finite number of elements, has still the
same power a. It has further been shewn that the composite aggregate
formed of a finite, or enumerable, number of enumerable aggregates has the
same power a. It follows, as an interesting case, that the set of all those
points of an n-dimensional space whose coordinates are rational numbers has




56-58]            The Power of an Aggregate                       71
the power a of the set of integral numbers, or of the rational numbers in
a given linear interval.
It is easily shewn that the power of the set of all the points in an interval
(a, b) is the same as that in any other finite interval, say (0, 1); for --   = x
establishes a (1, 1) correspondence between the points x of (a, b) and the
points x' of (0, 1). Again the relation      - =+ x', establishes a (1, 1)
NX2 + h2
correspondence between all real numbers, and those in the interval (-1, 1);
and thus the power of all real numbers is the same as that of all those in a
finite interval. This power is called the power of the continuum, and may be
denoted by c.
As regards unenumerable aggregates in general, it can be shewn that the
power of such an aggregate is unaltered by removing from the aggregate
any elements which form an enumerable aggregate. Let A denote the given
aggregate, and a the enumerable aggregate which is removed; and let B denote
the remaining aggregate, which cannot be enumerable, for otherwise (a, B), or
A, would be so also. From B, suppose an enumerable aggregate a' to be
removed, leaving the aggregate C, thus A = (a, a', C), B = (a', C). Now (a, a')
and a', being both enumerable, have the same power; and a (1, 1) correspondence therefore exists between their elements; and since A and B have the
aggregate C in common, it therefore follows that A and B have the same
power. As an example of this theorem, we see that the set of irrational points
in a given interval has the power c of the set of all numbers in the interval.
Again the set of transcendental numbers in a given interval has the power c
of the continuum; whereas the set of algebraical numbers in the same interval
has the power a.
The known infinite sets of points defined in accordance with the methods
usual in the theory of sets of points, in a line or in a continuum of any
number of dimensions, have either the power a or the power c; but it has not
yet been established that every possible set of points has one of these two
powers. Other aggregates have been contemplated which have a power
higher than c; these will be referred to later, in dealing with the theory
of functions.
58. The n-dimensional continuum has the power c of the one-dimensional
continuum*.
To prove this theorem, we use the fact that any irrational proper fraction
can be exhibited as an infinite continued fraction
11      1
an  - c+  +.... +al +...,
* Cantor, Crelle's Journal, vol. LXXXIV.




72                         Sets of Points                      [uH. II
where aC, a2, ai,... are determinate integers for any given value of x. Let
1o   1     1
al + cfn+l + -a2l+] +...,
I    I     1
X2 
a2 + ~a+2 + a2+2 2+...,..............................
1    1   1
ab + a2n +  34 +...,
thus, corresponding to any value of x, a set of irrational numbers x,, x2,.. xn
is uniquely determined, and conversely to any set of irrational numbers
x,, x2,... x,, a value of x is uniquely determined.
It has thus been shewn that the irrational points of the linear continuum
(0, 1) correspond uniquely to those points of the n-dimensional continuum in
which each coordinate is in the interval (0, 1), and is irrational. It has been
shewn in ~ 57, that the set of irrational values of xD, in the interval (0, 1) has
the same power as the set of all the numbers in this interval. Since this holds
also for x2, x3,... x,,, it follows that a (1, 1) correspondence can be established
between that set of points in the n-dimensional continuum, for which x1,
x2,... x, all have irrational values, and the set in which x,, x2,... xn have all
values rational or irrational; thus these sets have the same power. Hence
the set of all points of the n-dimensional continuum, in which each coordinate
is in the interval (0, 1), has the same power as the set of all points in the
linear interval (0, 1). It has thus been shewn that the n-dimensional continuum has the same power c as that of one dimension.
THE ARITHMETIC CONTINUUM.
59. The arithmetic continuum having been obtained by adjoining to
the set of rational numbers the set of all their limiting points, the question
arises how far it is legitimate to consider the complete set so obtained as
constituting a single object determined by means of the elements of which
it is composed. A finite set of numbers, or points, constitutes a single object
determined by means of its parts, in the sense, that those parts can be
exhaustively exhibited by means of a finite number of specifications representable by a finite number of symbols. An enumerable set of numbers,
or of points, in particular the set of rational numbers, is not determinate in
the sense that the elements of the set can be exhaustively exhibited; but
it is determinate in the sense that a table can be formed in which each
particular number of the set occupies a determinate place; and each particular number can be represented by means of a finite number of symbols.
Such a set may be regarded as an aggregate, or single object, in the same
sense in which the natural numbers 1, 2, 3,... may be regarded as forming




58, 59]           The Arithmetic Continuum                      73
an aggregate. When we come, however, to the case of the continuum, or
aggregate of all real numbers, the fact that this aggregate is unenumerable
introduces a new element into the question of the legitimacy of considering
the set of these numbers as forming a determinate whole, or as constituting
a single object of thought. The set of real numbers cannot be tabulated in
such a manner that no number fails to occur at some definite place in the
table. No set of rules or specifications can be given which suffice to determine
successively all the numbers of the set, and no finite set of symbols can
exhaustively exhibit the numbers. The only sense in which the numbers of
the set are determinate is that each such number is the limit of a convergent
sequence of numbers, taken from the unending table formed by the rational
numbers. It may fairly be doubted whether such a negative specification
of elements amounts to a valid synthetical definition of a determinate
aggregate; this point will however be further discussed in Chapter III., in
connection with the general theory of aggregates. It will there be shewn
that the arithmetic continuum has an order-type possessing definite characteristics which, in their totality, uniquely characterize it. This expresses
the only kind of unity which can appertain to the continuum, considered
as an arithmetic construction. If it be held that we possess an independent
knowledge of the existence of the geometrical continuum, derived from our
intuition of space, we may regard the function of the set of real numbers
to consist, not in a synthetical formation of the concept of the continuum,
but inversely in an analysis of the contents of the continuum. It is difficult
to see how precision can be introduced into the intuitional notion of the
continuum apart from some theory relating either to points or to infinitesimals; and the language employed in such a theory must be of a symbolical
character amounting to the use of some kind of arithmetical notation.
Regarding the geometrical continuum in this way as a single object of which
we have a direct knowledge obtained from our intuitions of space and time,
the reduction to a precise abstract form may be regarded as being made
upon the assumption that the system of rational numbers, with their limits
adjoined, is adequate to the analytical description of the continuum, in the
sense that each point in the continuum is represented uniquely by a single
real number, and that there is no point in the continuum which is not so
represented. This amounts to a definition, in a certain sense, of the contents
of the geometrical continuum. Such definition is not necessarily the only
possible definition, but it is a legitimate one, provided it suffices for the
purposes we have in view in Analysis and Geometry, and provided it does
not conflict with the concept of Continuity as derived from intuition. The
generic distinction between a continuous geometrical object, and a point,
or set of points, situated in that object, is not capable of direct arithmetic
representation. This does not, however, impair the efficiency of Arithmetical
Analysis in dealing with geometrical objects. In Cartesian geometry, for




74


Sets of Points


[CH. II


example, Analysis is really concerned only with the points that can be
determined in the geometrical objects with which it deals. This does not
mean that a continuous geometrical object is analysed into points which are
of necessity to be regarded as its " parts."
TRANSFINITE ORDINAL NUMBERS.
60. The theory of transfinite ordinal numbers had its origin* in the
investigation of the theory of sets of points. The general abstract theory of
such numbers, or order-types, will be deferred until the next Chapter; it is
necessary however to introduce here the conceptions connected with the
formation of these numbers, with a view to utilizing them in the theory of
sets of points.
Let Pi, P2,... P,,... denote a sequence of points in a given interval,
representing a sequence a,, a2, a3,... of increasing numbers, so that
al < t2 < C3... < an....
This sequence of points has a limiting point which is not one of the points of
the sequence, and is on the right of all those points; this limiting point we
may denote by P,. The symbol co may be regarded as denoting a new
ordinal number which comes after all the ordinal numbers 1, 2, 3,... n,...; it
is called the first transfinite ordinal number. The number co is not contained
in the sequence of finite ordinal numbers, but comes after all of them; and we
shall see that it may be taken as the first of a new sequence of ordinal
numbers, all of which must be regarded as ordinally greater than the finite
ordinal numbers.
PPW., -I  P 2 P+   Ad+
A   -  -  P2  P3 PF      P;. P2,+.2
FIG. 1.
Suppose that beyond the point P, there are other points which we wish
to regard as belonging to the same set as the points P,, P,,... P,,... P,;
then these points will be denoted by P,+,, P,+ 2,... P,+n,...; and if these
points are finite in number, there will be one of them P,+,,, which is the
last on the right. The indices of all the points of the set will be then
1, 2, 3,....n,... c, (o + 1, o + 2,... W - m;
and the numbers o, o + 1,... wt 4+ i are regarded as a set of transfinite ordinal
numbers, which commences with the first transfinite ordinal number o, and
contains the m succeeding transfinite ordinal numbers. It may however
happen that the set of points P,, P,,+1, P,+2,... has no last point. In that
case, assuming that the points are all contained in a finite interval, the set
* An account of Cantor's earliest presentation of this subject will be found in Math. Annalen,
vol. xxI.




59, 60]           Transfinite Ordinal Numbers                     75
has a limiting point which is not contained in the set itself; and this
limiting point we denote by PO,+ or P. 2, where 0. 2 is an ordinal number
which is not contained in the set co, C + 1, co + 2,..., but comes after the
numbers of the set.
If we wish to include further points which are on the right of Pt.2, we
must introduce numbers denoted by w. 2 + 1, co. 2 + 2,...; and, in case these
points form an infinite set in a finite interval, they will have a limiting point
which will be denoted by P.2, or Pw.3. We have now the ordinal
numbers
1, 2, 3,... w), a + 1, t+ 2,...  2,. 1.  2 +2,.21,.22... o.3.
If we proceed further in this manner it is clear that we may require numbers
eo. n, w. n + 1, c. n + 2,... co. n + 1,..., where n denotes any finite number.
Further, it may happen that the set of points P,, P.2, Pw.s3... Po.,...
is itself infinite, and has a limiting point on the right of all these points.
This point we denote by PO,; and the number c2 we consider to be a new
ordinal number which succeeds all the numbers co. n + m, where n and m
have all possible finite values.
Points on the right of P,, may be denoted by means of the indices
co2 + 1, o2 + 2, )2 + 3,...; and if these points are infinite in number, they may
have a limiting point P,,2+..
Points on the right of P,2+o may be denoted by the indices )2 + Co + 1,
)2 + t + 2,...; if these have a limiting point, it will be denoted by the
index o2 + c. 2. Proceeding in this manner, we may have points of which
the indices are t)2 + t. 3, 0o2 + t. 4.... If there is an infinite set of such
points, and the set has a limiting point, on the right of the set, this limiting
point will have 02 + Q)2 or O2. 2 for its index.
If we proceed still further, we see as before that we may have to
contemplate numbers of the form c2o. p + w. q + r, where p, q, r are finite;
afterwards co3, o3 + 1,..., 3.p + 02. q + Co. r + s, &c.  The general type of
ordinal numbers which can be obtained in this manner is represented by
i.r po +  l-1i. pi +... + o. p~ + Po; and it is clear that, for the representation
of points of a given set, such numbers may be required as indices.
It may happen that the set of points whose indices are co, c)2, 3,... is not
finite; then the limiting point of such set will be denoted by the index co*.
Starting afresh with this number, we may form numbers such as
W)Cto *~ 7+Uq —1.  ' %_l+... +Po.
If the points whose indices are otW, "OW, 3),... do not form a finite set, their
limiting point will be denoted by CoW.




76


Sets of Points


[CH. II


In a similar manner we may denote by e1 the number which comes after
the sequence c, w1o, cW, wwwo,...; and starting from el, we may similarly
proceed to form further numbers in endless succession.
61. All the ordinal numbers which can be formed in the manner above
described are formed by means of the application of Cantor's two principles
of generation (Erzeugungsprinzipien).
(1) After any number another immediately succeeding it is formed by the
addition of unity.
(2) After any endless seqnence of numbers, a new numnber is formed which
succeeds all the numbers in the sequence, and has no number immediately
preceding it.
All transfinite ordinal numbers which can be formed by means of these
two principles of generation are said to be ordinal numbers of the second
class. The finite ordinal numbers are said to be of the first class; they are
formed successively, starting with the number 1, by means of the first
principle of generation alone.
The numbers of the second class are of two essentially distinct species:
(1) non-limiting numbers, those numbers which have each a number immediately preceding them, and from which they are formed by the addition
of unity; for example o+ n, o p + o. q +, o +o+1: and (2) limiting
numbers, those which have no number immediately preceding them, from
which they are formed by the addition of unity; for example o, o2 + co,
cow + C02 are limiting numbers.
Any particular number of the second class can be denoted by a finite
number of symbols, but there is no upper limit to the number of symbols
required to denote such numbers.
Cantor has further postulated the existence of a number fl which comes
after all the numbers of the second class, and is the first number of a new
set which is called the third class. The number f2 cannot be obtained as the
number which succeeds a simple sequence, by means of the second principle
of generation; for every number which can be so obtained is itself a number
of the second class. This number 12 can be obtained only by means of a third
principle of generation, which postulates the existence of a new number
coming after all the numbers of the complex formed by the application
of the first and second principles of generation.  The validity of the
postulation of the existence of the number ~2, and of the higher numbers
of the third class will be discussed in Chapter III.
62. A fundamental property of the numbers of the second class may be
expressed as follows:



60-62]             Transfinite Ordinal Numbers                       77
Let Pi, P2, P3,... Pn,... Po, Po,+,... be an infinite set of points such that
either (1) there is a last point Pp, where 8/ is some number of the second class,
or (2) there is no last point, but every index occurs which is less than some
limiting number y of the second class, whereas the index y itself does not occur;
the set of points is then enumerable.
The sets
P1,    P2,.. P,...
P L 2o)+l n * * *  f,  **
P&,      P+1,... P+ 
Pi(d 2.  2(+1 ) * * * p. + 7 *2 *Pw.3, Pow.3+1, *..
P. I'   P W. (.rl,  - - *.....................
where every index less than Co2 occurs, form  an enumerable aggregate of
enumerable sets of points; and this has been shewn in ~ 54, to be itself an
enumerable set. Now consider the sets
PI P2, P3     **.. P    P.a+,.2.. Pc.2,....
02 PW     2+l  *. P -(2+.. Pc2++l, *...
P(o2.2, P2 z c2.2+1,  P '2. 2+fo,  **
PW2.3,  P(2.3+l,.1  P t2.3+o,  *....................................
such that in the first set there is every index less than ft2, in the second,
every index less than o)2. 2, and so on. Each of these sets is enumerable, and
there is an enumerable set of sets; hence the whole set, which contains every
index less than co3, is enumerable. In this manner it can be shewn that,
if every index less than o"n occurs, the set is enumerable. If the theorem
holds for sets which contain every index less than 8,, /2,, 3,..., then it holds
for a set which contains every index less than /, the limiting number of the
sequence /1, /2,.... For the points with indices less than /3, with indices
> /3 and < /2, with indices  /2 and < /3, &c. form an enumerable sequence
of enumerable sets; therefore by the theorem of ~ 54, the whole set with
indices < / is enumerable. Since the theorem holds for /, = t, /2= o2,
/3 = o03,... it holds for / = cow. By continual application of this method, since
any number can be reached by means of the two principles of generation,
and since every number is either a limiting number, or is obtained from one
by adding a finite number, we see that the general theorem holds.
It will now be shewn, conversely, that if a set of points P,, P2,...Pn,... P,,... Pp,... is enumerable, there must be some definite number y of the first
or of the second class, such that 7y does not occur among the indices of the points,
and such that every numbern less than y does so occur,




78


Sets of Points


[CH. II


In case y is a limiting number, there is no last point of the set; but if 7
is not a limiting number, there is a last point, viz. the one of which the index
is the number immediately preceding ry.
To prove the theorem, we observe that, since the given set of points is
enumerable, it may be placed in correspondence with a set of points Q1,
Q2,... Qn,... in which all the indices are numbers of the first class. Let us
suppose, that if possible, no number y exists, and let Pa, be the point of {P}
which corresponds to the point Q~ of {Q}. Let Qp, be the point of {Q} of
smallest index, such that the corresponding point of {P} has an index which
is >al; denote this index by a2. Then let Qv2 be that point of {Q}, of
smallest index, such that the corresponding point of {P} has an index > a2;
denote this index by a3. Proceeding in this manner, we have a set of points
Q1, Qp,, QP2,... Qp, 2   ". corresponding in order to a set of points Pa,, Pa,,
Pa,... Pal,... where ai< a, < a3... < a, <....  There exists a number a of
the second class, which is the limit of the sequence al, a2,... a,...; and by
hypothesis there exists a point Pa, which has a for index. Now the set {Q}
can contain no point which corresponds to Pa, because each point Qn corresponds to a point of {P} with an index less than a, and thus there is a
contradiction in the hypothesis that a occurs amongst the indices of the points
of {P}. Hence there exist numbers of the second class which do not occur
as indices in the set {P}, and these numbers form a set which is a part of the
aggregate of numbers of the second class. In this set there must be a lowest
number y, and this number y is the first which does not occur amongst the
indices of the set {P}. That every part of the aggregate of numbers of the
first and second classes, has a lowest number, will be shewn in Chapter iII.,
to be a consequence of the structure of the ordered aggregate.
EXAMPLES.
1. On a straight line AB, let us denote by P1, P2, P3,..., those points at which the
ratio AB/PB has the values 1, 2, 3,.... The point P1 coincides with A, and the point
B can only be represented by P.. Now take any one of the segments P.P,+; this may
for convenience be represented on an enlarged scale. Denote by Qrl, Q2, Q.3,..., the
points on PP +,,, at which P.Pr,. + QP,. + takes the values 1, 2, 3,...; thus Pr+2 car
only be represented by Qu,.. Supposing this to have been done with every segment
PPr +1 of AB, let us imagine all the points Q to be marked on AB, and to be numbered
from left to right.
In PiP2, we shall have          1, 2, 3,... co,
in P2P3 there will be          o +1, o +2,... o. 2,
and in PP4                co. 2+1, o.2+2,.... 3;
the point B can be represented only by 0c2. If now we proceed to take each segment
QrsQ., +, and to divide it in a similar manner, at points R for which Qr Qr, 8 + 1/RQ., +




62, 63]                   Transfinite Derivatives                             79
ha-s the values 1, 2, 3,..., and then imagine all the points R obtained in every such
segmeht QrsQ,8+1 to be marked on AB, and numbered as before, from left to right, it will
)e seen that all the numbers c2p + coq+r will be required, and that the point B can be
represented by 03. The points P1, P2,... P, will have for their ordinal numbers
41,:02, 32. 2, co2. 3,... o3; the point Q,. will be numbered 02. r+ o. s; the finite numbers
are all used up in the first sub-segment of AB. By proceeding to further subdivision, we
may exhibit on AB, the ordinal numbers o2pn +      -o+1.- 1  +... p, and the point B will
t.hen be represented by eo +1..2.. The properties of the integral numbers in relation to their prime factors may be
emnployed to rearrange the series 1, 2, 3,... n,.., so that the numbers may be made to
correspond with a series of ordinal numbers of the first and second classes.
First take the primes 1, 2, 3, 5, 7, 11,...; these correspond with the numbers of the
first class 1, 2, 3,... n,.... Then take the squares of the primes, omitting unity; we thus
have 22, 32, 52, 72, 112,..., corresponding to co, e+1, 0+2,... o+n,....
Wethen take the cubes of the primes,
23, 33, 53, 73, 113,..., corresponding to o). 2, o. 2+1,... o. 2+n,...,
and in general, 2'+1, 3r+, 5r+1,., corresponding to o. r, co. r+1,... r.+n,...  We
ma': then take the numbers ab which consist of the product of two prime factors; these
arranged in- asceiding order correspond to Co2, Cr2+,.. + 02+n,.... Next take the numbers
a2b2, which- i nsist of the squares of the last set; these correspond to 2 + co, 2 + o 1,....
We then take the successive sets of numbers of the forms a3b3, a4b4,...; we thus obtain
the iuniibers which may be taken to correspond with
Co2+. 2, o2+ o. 2 + 1,... (2 +o 3,... o2; 3  —,o..2.,
all of which are less than w2. 2. The sets of, numbers of the forms......(a..)2,... ()....a3b, (a3b)2,... (a3).......
may then be taken. Afterwards, we may proceed with the numbers which contain hree
different prime factors, and so on. It is clear that this mode of rearranging the integral
nuinbe s in their natural order, so that they correspond in the new order with ordinal
numbc 3 of the first and second classes, admits of great variety. In every case, there will.be ome lowest number of the second class, which is not employed in the correspondence
e   l ished.
THE TRANSFINITE DERIVATIVES OF A SET OF POINTS.
ij.     f. Gdenotes a set of points in the interval (a, b), it has been.S^tik'that.the derivatives G(), G(,... G0) are all closed sets, and that
all theo9intsi:of any one of these sets, after the first, are contained in the
precdingiset.   If G is of the second species, then G(n exists for all values
of,:n;,in this case the set D(G(), (2)... G...), which contains points
belo   h gig: tsoetvery G(), is denoted by G('), where o is the first transfinite
nuqmbere;-It1 will be shewn that G(0) contains one point at least, and is a
close&4:,:i It is defined to be the derivative* of G of order o.:f pa is a. point of G (1, P  a point of G(),...pn a point of G(, &c., the
poi.nts,,,,,. p..... form a set {pn} which has at least one limiting point p.
* See Cantor, Math. Annalen, vol. xvII.




80


Sets of Points


[CH. II


This point p belongs to Gn whatever value n has, because all except a finite
number of the set [pn} are points of G('2); and therefore p is a point of G,. Let
q1, q2,... qn,... be a sequence of points of G., in case G, contains more than
a finite number of points; and suppose this sequence to have the limiting;
point q. Then since all the points q1, q2,... q,... are points of the closed'
set G(n), the limiting point q is a point of G(T); and this holds for every
value of n, hence q is a point of G(-), and therefore G(,) is a closed set.
We can proceed to form the derivatives of G(*) in a similar manner to
that in which the derivatives G(1, G(2),... of G were formed. These successive
derivatives are denoted by G(~+1), G(+2),... G( +n),... and are regarded as the
derivatives of G of the transfinite orders o + 1, o + 2,... e + n,.... They have
the same properties as the derivatives of finite order, viz. that all the points
of each are points of G(), and that all the points of any one of them are points
of the preceding ones.
It may happen that one of the derivatives G(+~n) contains no points; then
the process of forming derivatives has come to an end, the last one being
G(w+n-l). If this is not the case, a repetition of the above reasoning shews tpat
the set D(G(w+1), G(@2),... G(w+fl),...) contains at least one poiit, and is
a closed set; this set is denoted by G(*' ), and is defined to be the derivative
of G of order co. 2. In the same manner we can proceed to form furlher
derivatives, whose orders are numbers of the second class.
In general, it ai, a2, a3,  a...  e,...  note a sequelce of iluttlberl  c the
second class, whose limiting number is /3, the same reasoning as before
shews that, if all the derivatives G(a,), G(a2),... G(*),... exist, then the set
D (G(a ), G(a2),... G(a),...) contains at least one point, and is a closed set... This
is denoted by G(0, and is defined to be the derivative of G of order /.
If we form the successive derivatives of the set G, whose orders areie:l
numbers of the first and. second classes, it may happen that there is a first
number y, of the first or second class, for which G(b) 0; but this nui-iber y
cannot be a limiting number of the second class.
It may, however, happen that no number y, of the first or second dass.
exists for which G(v) - 0, so that derivatives of G exist of orders correspondii;
to all the numbers of the first and second classes.  It will be shewn in ~ 73,
that if G(Y) does not vanish, for some number y, of the first or of the second
class, then there necessarily exists a number /3, of the first or second class,
such that G        + = G(+12) G( ).... This set G() is a perfect set, /end it is
frequently denoted by G("), where  2 is the first transfinite numbSer' of the
third class. The notation G(P) may however be employed, indepenan.tJy of
the acceptance of the theory of numbers of the third class.
Conversely, if G() does not exist, G(Y) must first vanish for some number
y of the first or second class, which number cannot be a limiting number.




63, 64]                     Sets of Intervals                           81
If G1, G2,,... G,... be any endless sequence of sets of points, such that
each set Gn is contained in the preceding one Gn_1, then the set D (G,, G2,...
Gn,...), if it exists, consists of those points each of which belongs to Gn for
every value of n, and this set may be denoted by G.  Commencing with G.,
a new sequence of sets G., Go+l, G,+2,... G+n)... may be considered, each one
being a part of the preceding one; the set of points each of which belongs
to all the sets of this sequence is D (G,, Go,+,,... G,)....), and may, when it
exists, be denoted by Gw.2.  In this manner further sets may be formed,
requiring as indices, higher numbers of the second class. An example
illustrating the fact that G. does not necessarily exist is
n ' n+   ' n+2' '
The case of the transfinite derivatives of a given set G, considered above,
is a special case of a sequence of sets each one of which contains the
next one.
EXAMPLES.
1*. Let G denote the enumerable set of points, each one of which is given by
1    1        1               1
2n    + 2nm   + +,+m 2+  + 2n, + + m2 +. + m,, 
when n has all positive integral values, excluding zero, and iml, m,... mn. have all positive
integral values including zero, independently of one another.
It is easily seen that in G('), the points 2,... all occur, and hence that G(-)
exists, and consists of the single point zero.
2-. Let G denote the enumerable set of points, each one of which is given by
1     1             1              1               1
1                         1
2mS+ 2"Wl   * + In*  2M + '.. +.+.  ' 2m + m2 + *- +  + + m +.. +  qm +m  + p+ 
2m, + m7 +... + m+ +r +p Q1+ q'* 2n m+ 'M- +   m+ +p +q +  '' + +... + 
where mn, m2,... mn,', p,, q, q... qp have all positive integral values, including zero. In
this case -G( +") consists of the single point zero.
3*. Let G denote the enumerable set of points, each one of which is given by
1     1       1             1             1
1                         1
where nn +  +... +   m +p h   qal n +  i +  + mm ++.  +  m +P +  q+  +  q2  + l e  +  p I 
where n, m, m,... m, p, l,... q, have all positive integral values. In this case G('-2)
exists, and consists of the single point zero.
SETS OF INTERVALS.
64. The properties of a set of intervals, which intervals are assigned in
any manner, are closely connected with the properties of sets of points, and
will therefore be considered here in some detail.
* These examples were given by Mittag-Leffler, Acta Math. vol. iv, p. 58.


H,


6




82


Sets of Points


[CH. II


If two intervals have only an end-point of each in common, they, are
said to abut on one another; and if the two intervals have more than one
point in common, they are said to overlap one another.
Every set of intervals, which is such that no two of the intervals overlap,
is an enumerable aggregate*.
First, suppose the set of non-overlapping intervals to lie in the finite
segment (a, b); and choose a sequence 1, e2,... En,... of positive numbers converging to the limit zero. The number of intervals of the given set which
are of length greater than, or equal to e,, is finite, since it cannot exceed
-(b- a). We can now arrange the intervals in order of magnitude, taking
En
first those which are g el, then those which are < e1 and  2 e, and so on,
there being only a finite number in each set. Therefore, since the set of
intervals can be arranged as a simply infinite aggregate, it is an enumerable set.
Next, suppose that the intervals are on an unlimited straight line in which
the position of any point is denoted by x. If we consider the correspondence
given by x'=, where the radical has always the positive sign, the
VX 2 + 1
unlimited straight line corresponds to the segment (-1, 1), in which the
point x' lies. The intervals of the given set correspond uniquely to intervals
of a non-overlapping set in the segment (-1, + 1), and this latter set is
enumerable; hence the given set is so also.
The theorem can be generalized so as to apply to the case of detached
portions of space of two, three, or any number of dimensions. If within
a finite portion of such space, there be a set of portions no two of which
overlap one another, though they may have portions of their boundaries
in common, the set of such portions is enumerably infinite if it be not
finite.
The theorem is proved, as in the case of intervals in a one-dimensional
space, from the consideration that there can only be a finite number of the
portions of volumet greater than, or, equal to En.
Since the points of unbounded space, say of three dimensions, can be
made to correspond with the points of a finite portion of space, by means
of the transformation
X/     +' -      y    ' 
the restriction that all the portions must be contained in a finite domain can
be removed.
* Cantor, Math. Annalen, vol. xx.
t More generally " measure," see ~ 81, below.




64-66]


Sets of I~ntervals


83


65. The theorem which is given in ~ 64, can now be applied to prove
that every isolated set of points is enumerable*.
Let P be a point of such a set. Since in a sufficiently small neighbourhood
of P, no other points of the set occur, take such a neighbourhood of length p,
and conceive such neighbourhoods to be chosen for every point of the set;
we now have a set of non-overlapping intervals which is enumerable, and
therefore the isolated set of points is also enumerable.
It has been shewn that any set of points G is made up of an isolated
aggregate, and of one which is a divisor of G'. It follows that, if the derivative
G' is enumerable, so also is G; but the converse does not hold.
Every set of points which is of the first species is enumerable. For, if s be its
order, G(s) contains only a finite number of points; hence G(S-1) is enumerable;
and therefore also G(S-2), G8(-3),... G are all enumerable sets.
A set of points of the second species is enumerable if one of its derivatives
be so. If any set G is not enumerable none of its derivatives is so.
66. Let us consider a given set of overlapping intervals contained
in the finite segment (a, b); it will be shewn that the given set can be
replaced by a set of non-overlapping intervals which is such, that every point
which is interior to any interval of either set is interior also to some interval
of the other set.
Taking any point P (x) which is an interior point of one or more intervals
of the given set, the points x' of the segment (x, b) can be divided into
two classes, those for which every point interior to the segment (x, x')
is an interior point of some interval of the given set, and those for which
this is not the case. This section of the numbers in the segment (x, b) defines
a single point x', such that (x, x') is the greatest segment on the right of x
which has every interior point of it also an interior point of the given set
of intervals. Similarly a definite segment (x", x) on the left of x, can be
found, which has the corresponding property. Therefore the interval (x", x')
is one of the required intervals. If now we take any point in either of the
parts of (a, b) complementary to (x", x'), which is an interior point of an
interval of the given set, we may proceed as before to construct an interval
of the required set which contains that point; and so on, until we have a
set of non-overlapping intervals which contain, as interior points, every point
that is interior to any interval of the given set. It has thus been proved
that:
Every set of intervals contained in a finite segment can be replaced by a set
of non-overlapping intervals of which the interior points are the same as those
of the given set,
* Cantor, Math. Annalen, vol. xxi.
6-2




84


Sets of Points


[OH. II


The new set may be spoken of as the set of non-overlapping open
intervals equivalent to the given set of open intervals.
An open interval PQ is defined as in ~ 40, to consist of the aggregate
of points interior to PQ, excluding the end-points P, Q.
The properties of any set of open intervals in a finite segment thus
depend upon those of a non-overlapping set of such intervals, and we
proceed to the consideration of the latter.
Every point of (a, b) which is not interior to an interval of the nonoverlapping set is either
(1) a common end-point of two intervals of the given set; or
(2) a point interior to, or at an end of, an interval not belonging to
the given set, this interval containing no point which is interior to any
interval of the set; or
(3) a limiting point, on both sides, of end-points of intervals of the
set; or
(4) an end-point of an interval of the given set, and also a limiting
point, on one side, of end-points of intervals of the given set.
If either a or b is an end-point of an interval, we reckon that point as
belonging to the points (1).
The points described in (2) or (3) may be described as external points
of the given set; and if a or b is a limiting point of end-points, it will be
reckoned as an external point.
The points described in (4) may be spoken of as semi-external* points.
The following theorem will now be established:Those points of the segment (a, b), which are not points of a given set
of non-overlapping open intervals, form a closed set of points.
The closed set includes all the end-points of the given set of intervals,
and all the external points.
To prove this theorem, we observe that no limiting point of the set
of points, complementary to the set of open intervals, can be interior to
an interval of the given set. For if P be such an interior point, a neighbourhood of P exists, viz. the interval in which it is contained, within
which there are no points of the set; and thus P cannot be a limiting
point of the set. All the limiting points of the set must therefore belong to
the set itself, which is consequently closed.
The closed set which is complementary to a set of non-overlapping open
intervals contains all the end-points of the intervals, all those points which,
not being end-points, are limiting points on both sides of end-points of


* This term is due to W. H. Young, Proc. Lond. Math. Soc. vol. xxxv, p. 250.




66, 67]                     Sets of Intervals                           85
intervals, and also the points interior to the complementary intervals, in case
such complementary intervals exist.
In case there are no complementary intervals, then the closed set of points
defined as the set complementary to a given set of open intervals, is a nondense closed set.
67. It will now be shewn that*, unless a given set of non-overlacpping
intervals is a finite set, there must be at least one external or semi-external
point; in other words the whole interval (a, b) cannot be filled up by an
indefinitely great number of non-overlapping intervals, each one of which
abuts on the next, without leaving at least one point over, which is neither
interior to an interval nor is an end-point of two intervals, the points a, b
being regarded as end-points of two intervals if they are end-points of one
interval of the given set.
If there be any complementary intervals, then the points of these
intervals are all external points, and we therefore need only consider the
case in which no such complementary intervals exist. We observe that,
when the number of intervals is not finite, their end-points must have at
least one limiting point P. Now this point P cannot be interior to one of
the given intervals; for, if it were so, it would have a neighbourhood, viz. the
interval to which it is interior, within which are no end-points. Neither can
P be a common end-point of two intervals; for it would then have a neighbourhood on the right, and also one on the left, within which there is no
end-point except P itself. The point P must consequently either be an
external point, i.e. one which is not an end-point but is a limiting point,
on both sides, of end-points; or else it must be an end-point of one interval,
and a limiting point, on one side, of end-points. If a, or b is not an endpoint, it is regarded as an external point. It will subsequently be shewn
that the external and semi-external points form a set which may be either
finite, or of cardinal number a, or of cardinal number c.
EXAMPLES.
1. In the interval (0, 1) take the interv  (,    ), ( 2, 2)...( ',
and also the intervals obtained by reflecting these intervals in the point ~. The point ~ is
external to all the intervals, and yet the limiting sum of the intervals is equal to 1, the
length of the whole interval (0, 1) in which the enumerable set of intervals is contained.
If instead of reflecting the intervals in the point ~, we take the interval (1, 1), the
point - is now a semi-external point, and the limiting sum of the intervals is the same as
before.
2. Take the set (i, 1), (0, )  (      2.. of intervals and divide each
* This theorem was given by W. H. Young, Proc. Lond. Math. Soc. vol. xxxv, p. 251.
t See W. H. Young, Proc. Lond. Mlath. Soc. vol. xxxv, pp. 249-251.




86


Sets of Points


[CH. 11


interval into a set of sub-intervals similar to the whole. We now have a new enumerable
set of intervals which has no external points, but of which the semi-external points form an
enumerable set i, ~,, 1,....
68. If a set of intervals\in (a; b) is such that every point of (a, b) is
an interior point of at least one interval (the end-points a, b being each an
end-point of at least one interval), then a finite n6mber of the intervals can be
selected which has the same property as the whole set.
This theorem, which is known as the Heine-Borel theorem*, is of considerable importance in the theory of functions, and may be proved as
follows:Denoting the points a, b, by A, B, we may select an interval Aq, which
has A as end-point; then select an interval p2q2, of which q, is an interior
point; then p3q3, of which q2 is an interior point, and so on; and consider
a.                                 b
A     _P    _2  q  Pa  q2  -n  qs                              - q'  B
FIG. 2.
the points q,, q2, 3,... thus constructed.  If one of these points q, coincides
with B, the finite set of intervals Aq,, p2q2,... pnq  required has been found.
If q, does not coincide with B for any value of n, then the infinite set of
points q, q2,... qn,... has a limiting point q' which is on the right of all of
them. Let us now suppose that it is impossible to select a finite number of
the intervals in the manner described, so that the end-point of the last is
at B. Then whatever particular selection of intervals we make as above, we
obtain a point q' on the right of the intervals, the position of q' depending
on the selection made. The set of points {q'} which has thus been obtained,
has either (1) a limiting point q on the right of all the points of Iq'}, or
(2) an extreme point q, belonging to the set, on the right of all the other
points of the set; and in either case q may or may not coincide with B. In
case (1), the point q is interior to an interval a/3 of the given set, or else is
an end-point of such, /3, q then coinciding with B. We can now choose a set
of intervals Aq,, p2q2,... pnqn,... for which the limiting point q' lies within
aq, since q is the limiting point of {q'}; and then only a finite number of the
points
I ql, q2,..  qn,...
lie outside aq'. Let then q, be the first of them which lies inside aq', and
consider the set of intervals Aq,, p2q2,... pnqn, a/. If q and / coincide, and
are therefore both coincident with B, we have here a finite set of intervals
such as the theorem   requires, and this is contrary to the hypothesis made
that no such finite set exists. If, on the other hand, q and / do not coincide,
* Borel, Ann. de I'ec. norm. (3) xnI, p. 51. See also Borel's Legons sur la theorie des
fonctions, p. 42.




68]


Sets of Intervals


87


it is impossible that q should be the limit on the right of the limiting points
of all the possible sets q1, q2,... qn, qn+,...; for we may take qn+i to coincide
with /, which is itself on the right of q. We have therefore again a contradiction. In case (2), there is one set of intervals
Aq1, pq2,.. pnqn,..
such that q is the limiting point of q1, q2,... q,...; and the position of q'
for any other set is on the left of q.  Now q is interior to an interval al/ of
the given set, and only a finite number of the points q1, q2,... qn,... is on
the left of a. Let qn be the first which is inside a/3; then the set
Aq1, p2q2,... pnqn, a/
is a finite set such as the theorem requires, in case /3 coincides with B. But
if /3 does not coincide with B, it is part of an infinite set for which the limit
of ql, q2,... qn, /,... is on the right of q, which is contrary to the supposition
that q has the extreme position on the right for all points of the set {q'}.
There is therefore, as in the other case, a contradiction in supposing that
B cannot be reached after taking a finite number of intervals. It will be
observed, that the set of intervals contemplated in the theorem is not
necessarily enumerable.
The theorem   may be stated in a somewhat different form, in which it is
capable of being proved in a simple manner.
Let us suppose that with each point of (a, b) is associated an interval of
which the point is an interior point, the intervals associated with a, or b,
extending beyond (a, b). Let the associated interval be called the proper?intterval of the point.  Further, let any interval be provisionally called a
a3itable interval, when it is included in the proper interval of some point
within or upon the boundary of itself.
The theorem   may then be stated*, that the interval (a, b) can be divided
*into a jfinite number of suitable intervals.
Fol, let the interval (a, b) be halved; if one of the halves is not suitable,
e it be halved; and so on. This halving process, which is to be applied to
every interval not already suitable, indefinitely, will terminate after a finite
number of steps.   For otherwise, let us consider an indefinitely continued
sequence of intervals, each half of its predecessor, and no one of them suitable.
These intervals determine a single point within or upon the boundary of,.very orle of them.  Let us consider the proper interval of this point; the:;equence of intervals of which the point is the limiting point, will, from and;<fter some fixed member of the sequence, all lie within this proper interval of
Atce point. This is contrary to the hypothesis that the sequence of unsuitable
itltervals is indefinitely continued.
Tlhe iheorem stated in this form is really contained, in Goursat's proof of Cauchy's theorem;
s(, Trans, Amer. Math. Soc., vol. I, p. 15.




88


Sets of Points


[CH. II


69. The Heine-Borel theorem can be extended to the case of sets in
two, three, or any number of dimensions. In the case of a set in two dimensions, we may suppose for simplicity that the set of areas is contained in a
rectangular area ABDC. We suppose that there exists a set of closed areas,
which may be, for example, all circles, or all rectangles, such that each point
Cl                                                         D
A        q1   q2      q3                 qn    q    q    B
FIG. 3.
inside ABDC is interior to one at least of the areas, and that each point on
the boundary of ABDC is interior to the straight boundary of at least one
such area. Since all the points of AC are interior, in the sense explained, to
areas of the given set, and since these areas are bounded by intervals on AC,
the Heine-Borel theorem proved above, shews that a finite number of areas
can be selected such that all the points on AC are interior points of them.
A straight line qq1' can then be found such that all the points interior to the
area Aqlq,'C are interior points of the areas which have been already
selected. Next we see in a similar manner that all the points of qlql' may
be enclosed as interior points of a properly selected finite set of the given
areas; we then see that a point q2 exists to the right of q, such that the
areas already determined enclose all the interior points of q1 q'q2'q2 as
internal points; and proceeding in this manner, we obtain a set of intervals
Aqj, q1q2, q2q3,... on AB. Now the point B must be reached at the end of a
finite number of stages of this process; for we may shew by precisely the
same reasoning as before, that it is impossible but that the point B be
reached at a finite stage of the process of taking in new finite sets selected
from the given set of areas.
Assuming the truth of the theorem for sets of areas in two dimensions;
it may be extended, in an analogous manner, to a three-dimensional space
and so on to spaces of any number of dimensions.
It is clear that Goursat's form of the Heine-Borel theorem may be proved
in the case of sets of any number of dimensions, exactly as in the proof giveia
in ~ 68, for the case of linear sets. The division of a rectangular cell of /,,
dimensions into 21 equal rectangular cells, will replace the process of halving
applicable to linear intervals.




69-71]


Sets of Intervals


89


70. The following theorem will now be proved:If any unenumerable set of overlapping intervals in (a, b) be given, then
an enumerable set can be selected out of the intervals of the given set, of which
the interior points are the same as those of the given set.
It has been shewn in ~ 66, that the given set can be replaced by a
non-overlapping set of intervals with the same interior points. An interval
of this second set is however not in general an interval of the given set.
Let PQ be an interval of the equivalent non-overlapping set; then every
internal point of PQ is an internal point of one interval at least of the given
set. The point P is either an end-point of some interval Pp of the given
set, or else it is a limiting point of end-points of an infinite number of
intervals of the given set. In the latter case we can choose an enumerable
sequence P.pl, P2p2, P3p3,... of intervals of the given set such that P is the
limiting point of the sequence of points Pi, P2,... Pn,.... Similarly, unless
Q is an end-point of an interval qQ of the given set, it is the limiting point
of a sequence Q, Q2,... Q,,... of end-points of intervals qI Q, q2Q2,... QQ,...
of the given set. Consider the intervals PiQ1, P2Q2,... PQn,..., where
P1, P2,... may be taken all to coincide with P in case the interval Pp exists,
a similar convention being made as regards Q. Since every point of PQ, is
interior to some interval of the given set, therefore in accordance with the
Heine-Borel theorem, a finite number of intervals of the given set can be
selected so that every point of P1Qj is interior to one at least of them. Let
a similar selection of a finite set of intervals be made for each of the intervals
P2Q2, P3Q3,... PnQn...; we have then altogether an enumerable set of
finite sets of intervals. The totality of these intervals forms a finite, or an
enumerable, set of intervals selected from the given set, which contains every
point in the interior of PQ as an interior point. Applying the same process
to each interval PQ of the equivalent non-overlapping set, and remembering
both, that the intervals PQ form a finite or enumerable set, and that an
enumerable set of finite or enumerable sets is itself enumerable, we derive
the conclusion that an enumerable set of intervals can be selected from the
given set such that the internal points are identical with those of the given set.
71. The Heine-Borel theorem can be extended to the case where the
points, which are to be internal to a finite number of intervals selected from
a given set, are a given closed set of points, instead of the whole set of points
of the segment in which the intervals lie.
Let a given set of intervals in (a, b) be such that every point of a given
closed set of points is interior to one at least of the given intervals. Consider
the set of non-overlapping intervals equivalent to the given set; this set
must be finite; for, if not, it has at least one external or semi-external point
P which is a limiting point of the end-points of the intervals. Then any




90


Sets of Points


[CH. II


arbitrarily small neighbourhood of P contains an indefinitely great number of
end-points of intervals, and therefore also of points of the given closed set;
and P would therefore be a limiting point of the closed set, but this is
impossible, as P does not belong to that set. Let pq be one of this finite
number of intervals of the equivalent non-overlapping set; then the part of
the given closed set of points which is in pq is itself closed. In pq take an
interval pq' which contains this closed part of the given set of points in its
interior: then by the Heine-Borel theorem      a finite number of intervals can
be selected from   the given set of intervals which contains every point of
p'q' as an internal point; and therefore contains the part of the closed set of
points which is interior to p'q'. Applying this process to each of the finite
number of intervals pq we have the following theorem*:Having given a closed set of points in (a, b), and a set of intervals such that
each point of the closed set is interior to one interval- at least of the set, a
finite number of intervals can be selected from  the given set which is also such
that every point of the closed set of points is interior to one at least of these
intervals.
The proof of Goursat's form of the Heine-Borel theorem can be modified
so as to apply to this case.    We have only to neglect, in the proof of ~ 68,
those sub-intervals which do not contain any of the points of the given
closed set.
NON-DENSE CLOSED AND PERFECT SETS.
72. It has been shewn in ~ 66, that if an infinite number of nonoverlapping intervals be contained in (a, b), the set of points which is complementary to the internal points of the intervals forms a closed set. If no
interval whatever can be found in (a, b) every point of which belongs to the
closed set, the given set of intervals is everywhere-dense, and the closed set
is in no interval everywhere-dense, and is therefore said to be non-dense
in (a, b).  We shall now prove the converse theoremt that:Every non-dense closed set of points consists of the end-points of a set of
non-overlapping intervals which is everywhere-dense in the domain, and of the
limiting points of such end-points.
* See W. H. Young, Proc. Lond. Math. Soc. vol. xxxv, p. 387; also Borel, Comptes Rendus,
January, 1905. For a further extension of the theorem, see W. H. Young, Messenger of Math.
vol. xxxni, p. 129, and also Proc. Lond. Math. Soc. Ser. 2, vol. ii, p. 67.
+ This relation between everywhere-dense sets of intervals and closed sets was discovered by
Du Bois Reymond and by Harnack. See Du Bois Reymond's Allgemeine Functionentheorie (1882),
p. 188; also Math. Annalen, vol. xvi, p. 128, where everywhere-dense sets of intervals are introduced. See also Harnack, Math. Annalen, vol. xix, p. 239, and Bendixson, Acta Math. vol. 11,
p. 416, and Ofv. af. Svensk. Vet. Forh. vol. xxxix, 2, p. 31. Proofs of the fundamental theorems
based on the amalgamation of abutting intervals have been given recently by W. H. Young,
Proc. Lond. Math. Soc. Ser. 2, vol. I, p. 240, and by Schoenflies, Gottinger Nachrichten, 1903.
The proof given in the text, in ~ 73, is based upon the latter proof.




71, 72]               Non-dense closed sets                      91
In any arbitrarily chosen interval a point P can be found which does not
belong to a given non-dense closed set G, and an interval pq containing P in
its interior can be found such that no internal point of pq belongs to G. For
if no such interval could be found, P would be a limiting point of G, which
is impossible. Let pq be the greatest interval containing P for which this
holds, then p, q are limiting points of G, and therefore belong to G.
We now proceed to construct intervals such as pq in the remaining parts
of the domain. Then, when every possible such interval has been constructed,
there are no intervals complementary to them; and all the end-points of the
intervals, together with the limiting points of such end-points, are the only
points which are not internal to the intervals. These points are consequently
the points of G.
The points of G consist in general of three classes:
(1) those which are common end-points of two intervals abutting on one
another;
(2) semi-external points (see ~ 66), which are end-points of one interval
and also limiting points on one side, of end-points; and
(3) external points, viz. such as are not end-points of intervals but are
limiting points, on both sides, of end-points.
An end-point of the domain of the set may be regarded as belonging to
(1) or (3) according as it is, or is not, an end-point of an interval.
Those points which belong to (1) are clearly isolated points of G. Hence
if no such points exist, every point of G is a limiting point; and therefore G
is perfect. The theorem has thus been proved that:Every non-dense perfect set G consists of the end-points of an everywheredense set of non-overlapping intervals no two of which abut on one another,
together with the limiting points of these end-points.
The end-points of the domain are points of G, but not end-points of
intervals.
If the set G is such that no semi-external points exist, then every interval
abuts on another one at both its ends. In this case, all the points of G
are either end-points of adjacent intervals, or limiting points, on both sides, of
a sequence of such end-points, unless a or b be a limiting point, in which case
it belongs to G. The end-points have the same cardinal number a as the
rational numbers, since the set of intervals is enumerable. Moreover the
external points form a finite set, or an enumerable set; because to each such
external point there corresponds an enumerable set of end-points of which it
is the limiting point, and in this correspondence any one end-point can
correspond to at most two limiting points, one on each side of it. We thus
have the theorem that:



92


Sets of Points


[CH. II


A non-dense closed set is enumerable if its complementary intervals are
such that every one of them abuts on another one at each of its ends.
73. Every non-dense closed set is, in general, made up of an enumerable
set and of a perfect set.
Let the intervals complementary to the set G be arranged in enumerable
order, that of descending magnitude; we may denote them by 81, 82,... 3n,....
If G is not perfect, it contains isolated points, each of which is the common
end-point of two adjacent intervals; let 8p, be the first of the intervals {8}
at an end of which there is such a point; let 8p, be the interval which abuts
on 611 at that end. It may happen that the other end-point of 8p, is also a
common end-point of two intervals. If so, let Sp, be the interval which abuts
on 8p,, and so on: after a finite, or enumerable set, of such intervals
Sp,  p"p,,,p'....
we must arrive at an interval of which the end-point does not belong to
G,, the set of isolated points of G, or else at an end-point of the domain of G;
unless G is an enumerable set. It may happen that sp, at its other end
abuts on another interval; in that case we proceed, in the same manner as
before, to find the intervals sq,, q,,,... each of which abuts on another one.
Now conceive all the intervals;p,, p,,,..., and if they exist, q,', Sq",... to be
amalgamated with p,1 into one interval pl(1), by removing all the common
end-points. If any isolated points of G now remain, let 8P2 be the first
interval of \8] after Sp, of which an end-point is such a point; proceed as
before, we then have an interval Sp2(1 formed by amalgamating a finite or
enumerable set of intervals. We proceed in this way, and thus form a set of
intervals 8,p(1), p3(P,... no end-points of which are points of G,.
Since G = G, + G(), where G(1) is the derivative of G, the set of intervals
{S1)} complementary to G(1) consists of the intervals 8p,(1), p(1)),... and of any
intervals of {8} which remain after such intervals as Sp,, vp,,,... 'q,, Sq,,,... have
been removed, and the p(1) substituted for the $p.
We proceed in a similar manner with G() =- G,(1) + G,(, again removing a
finite or enumerable number of the set {(1)}, and again with G(, and so on.
It may happen that the process comes to an end after a number n of such
stages, either if G,n) does not exist, in which case G(n) = G(n+l, and thus G()
is perfect; or else, if G(n) does not exist, in which case, G being the sum of a
finite number of enumerable sets WG,), is itself enumerable. If the process
does not come to an end for any finite value of n, we form the derivative
G() = D (G(), G',... G(),...), which contains all the points common to all
the derivatives of G of finite order. This set has been shewn in ~ 63, to exist,
and to be a closed set; G() is then resolved as before into G,(O) + G(w+1), and
we proceed further as before.




72-74]


Non-dense closed sets


93


We obtain, by proceeding in this manner,
G = G, + G,() +... + G,() + G,(+) +... + G,( + G+),
where /3 is a number of the first or second class. It will now be shewn that
there must be some definite number / of the first or second class, for which
this process comes to an end, either by G, ( containing no points, in which
case GO(  = G('+1), so that G(O is perfect; or else by G(+1) containing no points,
in which case G being the sum of an enumerable set of finite, or enumerable,
sets, is itself enumerable. The {8} contain all the indices 1, 2, 3,... n,...;
from these indices we must remove a finite, or an enumerably infinite number,
to obtain those indices which occur in the {8(1 }; and again an enumerable set of
indices must be removed from those which occur in the {8(1)}, to obtain those
which occur in the {8(2)}. Now as the indices 1, 2, 3,... n,... are enumerable, the
process of removing successively a finite, or enumerably infinite, set of them
must cease for some order 8 of 8(s, for otherwise a more than enumerable
infinity of indices could be removed from the set 1, 2, 3,... n,... which is
impossible; hence for some fixed number /3 of the second class all the
indices must have been removed.
It has thus been shewn that, unless the given set G is enumerable, for
some number /3 of the first or second class, G() = G(l+1); and therefore G(O is
perfect. Thus G has been resolved into an enumerable set and a perfect one.
If for any value of /, G( _= 0, the set G is enumerable.
74. The following theorem*, more general than that of ~ 73, includes the
latter as a particular case. The proof here given may be taken as alternative
to that of ~ 73.
If P   2, P,... P,... PP,... Pa,... are all closed sets of points such that
(1) if aot < ct, all the points of Pa2 belong to Pa1, and (2) if in any interval, any
set Pa contains only a finite number of points, the set Pa+i contains no points in
that interval; then either Pp must vanish for some definite number /3 of the
first or second class, or else there is a definite number /3 such that Pa is a
perfect set.
If for some number /, the set Pa vanishes, then Py vanishes for all values
of y which are > /.
Let us now suppose that there exists no number / such that P, vanishes.
In this case there exists a set of points which may be denoted by Pa, such
that each point of the set belongs to Pa whatever number / may be. The
set Pa is closed, for if p be a limiting point of the set, it is the limit
of a sequence of points contained in Pp, whatever number /3 may be;
hence p belongs to Pp, whatever /3 may be, and thus p itself belongs
to Po.


* See Baire, Annali di Mat., Ser. 3, vol. In.




94


Sets of Points


[CH. II


It will now be shewn that Pa contains no isolated points, and is therefore
dense-in-itself. If Pa contains an isolated point p, a neighbourhood of p can
be found which contains no point of Pa except p; let Q be that part of P1
which is contained in this neighbourhood. In the neighbourhood considered,
let us suppose a sequence of intervals f8, 8, *... n,... constructed, each one
containing the next one and the point p, and such that,, converges to zero
as n is indefinitely increased. Let Q(n) denote that part of Q which lies in 8,,
but not in an+1, then Q= Q() + Q(2) +... + Q() +... +p. There must exist a
number /,, of the first or of the second class, for which Q(1) contains no point of
Pa,; otherwise Q, would contain points which belong to Pa, and this is not
the case. Similarly, there exist numbers /2, /3,... /,,... such that Q(2)
contains no points of P,2, and Q(l) contains no points of P3,, etc. Of the
numbers A3,, 32,... 3n,..., let 7y be the first which is > /3, then let 72 be the
first which is greater than fY, and so on; we have therefore a sequence
1i, 72,... 7n,... of increasing numbers all of which belong to the set /3,
32,... Pn, **.. This sequence ly/, 72,... 7y,... is either finite, with say 7 as
the last, or else there is a limiting number y of the second class which is
greater than all of them, and therefore greater than all the numbers
/, 2a,... /3n,...  The set Q can have no point except p which belongs
to Py, hence since Py contains only one point in a certain interval, P,+'
contains no point in that interval, and does not contain p, which is contrary
to the hypothesis.
It has now been shewn that P0 is closed and dense-in-itself; it is therefore
perfect. Let us next consider the enumerable set of intervals which are complementary to Pa. For any one of these intervals there exists a number y
such that PY contains no point in the interior of the interval. As before, it is
seen that there exists a number, of the first or the second class, which is greater
than all these numbers y; if this number be /, the set Pa contains no points
which do not belong to Pa. It is thus seen that Pa is perfect, and
P    P=+1 =... = P.
The theorem has now been completely established.
75. Every perfect set* has the cardinal number c of the continuum; and
every closed infinite set has the cardinal number c, or else the cardinal number
a of the rational numbers.
Let the intervals whose internal points are the set C(G), the complement
of the perfect set G, be denoted by [6}; and let A denote the greatest, or one
of the greatest in case of equality, of the intervals {8}. Let 1, the whole
interval (a, b) in which G lies, be divided into the three parts 10, A, lx so that
I = lo + A + 1, where lo is on the left, and 1l on the right of A the greatest
interval of {8}. Denote the greatest of the intervals {3} in lo, by A0, and the
* Cantor, Math. Annalen, vol. xxiii.




74, 75]


Non-dense closed sets


95


greatest in 11, by A,; then the interval lo is divided by means of A0 into three
parts 100, A0, 1, in order from left to right, and the interval 11 is divided by
means of A, similarly into o10, Al, ln. Proceeding in this manner to a
further subdivision, let Apq be the greatest of the intervals {8} which lie in
lpq, where p, q each has one of the values 0 or 1; then Ipq is divided into
three parts pqo, Apq, pq,,,, and so on indefinitely. The intervals {3} are thus
arranged in the order A, Ao, Al, Aoo, Ao0, A1o, An,... and each interval of {3}
occurs at a definite place in the sequence. Consider a sequence of intervals,  p, Ipq,  pqr, pq,*
where p, q, r,... all have definite values each of which is either 0 or 1. Each
of these intervals is contained in the preceding one, and has one end-point
in common with it; and the sequence determines a single point P which is
interior to all the intervals of the sequence, unless, from and after some fixed
index, all the indices are identical, in which case P is a common end-point of
all the intervals after a fixed one. Hence since the point P is not interior
to any of the intervals {8}, it is a point of G. Conversely, every point of G
can be so determined by means of a sequence of intervals; for every point of G
belongs either to to or to 11, and also to one of the four intervals loo, o10, 110, ln,
and so on. The point P is the limiting point of the end-points of the
intervals Ap, Apq, Apqr,... with the indices the same as those of the sequence
1p, lpq, 1pq,.,... which determines the point.
Every number of the continuum (0, 1) is expressible in the dyad scale by
means of a sequence 'p, pq, pqr,... where each of the numbers p, q, r,... is
either 0 or 1; and all numbers are expressed uniquely in this manner, except
those for which all the digits after some fixed one are 1, these numbers being
also expressible by a sequence in which only 0 occurs after some fixed place.
The numbers last mentioned correspond as indices of lp,  lpq, pq,... to a point
of G which is an end-point of one of the intervals {8}; but in every other case
a number in the dyad scale corresponds to a point of G which is not an endpoint of the intervals {8}. Since the set of numbers of the continuum (0, 1)
has the cardinal number c, it follows that the points of G form a set of the
same cardinal number, because each point of G corresponds uniquely to a
single number of the continuum, except that two points of G which are endpoints of one interval correspond to a single number of the continuum.
Every closed set which is not enumerable has been shewn to contain a perfect
set as component; such a set has therefore the cardinal number c.
It will appear from the theory of order-types which will be discussed in
the next Chapter, that the set of intervals {8} which define a perfect set G,
when taken in their order of position from left to right, have an order-type
which is the same as V the order-type of the rational numbers which lie
between 0 and 1, excluding 0 and 1 themselves, taken in their natural order




96                             Sets of Points                          [CH. II
in the continuum. It follows that a correspondence can be established
between the intervals and the rational numbers, in which any two intervals
correspond to two rational numbers that have the same order. If we take
each rational number to correspond to the end-points of the corresponding
interval, then each irrational number corresponds to a point of G which is a
limiting point of end-points of intervals.
EXAMPLES.
1. Let x be a number given by x=      3    +  3+... +C-  where the numbers
C, C2,... C,,, have each one of the values 0, 2, and n has every integral value, and may
also be indefinitely great. The set {x} is a non-dense perfect set.
No number of the set lies between
07 + C2       0    2     2        o       r
3   32     + +   3   + — +3 2+  '" or. +  +... + -,and                        C +    C+... +   2
3   32  '    3" 
these two numbers determine a complementary interval of the set, the interval being of
length 2. The number of complementary intervals of length - is 2"-1, hence the sum
oo 2n-1
of all the complementary intervals is 2   which is unity. It is clear that the set of
n=l 3
complementary intervals is everywhere-dense, and thus the set of points is non-dense.
This example was constructed* by Cantor, and is the first example of a perfect non-dense
set which has been purposely constructed.
2. Let us suppose that the numbers of the interval (0, 1) are expressed in the dyad
scale, in the form 'alaa3... an...; where each a is either 0 or 1. Each number for
which the a's all vanish, after some fixed one a,, which must be 1, is also representable as
an unending radix fraction, in which an is 0, and all the subsequent digits are 1. Let the
numbers now be interpreted as if they were in the decimal scale. To each irrational
number in the dyad scale, there corresponds a single number in the decimal scale, represented by the same digits. Of each rational number, there is a double representation in
the dyad scale, and there correspond two numbers in the decimal scale, which define a
complementary interval of the set of points which represents the numbers in the decimal
scale. A perfect non-dense set of points is thus defined.
3. Taking a positive integer m (>2), let the interval (0, 1) be divided into m equal
parts, and exempt the last part from further subdivision. Divide each of the remaining
m - 1 intervals into m equal parts, and in each case exempt the last part from further subdivision. Let this operation be continued indefinitely. The points of division form a
non-dense set; for if an interval d be taken anywhere in the interval (0, 1), k may be so
chosen that 1 <,d and     a segment, entirely within d, can be determined. This
k   2 a   a, I 


* See Math. Annalen, vol. xxI, p. 590.




75]                        Non-dense closed sets                              97
segment is either an exempted interval, or its mth part is one. The end-points of the
intervals, together with their limiting points, form a non-dense closed set, of cardinal
number c.
4. As in* Ex. 3, let the interval (0, 1) be divided into m equal parts, and the
last be exempted from further division. Then let the remaining m -1 parts each be
divided into m2 equal parts, the last of each being exempted from further division. Let
the remaining parts be then divided into m3 equal parts, the last of these in each case
being exempted from further division. If this process be carried on indefinitely, the endpoints of the divisions together with their limiting points, form a non-dense closed set, of
cardinal number c.
5. Let k, k2,... kn,... be a sequence of positive integers each of which is greater than
unity, and defined according to any law.
It can be shewnt that every irrational number x, in (0, 1) can be uniquely represented
in the form
-- klk k... + l2.  n +
where c,<k,,, and not all of the numbers c,, c +,... are zero, for any value of n.
It can further be shewn that
ki   k k2      kk2...k "'
where r1 =k- 1- c,1. If, from and after a certain value of n, the condition cn =k,- 1, is
always satisfied, then all the 77, vanish, and x is rational. It thus appears that the rational
numbers are capable of a double representation in the form
-C1   2            Cn
X=   + k'',k -   1+k2,.. n;
(1) by the vanishing of all the c, after some fixed one, and (2) by the condition c- ==k- l
being satisfied from and after some fixed value of n.
If we now take those values of x, for which every c does not exceed some fixed integer X,
these values of x form a non-dense perfect set GX. It is easily seen that the interval of
which the end-points are
1    c            C            X            X
k1  k12 i+ * *.lk2....k, + lkkk2... + 2 
kx_~M..  c^, +  c~
and                         k   - + C2 +... o  k, +
k  k k,       kk2... n
contains no points of the set in its interior, although these points belong to the set.
A particular case of this set consists of the numbers given by
CX  l +  2 +  c3 +   +c n +
10  101.2  101.2.3     1 076!
where every c is _ 9. This set consists of the transcendental numbers first defined by
Liouville t.
* See H. J. S. Smith, Proc. Lond. Math. Soc. vol. vi, 1870.
+ Broden, Math. Ann. vol. LI.
+ Liouville's Journal, voQl. _xv p. 133.
H.                                                                 '7w~~v   Vtr CI


H.


7




98


- Sets of- Points


[CH. II


PROPERTIES OF THE DERIVATIVES OF SETS.
76. If a set is dense in any sub-interval of the domain in which it is
contained, its derivative G' contains every point of the sub-interval, and is
identical, so far as such sub-interval is concerned, with the totality of the
points of the sub-interval; we confine ourselves therefore to the case in
which G is a non-dense set, and consequently its derivatives are also nondense.
The derivatives of transfinite orders have been defined in ~ 63; and it
was there shewn that there is either a first derivative whose order is some
number of the first class, or non-limiting number of the second class, or else
that derivatives of all such orders exist, and have a set of points G(a) in
common.
It was shewn in ~ 73, that G(1) being a non-dense closed set, two cases
arise:
(1) If G(') is enumerable, in which case G is also enumerable, then G()
vanishes for some number /9 of the first or the second class. A set of this
kind is called a reducible set.
(2) If G(1) is not enumerable, then there exists some number,, of the
first or second class, for which G() is a perfect set, and is consequently
identical with G1+'), and with G(a) as defined in ~ 63. The set G(1) is the
sum of an enumerable set and the perfect set G(). A set G which has this
property, is said to be irreducible.
It should be observed that when G(1) is unenumerable, and consequently
of cardinal number c, the same as the cardinal number of its perfect component, we are unable to make any inference as to the cardinal number of G
itself. This may be a or c, or other cardinal number between the two, in case
such a number exists.
THE CONTENT AND THE MEASURE OF SETS OF POINTS.
77. The theory of the content of a set of points in a finite linear domain
was originated by Hankel*, and further developed by Harnack, Stolz, and by
Cantort, who extended the conception to the case of sets of points in a domain
of any number of dimensions.
Suppose we have a linear set of points G in the finite interval (a, b), and
conceive the interval to be divided into any finite number n? of sub-intervals,
the greatest of which is A,; let the sum of those v, sub-intervals which
have points of G either as interior points, or as end-points, be denoted by
* See Hankel, Math. Annalen, vol. xx; Stolz, Math. Annalen, vol. xxII; Harnack, Math.
Annalen, vol. xxv; Pasch, Math. Annalen, vol. xxx.
t Math. Annalen, vol. xxiiI.




76, 77]


Content and Measure


99


Snlv where v,: n,; then Snv,,,  b-a. Now suppose each sub-interval to be
again divided into any number of parts, so that the whole interval (a, b) is
now divided into n2 sub-intervals (n2 > n,) of which the greatest is A2; and of
these suppose 2v (5 n2) to contain points of G as interior points or at their
ends. Let SnV denote the sum of the v2 intervals; thus Sn,22  -SV b - a.
Proceed in this manner to make further sub-divisions, so that at any stage
there are n,. sub-intervals of (a, b), of which the greatest is of length A,., and
such that Sn rV. is the sum of those vP. intervals (vy - no.) which contain points
of G. Let the process of further sub-division be continued indefinitely in
any prescribed manner.which is subject to the condition that A,, A,... Ar,,...
is a sequence which has the limit zero. Then the numbers
S    >X S7  >"        S
fSlni V1 -l2 V2  S3, s  "s Sr V' 
have a definite limit X to which Snvr is arbitrarily near, for a sufficiently
great value of r; and X may be equal to b - a, or to zero, or to a number
between 0 and b - a.
It will now be shewn that the number E is independent of the original
mode of sub-division of the interval (a, b), and of the mode in which the
further sub-division is carried on, the sole restriction on the mode of formation
of successive sub-divisions being that A,, the greatest sub-interval of the
rth sub-division, must have the limit zero when r increases indefinitely.
Considering two different processes of sub-division, suppose 2, 2' to be
the limits, in the two cases, of the sums of those sub-intervals which
contain points of G.   Suppose the first system  of sub-division so far
advanced that Srv -  < e, where e is an arbitrarily chosen positive number;
and let the second system of sub-division be so far advanced that A',< d,
where d is an arbitrarily chosen positive number. If we conceive the two
sets of points of division to coexist, we then have a further sub-division of
(a, b), which may be considered as a continuation of either of the subdivisions corresponding to SnrVr or to XSn' v'S  Suppose. is the sum of those
of the new sub-intervals which contain points of G, then X < Snr.. Of
the sub-intervals in S',n V'gS there can be at most r - 1 which are not
sub-intervals of 2; hence S',n'is <: + n, d, and thus
',< Sl''s,  < Sn,,,r. + nd <. + e + n,.d.
Now e is arbitrarily small, and d is also arbitrarily small and independent
of nr; thus  ' < E. Similarly it can be proved that  Y, 2'; and thus E and
A' must be equal.
We have now established the following theorem:If G be any given set of points in the interval (a, b), there corresponds to G
a definite number X, which is such that all the points of G can be included in
7-2




100


Sets of Points


[CH. II


a definite number of intervals whose sum exceeds S by less than an arbitrarily
chosen positive number e, the number of the intervals depending on e.
The number S is called the content of the set G, and the content may have
any value between the two numbers 0, b - a, both inclusive.
Those sets of points for which the content is zero are of special importance
in the Theory of Functions. A set of zero content is said to be an unextended, or a discrete, or an integrable set of points.
78. The content of a set of points is the same as the content of its
derivative.
Let S' be the content of G' the derivative of a set G; then the points of
G' may be included in the interiors of a finite set of intervals whose sum is
less than A'+ 8, where 8 is an arbitrarily chosen positive number. There
can only be a finite number of points of G which do not fall within the
intervals that include the points of G' in their interiors; and this finite
number of points may be included in intervals whose sum is arbitrarily
small, say e. All the points of G are now included in a finite number of
intervals whose sum is less than s'+  + e; and a series of diminishing
values may be assigned to 8 and e, each sequence having the limit zero; and
therefore both 2' + 8 + e and  ' + 8 converge to the value E'; which proves
the theorem.
It follows from this theorem, that the content of any set is the same as
that of any of its successive derivatives. In the case of a set which is of the
first species, one of the derivatives contains only a finite number of points,
and consequently the set must be of zero content.
79. A definition of the content of a set of points has been given by
Cantor* which, though differing in form from that of Hankel and Harnack, is
in reality equivalent to it. Instead of enclosing the points of the set G in
a finite number of intervals, Cantor encloses each point of G in an interval
2p of which the point is the middle point, the number p being the same for
each point of the set, those parts of intervals 2p which do not lie within
(a, b) being disregarded. We have in this manner obtained an infinite
number of overlapping intervals which contain all the points of G, and, as is
clear, all the points of G', which is a closed set. If we replace this set of
intervals by the set of non-overlapping intervals with the same interior
points, each interval of this latter set is _ 2p. The set, which is non-overlapping, and equivalent to the infinite set, is consequently a finite set, the
sum of whose lengths may be denoted by I (p, G). When p is diminished
indefinitely, the number II (p, G), which cannot increase as p is diminished,
must have a definite lower limit, which defines the content of either of the


* Math. Annalet, vol. xxIIi.




77-80]


Content and Measure


101


sets G and G'. Since the infinite set of intervals which has been employed
only covers a finite number of detached lengths, this definition is equivalent
to that of Hankel and Harnack. Cantor applies this definition to the case
of a set of points in a p-dimensional continuum, by enclosing each point in
a "sphere" of radius p with its centre at the point; the content is then the
lower limit of the volume of the continuum contained within the spheres.
The essential point in the above definition of the content of a set of
points is that all the points are enclosed in a finite number of intervals
which therefore enclose all the limiting points*; and the lower limit of the
sum of these intervals is taken as defining the content of the set. If the
points are enclosed from the commencement in an infinite number of intervals
which are of unequal length, in accordance with some prescribed law, and
the lengths of these intervals are then diminished, each one in a prescribed
manner tending to the limit zero, then the limit of the sum of those parts of
the interval which are included in the infinite set of intervals is not necessarily equal to the content as above defined. For example, let us consider the
set of rational points in the interval (0, 1). These points can be arranged in
enumerable order Pi, P2, Ps,...: now enclose Pi in an interval of length
I                    1                                1
e, P2 in an interval 2 e, &c., Pn in an interval of length - e, and so on;
the total length covered by these intervals cannot exceed e 21 or e, and this
has the limit zero, as e is diminished towards zero. On the other hand, the
content of the set of rational points is the same as that of the derived set;
but this consists of all the points of the interval (0, 1), and is therefore unity.
In general, any enumerable set of points can be enclosed in an infinite
number of intervals, which covers a length that is arbitrarily small, and has
the limit zero; whereas the content of the set is not in general zero.
80. A completely satisfactory definition of the content of a set of points
of the most general character should satisfy the condition of affording a
consistent generalization of the notion of the length of a continuous linear
set of points, or of the notions of area and volume, in the case of sets of
points in two or three dimensions. In the case of closed sets, the definition
given above leaves nothing to be desired in this respect; but in the case of
open sets, the definition leads to consequences which are at variance with the
fundamental properties of lengths, areas, and volumes, as understood for the
case of continuous domains. If G1, G2 are two complementary sets of points
in the continuous interval (0, 1), then, in order that the contents of the sets
G1, G2 may accord with a generalization of the notion of length, their sum
should be unity; however, when G? and G2 are unclosed, this condition is in


* See Harnack, Math. Annalen, vol. xxiii, p. 241.




102


Sets of Points


[CH. II


general not satisfied by the definition given above. For example, if Gi consists
of the rational points, and G2 of the irrational points, each of the two sets
Gi, G2 has its content unity, the same as that of the continuum (0, 1) itself.
Again, let us consider an everywhere-dense set of non-overlapping intervals
contained in (0, 1); then the internal points of these intervals form an open set
Gi, of which the derivative consists of all the points of the continuum (0, 1);
the external and the end-points of the intervals forming a non-dense closed
set G2. It will be shewn subsequently that the everywhere-dense set of
non-overlapping intervals can be so chosen that the limit of the sum of their
lengths is an arbitrary number 1, where I is subject to the condition 0 < I <1;
whereas the content of the set G1 is, in accordance with the definition given
above, always unity, and therefore may differ from the sum of the contents of
the sets of points contained in the separate intervals. To obtain the content
of the closed set G2, cut off, from each of the intervals which define GI, the
1~~~~~1
2- th part of its length at each end; the limiting sum of the intervals so
restricted is 1  - -).   Of these restricted intervals, a finite number can
be so taken that their sum is >  1- -     e, and < I  -), where e is an
arbitrarily chosen positive number. All the points of G2 are now enclosed in the
finite set of intervals which is complementary to the finite set of restricted
intervals. The sum of these complementary intervals is < 1 — (l -) + e
and > 1 -  ( — ); the sum has for its lower limit the number 1 - 1, which
is therefore the content of G2. The sum of the contents of G1, G2 is therefore not equal to unity, which is nevertheless the content of
G,+ G2(0, 1).
81. For the reasons which have been explained and illustrated, the
definition of the content of a set of points in the form given either by Hankel
or by Cantor is appropriate only in the case of closed sets.
A theory has been developed by Borel* and Lebesguet, and also by W. H.
Young+, of which the aim is to attach to each set of points a definite number
called its measure, which shall be such as to form a natural extension of the
notion of the length of a continuous interval, or of the notions of area and
volume in higher dimensions.  In this theory, certain postulates are made, by
* See his Legons sur la theorie des fonctions.
t See the memoir "Integrale, Longeur, Aire " in the Ann. di Mat. Ser. mI, vol. vii (1902).
+ " Open sets and the theory of content," Lond. 'Iath. Soc. Proc. Ser. iI, vol. II, where a
similar theory has been developed independently of the work of Lebesgue. Here the term
"content" is used for Lebesgue's "measure"; the latter term has been adopted in the text in
order to avoid confusion with the term "content " as used by Hankel, Harnack, and Cantor.




80, 81]


Content and Mieasure


103


means of which definite measures are assigned to successive classes of sets of
points. The question whether every set of points, however defined, has a
measure, is left open, but it appears that all sets of points which have in
point of fact been defined have definite measures; such sets are said to be
measurable.
The problem of assigning definite measures to sets of points is taken to
require that the measure of a set shall satisfy the following conditions:(1) Sets containing an infinite number of points exist of which the
measure is not zero.
(2) Two congruent sets have the same measure.
Two sets are said to be congruent when, corresponding to every pair
P, Q of points of the first set, there exist corresponding points P', Q' of the
second set, such that the distance of P from Q is the same as that of P'
from Q'. The second set is therefore the first in a displaced position.
(3) The measure of the sum of a finite number of sets, or the limiting
sum of an enumerably infinite number of sets of points, is the sum, or the
limit of the sum, of the measures of the different sets, provided that no two
of the given sets have a point in common.
If G is any given set of points, let the points of G be enclosed in a finite
or an infinite number of intervals; it has been shewn that these intervals
are equivalent to an enumerably infinite, or a finite set of intervals which do
not overlap. The sum, or the limit of the sum, of these non-overlapping
intervals has a positive value depending upon the mode in which the intervals
are constructedc This sum has a lower limit, when every possible choice of
the intervals which enclose the given set is taken account of; and this lower
limit me (G) is defined to be the exterior measure of the given set G. Every
set of points G has an exterior measure; and the points of the set can always
be enclosed in an enumerably infinite, or finite, number of intervals, whose
sum does not exceed me (G) + e, where e is an arbitrarily chosen positive
number.
Let C(G) denote the set which is complementary to G relatively to the
interval (a, b) of length 1, in which G is contained; if me {C(G)} denotes
the exterior measure of C (G), then I - me C (G)}, which may also be denoted
by miG, is defined to be the interior measure of G.
An equivalent definition of the interior measure of a set is the following*:The interior measure of a set is the upper limit of the content of its closed
components.
For if the complementary set C (G) be enclosed in a set of non-overlapping intervals whose sum, or limiting sum, is me {C (G)} + e, then the
* See W. H. Young, loc. cit., p. 28.




104


Sets of Points


[cII. II


closed set formed by the end-points of these intervals and the exterior points
is a component of G, and has a content I - m, 0 (G)} - e. Since e is arbitrarily
small, this content is less than 1 -me tC (G)} by less than any arbitrarily
chosen small number; and thus    - me {C(G)} is its upper limit.
Every set of points G has both an exterior and an interior measure. When
the two are equal, the set G is said to be measurable, and the number
me (G) = mi (G)
is defined to be the measure of G. The measure of a measurable set G may be
denoted by m (G).
It will be shewn that this definition satisfies the conditions which have
been stated above, whenever it is applicable; and it will be shewn that the
sets which ordinarily arise are measurable in accordance with this definition.
Whether sets exist for which the external and internal measures are unequal,
and, if so, whether it is possible to give a definition of the measure of such a
set, so as to still satisfy the conditions given above, are questions which will
not be here discussed.
A single point is clearly measurable, and has a zero measure. The points
in a single continuous interval, or in a finite number of such intervals, are at
once seen to be measurable, whether the intervals are open or closed; and
the measure is the length of the interval, or the sum of the lengths of the
intervals.
The condition that any set of points G is measurable, in the sense defined
above, may be stated as follows:A set of points is measurable if its points can be enclosed in a finite, or
enumerably infinite, number of non-overlapping intervals a, and the complementary set can similarly be enclosed in intervals /3, such that the sum, o0
limiting sumr, of the common parts of a and /3 is arbitrarily small.
Any enumerable set of points is measurable, and has zero for its measure.
For if P1, P2,... Pn,... are the points, they may be enclosed in intervals
11         1
of lengths 2 e,  e,... 2e,..., and the sum of the lengths of the equivalent
non-overlapping intervals is < e, which is arbitrarily small. The exterior
measure being zero, the set is measurable.
The points contained in the interior of an enumerably infinite set of nonoverlapping intervals form a measurable set, whose measure is the limiting sum
of the intervals.
This theorem has been shewn above to be not in general true of the
content of such a set, as the content is defined in ~ 77.
If X is the length of one of the intervals, cut off - of X from each end;
2n




81, 82]


Content and Measure


105


we have then a curtailed interval of length  (1 -  ); do the same, taking
the same value of n, with each interval of the set. Then, if we remove all
these curtailed intervals from (a, b), we have all the points complementary to
the given set enclosed in a set of intervals, such that the part which is common
to them and to the given set of intervals is at most - EX, which may be
n
made as small as we please by taking n large enough. The condition that
the given set of points is measurable is therefore satisfied; and it follows that
the closed set which is complementary to the given set is also measurable.
It will hereafter be proved that every closed set is definable as the complementary set of the points interior to an enumerable set of intervals.
From this it follows that every closed set is measurable, and that its measure
is identical with the content as defined in ~ 77. The measure of the set of
points interior to the intervals is however not in general identical with its
content, as we have shewn in ~ 80.
82. It will now be proved that if G1, G2,... G,... are a finite, or enumerably infinite, number of sets which are measurable, then M (G1, G2,... Gn,...)
is measurable; and, if no point belongs to more than one of the sets, the measure
of G, + G2 +... + Gn +... is the sum of the measures of G,, G2,... G,,...;
the definition of the measure thus satisfying the condition (3).
Let G, and its complement C (G,) be enclosed in sets of intervals al, and /,,
each of which sets is non-overlapping, and such that the total length of the
parts common to a,, /3, is the arbitrarily small number e,. Let G2 and
C (G2) be similarly enclosed in sets of intervals a2, 32 which have a common
part e2, arbitrarily small; and let a2', 82 be the parts of a2, 32 which they
have in common with /,. For G3 and C(G,), we similarly take a,, /3, which
have a common part e,, arbitrarily small; a,', /3' are the parts which a,, /3,
have in common with /3': and so on.
The points of M(G,, G2, G,...) can be enclosed in the intervals
al, 2, a, s,;
moreover CI{M(G,, G0, G,,...)} has all its points enclosed in /3,', whatever
value l may have; therefore the two sets of intervals have the common part,1 + 2 +... + e,  a + m (a+)+  (a,+2)+....
The series Em (a') being convergent, since each term is positive, and the sum
has a finite upper limit, t can be chosen such that m (a'+,) +... is less than e,
where e is an arbitrarily small number; and e,, 62,... may be chosen so that
61 + 62 +... < e: therefore the common part of the two sets of intervals is < 2e,
and thus M (G,, G2, G,...) is measurable.




106


Sets of Points


[CH. II


If G1, G2, G3,... have no points in common, we see that
m (G1 + G2 + G +...) differs from  m (a+) + Wn (a2) + m (a(3) +...
by less than e1 + e2 + 63 +...; and therefore
m (G, + G+     G3 +...)= m (GI) + m (G) +...,
and thus the measure of the sum of a finite, or enumerably infinite, number
of measurable sets satisfies the condition (3).
If one set G0, which is measurable, contains another set G2 which has the
same property, then G0 - G2 is measurable.
The complement of G - G2 consists of G2 together with C(G0), hence
G - G2 is measurable. Further, since
G1 = (G - G2) + 02, we have m (G - G,) = m (G1)- m (G2).
If G, G2, G3,... are a finite, or enumerably infinite, number of measurable
sets, then the set D(G0, 0G, G3,...) of points common to all of them is
measurable.
For the complement of D (G1, G2,...) is M {C (G0), C(G),...}, and since
C (Gi), C(G.),... are all measurable, it follows that the complement of
D (G), G2,...) is measurable, and therefore that the set itself is measurable.
If * G, G2, G3,... Gl,,... are an enumerably infinite number of measurable
sets, and H is the set of points each of which belongs to an infinite number of
the given sets, then H is measurable.
For the set C (H) complementary to H, consists of those points which
belong to none, or only to a finite number, of the sets Gi, G,... On,...; and
hence C(H) consists of the points which belong to one or more of the sets
L,, L2,... Ln,..., where Ln denotes the set D tC(G,), C(Gl+),...} which
consists of the points common to all the sets C(Gn), C (Gn+),....     The
sets Ln are all measurable, hence C(H) is measurable; and therefore H is
measurable.
If* GI, G2, G3,... Gn,... are an enumerably infinite number of measurable
sets, and K is the set of points each of which belongs to all the sets Gn, Gn+,...,
where n has a definite value for each point of the set K; then the set K is
measurable.
For the set C(K), which is complementary to K, is the set of points
each of which belongs to an infinite number of the measurable sets
C((Gi), C(),... C(Gn),...; and hence, by the last theorem, C(K) is
measurable. Therefore K is a measurable set.
* See Borel's Leqons sur les fonctions de variables reelles, p. 18. The set H is named, by
Borel, the "ensemble limite complet," and the set K the "ensemble limite restreint," of the
given sequence of sets.




82-84]


Content and Measure


107


83. All the sets which we have so far proved to be measurable were
obtained from two fundamental sets, the single point, and the single interval,
open or closed, by taking a finite, or enumerably infinite, number of these
fundamental sets, and by taking the set common to a finite, or enumerably
infinite, number of the sets so obtained, or by taking the complements of the
measurable sets so obtained.
It can be shewn that measurable sets may exist which are not definable
in the manner we have described. It will be subsequently shewn that perfect
sets exist whose measure is zero; any component whatever of such a set has
its external measure zero, and is therefore measurable. The cardinal number
of all such components is usually regarded as greater than the cardinal number
of the continuum, which is the cardinal number of all the sets obtainable in
the manner indicated above. It follows that other measurable sets may be
definable besides those obtained by the processes we have described. Whether
every set which can be defined is measurable, is a point which has not been
settled.
If G be any measurable set whatever, the points of G can be enclosed in
a set of intervals a,, whose sum is m (G) + t,; where e, is one of a sequence
of positive decreasing numbers which converge to zero. The set'G1 of the
points which are common to all the sets of intervals al, a2,... a,,... is
measurable, and its measure is m(G); also the set G1 contains G as a component. The set GI - G has measure zero, and its points can be enclosed in
a set of intervals /, contained in a,, and of measure eL. The set H of points
common to all the /3, is measurable, and of measure zero; and the set
G2=G -H is measurable, and has m(G) for measure. It has thus been
shewn that every measurable set G is contained in another measurable set
G., and also contains a third measurable set G2; where G,, G2 are measurable
sets of the type definable as the set common to an enumerable number of sets
of intervals. The measures of G1, G2 are both m (G).
84. A definition has been employed by Jordan*, and by Peanot, of the
measure of a set of points, which differs from the one which has been
developed above. It is applicable to sets of points in space of any number of
dimensions.
Let G be a set of points in a domain E, and let C(G) denote the set
complementary to G; then a point of G which is not a limiting point of C(G)
is said to be an interior point of G; and a point of C(G) which is not a
limiting point of G is said to be an interior point of C (G).


* Liouville (4), vol. viII (1892); also Cours d'Analyse, vol. I, p. 28.
t Applicazioni geom. del. calc. infinit. (1887), p. 153.




108


Sets of Points


[CH. II


Every point of E, which is an interior point neither of G nor of C (G), is
said to be a point of the frontier of G. Every such point is either a point
of G, which is at the same time a limiting point of C(G), or else it is a point
of C (G), which is also a limiting point of G; and the aggregate of all such
points constitutes the frontier of G.
Divide E into any finite number of continuous parts, consisting of linear
intervals or of rectangular cells, some of which may if necessary extend
beyond the domain E; and let S, be the sum of those parts which are
such that every point of each of them is an interior point of G. When the
number of the continuous parts is increased indefinitely, in such a manner
that the greatest of them converges to the limit zero, it can be shewn that ES
converges to a definite limit S1. If the sum ~2 of those parts of E is taken,
each of which contains at least one interior point of G, or a point on the
frontier of G, it can be shewn that E2 converges to a fixed number S2.
The number S1 is called the interior extent of G, and the number S2 is
called the exterior extent of G; when S2 is equal to S1, the set G is said to
be measurable, and S1 = S2 is its measure. The exterior extent of a set G is
identical with its content. In accordance with this definition, a set which
does not contain any part which is a continuum has its interior extent zero;
and such a set is only measurable when its content is also zero.
It can be shewn that, in those cases in which a set is measurable, both in
accordance with this definition and with that of Borel, the measure is the
same in the two cases. A set which is measurable in accordance with
Jordan's definition is also measurable in accordance with that of Borel; but
the converse does not hold. The definition of Borel will accordingly be
employed, as being the more widely applicable of the two definitions.
THE CONTENT OF CLOSED SETS.
85. The content of a non-dense closed set is zero, in case the set is
enumerable; and in case the set is unenumerable, its content may be zero, or
may have any value less than the length of the whole interval in which the set
is contained.
The content of a closed set being the same as its measure, its content is
the sum of the content of its perfect component and the measure of its enumerable component; and this last is zero. If the set is enumerable, it has
no perfect component, and therefore its content is zero. Since the content
of any closed set is the same as that of its perfect component, it will suffice
to consider the content of a non-dense perfect set.
Let I be the whole length of the interval in which the set is contained,
the end-points of which interval belong to the set; let 81,, 8,... be the




84-86]              The content of closed sets                    109
intervals complementary to the set, arranged in descending order of magnitude.
Of these intervals no two abut, and the set of intervals is everywhere-dense.
Let        81 = xl,  =   (-,     -82 =  2 ( - 3 =  (1 - 1 - 82),..
8n = M1 (1 - 81 - 82 --  81-1)...;
thus Xi, X2, X,... are proper fractions.
We have     82 = X2 (1 - X) 1, 83 = X3 (1- X) (1 - X) 1,...
8n = XA, (1 - X) (1 - \,)... (1 - Xn_) l,
hence        -(81 + 8+... + 8n) =   - \)(1 -\)... (1 -  n) 1
The content of the set is therefore* I multiplied by the limit of the product
(1-\) ((1 -X2)... (1 - -Xn).
The values of X,, X,... Xn,... can be so chosen that the content is zero;
for example, we may take X = X2=... =   = *... Those perfect sets, and
the related closed ones, which have content zero, are of special importance in
the Theory of Functions.
The values of XI, X,... may be so chosen that the content of the set is
arbitrarily nearly equal to 1. For example, let
X\ = 0, \2 = 0/22, X3 = 0/3,... X, = /n...
where 6 is a fixed positive fraction, then the content of the set is
I sin (7r V/0)/7r V0,
and this may be made as nearly equal to 1 as we please, by choosing a
sufficiently small value of 0. That in the interval I we may place an indefinitely great number of non-abutting intervals whose sum is arbitrarily
small, and so that no interval exists whose points are all external to the set
of intervals, is one of the paradoxes of the subject.
86. A closed set of the most general type is obtained by adding to a
non-dense closed set the internal points of some of the complementary
intervals. For, if a closed set be everywhere-dense in any interval (a,,8)
contained in (a, b), the interval in which the set exists, it is clear that every
point of (a, 3) belongs to the closed set. If the interior points of (a, /S) be
removed from the set, the remaining set is still closed. We may conceive
this process of removing the interior points of intervals in which the closed
set is everywhere-dense, to be continued, until a closed set remains which is
dense in no interval. It has been shewn in ~ 72, that every non-dense
closed set is definable as the end-points of an everywhere-dense enumerable
set of non-overlapping intervals, together with the limiting points of these
end-points. It has thus been shown that every closed set is definable as the
complementary set of the points interior to a finite, or enumerable, set of nonoverlapping intervals, not necessarily everywhere-dense.
* Harnack, Math. Annalen, vol. xix.




110


Sets of Points


[CH. II


It follows from this result, that the content of a closed set which is dense
in some parts of the interval, is the sum of the content of the non-dense
closed set from which it can be derived, and of the lengths of those intervals
all the points of which belong to the closed set.
EXAMPLES.
1. The perfect set of points defined by x= C +2 +... +  +..., where the numbers
c, c2,... have each one of the values 0, 2 (see Ex. 1, ~ 75), has the content zero. For the
limit of the sum of the complementary intervals is unity.
2. The non-dense closed set considered in Ex. 3, ~ 75, has the content zero. For,
after k operations, the sum of the exempted segments is
1         (in.- 1)2    (" - 1)k -l    rn-i1
— +(                      - + +(-  or 1- -
m           m3            mk              rn /
When k is increased indefinitely, the limit of the sum of the free intervals is 1.
3. The non-dense closed set considered in Ex. 4, ~ 75, has a content between 0 and 1.
After k operations, the sum of the exempted segments is
1  m-1    (mn-l)(mn2-1)     (?,-1)(m2-1)...(mk -  1)
m    m3 +      m6     +... ++l)
ffb3       b6 '", /,  IV-  1  / 1
or                     1-(1-m) ( 1- 2)...(1 -2k).
The limit of the sum of the exempted intervals is 1-   (1-  ) and therefore the
content of the set of points is I ( -  ), which is between 0 and 1, depending upon the
value of m. By taking m sufficiently great, the content of the set may be made arbitrarily
near to unity.
ASCENDING SEQUENCES OF CLOSED SETS.
87. Let G,, G2,... GL,... denote a sequence of non-dense closed sets contained in the interval (a, b), such that each set contains the preceding one as a
component; then the set G,, the limiting set of the sequence, is defined as a
set such that any point P belonging to it is a point of some definite Gn, and
consequently also of all the subsequent sets of the sequence; and, further,
that every point which belongs to any G, is a point of G,.
The limiting set G, is not necessarily a closed set; and it- may, or may
not be everywhere-dense in (a, b). Thus its derivative G,0 may be a nondense closed set, or it may be the continuum (a, b). To make it clear that
Go may have limiting points which do not belong to any G,, we observe that,
if PI Q, P2Q2,... PnQn,.... are complementary intervals of G,, 2,... Gn,... respectively, such that each interval contains the next as an interior interval,




86, 87]


Ascending sequences of closed sets


Ill


the lengths of the intervals may converge to zero; in that case there exists
a point p which is interior to all the above intervals, but does not belong to
any of the sets G, G2,... G,,.... Since this point p is interior to a complementary interval of each set, it is the limiting point of each of the
sequences P, P2,... P,..., Q,, Q2,.. Q,,... of points all of which belong
to G,: it thus appears that p is a point of G,', but not of G,; or G, is an
open set.
It may happen that the sequence of intervals P1QI, PQ2,... PnQ,... is
such that the limit of PQ,, is not zero. In that case PI, P2,... Pn,... may
have a limiting point p different from any P,; and QI, Q2,... Q,,,... a different
limiting point q different from any Qn. The points p and q are in this case
not points of G,, but are points of Go'; and the open interval pq is a complementary interval of Go'.
If, from and after some value n, of n, all the points Pn are coincident with
p, then p is a point of G,; and a similar remark applies to q.
To shew that the set Go may be everywhere-dense in (a, b), let G1 bea
perfect set; in each complementary interval of G, place a perfect set similar
to G1, i.e. identical with G1, except that the distances of every pair of points
are reduced in the ratio of the length of the complementary interval to that of
(a, b): we have now a new perfect set G2. Place in each complementary
interval of G2 a set similar, in the same sense, to G,; we thus obtain G3; and
so on indefinitely. The resulting limiting set G. is everywhere-dense: for, if
the greatest interval in G, is 0 times the length of (a, b), then the greatest
interval in G,, is 0n times the length of (a, b); and this is arbitrarily small
as n increases. Hence, in any interval whatever taken in (a, b), for a sufficiently great value of n, complementary intervals, and therefore points of
Gn, are contained; and therefore G. is everywhere-dense.
In the case in which G. is closed, it will subsequently appear that it must
be non-dense. In this case a complementary interval of G. is either the
whole, or a part, of a complementary interval of Gn, whatever value n may
have; for, if pq be such a complementary interval of G., no interior point
of pq belongs to any Gn. For some value n, of n, p belongs to G,,1 but not to
Gn-,_; and for some value n2 of n, q belongs to G,,, but not to Gn2_,; and
therefore, if m is the greater of the numbers n, and n, pq is a complementary
interval of G,,, for every n which is _ im.
When G, is open but non-dense, it can be shewn* that, corresponding to
ai.y complementary interval pq of G,', a number m can be found such that, for
every n which is _ m, Gn has a complementary interval p'q' which contains pq,
and exceeds it by less than an arbitrarily small number rq.


* W. H. Young, Proc. Lond. Math. Soc. vol. xxxv, p. 275.




112


Sets of Points


[CH. II


Increase pq on each side by ~q, and let PQ be the interval so increased.
If p and q are not both points of G,, an interval p'q' contained in PQ, and
containing pq, can be found such that p', q' are both points of G,. Now
p', q' are both points of some set Gm; then either p'q' is a complementary
interval of G,, or else Gm has a complementary interval whose end-points lie
in (p', p) and (q, q') respectively. In either case Gm has a complementary
interval which exceeds pq by less than r, and contains pq.
If e, a- are arbitrarily small positive numbers chosen independently of one
another, an integer m can be found such that, provided n _ m, the difference
between the sum s,n (e) of those complementary intervals of G, each of which
is _ e, and of those of G,' which are _ e, is < a.
Let s be the number of such complementary intervals of G.'; then for each
such interval a value of n can be found such that G, has a complementary
interval which contains that of G.', and exceeds it by less than a/s. Hence
a value v can be found of n, such that G, has a number of complementary
intervals all _ e, and whose sum exceeds the sum of those of G.' which are
_ e, by less than -.
It may however happen that G, has other complementary intervals which
are - e, but of course only a finite number of such intervals. Let PQ be such
an interval of G,; then PQ contains no interval of GJ' which is _ e. In PQ
we can take a finite number of points of G,, say pi, p2,... p,,, such that
Ppj1, Plp2, p2P3,... are each less than e. If we treat each of the finite number
of intervals of G,, such as PQ, in a similar manner, there exists a value nm
of n (m > v) such that all the points p for every interval PQ are points of Gm;
then the set Gm has no complementary intervals which are _ e, except such
as contain the intervals of G,' which are _ e; and this proves the theorem.
It is clear that, in the case when G, is closed, the above theorem reduces
to the simpler form, that corresponding to an arbitrary e, a number m can be
found, such that, for n -_ m, those intervals complementary to G,,, which are
_ e, are identical with those of G,, which are _ e.
88. The set G. may be regarded as the sum of the sets G1, G, -G1,
3-G G,... each of which contains no points which belong to the preceding
ones. Since G., G2, Gs,... are measurable sets, it follows from ~ 82, that
G2 - G1, G3-G2,... are also measurable; and thus that G, is measurable,
its measure being the limiting sum of the measures m (G), m (G2-G,),
m (G- G2).... Now it has been proved that
Do (Gn - Gn-) = n (Gn)- m (G,,_-),
and therefore the limiting sum of the measures of the sets is the limit of
m(Gn), which is a number that does not decrease as n increases. It th-us
appears that n (G) is the limit of m (Gn) as n is indefinitely increased; and
hence it has been proved that:



87, 88]


Sequences of closed sets


113


The measure of the limit of a sequence of non-dense closed sets is the upper
limit of the measures of the sets of the sequence.
The measure of a closed set being identical with its content, we obtain
Osgood's theorem* that:If the limit of a sequence of closed sets is itself a closed set, then the
content of the limiting set is the upper limit of the contents of the sets of the
sequence.
It should be observed that, when Go is not closed, it is in general not true
that the content of G,, or of G', is the limit of the content of Gn. For
example, if all the sets G, hlave zero content, the points of each Gn can be
enclosed in a finite number of intervals of arbitrarily small sum; but this is
not in general true of G,, unless G. is closed. The points of G, can however
in this case be enclosed in an indefinitely great number of intervals whose
limiting sum is arbitrarily small.
The content of any closed component of a measurable set G. cannot exceed
the measure of G.; we have therefore the theorem that:If Go, is the limiting set of a sequence of non-dense closed sets G,, G,... G,..
each one of which contains the preceding one, then no closed component of G, can
have content greater than the limit of the content of Gn.
An important particular case of this theorem arises when all the sets G,
have zero content; in that case every closed component of G(, has zero
content.
If e is an arbitrarily small number, and s, (e), s (e) are the sums of those
complementary intervals of G?, G,', respectively, each of which is _e; and
Rn(e), R(e) the sums of those complementary intervals of Gn, G,' each of
which is < e, we have
n(e) + R,(e) -=   {C (Gn)}, s(e)+ R(e)= m C(G.')};
hence          m {GW. - Gn} = {Sn( - (e) - ()} + {Rn () - R (e)}.
Now, as we 'have above shewn, if n  mn, where m depends on e,
Sn (e)- s (e) < -;
and we can choose e so that R(e) is as small as we please. Therefore we
see thatt the necessary and sufficient condition for the measure of G.' being
the same as that of G,. is that e can be so chosen that, from and after some
fixed value of n, Rn (e) may be less than an assigned arbitrarily small number.
If R, (e) has not the limit zero, when n is indefinitely increased, it is certain
that G, is unclosed, and has a measure less than the content of G,'.
* American Journal of Math., vol. xix, p. 178.
t W. H. Young, Proc. Lond. Math. Soc. vol. xxxv, p. 284.


H.


8




114


Sets of Points


[CH. II


SETS OF THE FIRST AND OF THE SECOND CATEGORY.
89. If PI, P2,... Pn,... is a sequence of non-dense closed sets, the set
M (PI,, P,...,... ), which contains all points belonging to one at least of the
sets, is said to be a set of the first category.
A set of the first category can be exhibited as the limit of a sequence of
non-dense closed sets each of which contains the preceding one. For such
a sequence is
P1, M(P, P2), M (P, P2, P3),...(P, P2... P...)....
It is clear that a set of the first category is of cardinal number a or c, the
former in case all the sets P are enumerable, and the latter in case some or
all are unenumerable.
It has been shewn in ~ 87 that a set of the first category may be everywhere-dense in its domain; or it may be non-dense.
A set which is complementary to a set of the first category* is said to be
of the second category.
It will be shewn that such a set is not of the first category.
In the first place, the set complementary to a set of the first category is
everywhere-dense. For if (a, 3) is any interval of the domain, there exists in
the interior of (a, /3) an interval (a1, fi), which contains no points of PI; and in
(a, /3,) is contained an interval (a2, /3) which contains no points of M (P1, P2);
and so on: there is consequently a point interior to all the intervals (a, /3),
(a1, 1/), (a2, 2),... (a,, /3n),... which is not a point of M (Pi, P,... P,,...);
hence the complementary set is everywhere-dense. It follows from      this
result that the continuum (a, b) is not a set of the first category.
Next, suppose if possible that the set complementary to
M (PI, P2,... Pn,...)
is itself the limit of a sequence Q1, Q,... Qn,... of non-dense closed sets. The
sets P1 + Q2, P2 + Q2,... P,, + Qn,... are all closed non-dense sets, and their
limiting sum, which is of the first category, is identical with the continuum;
but this we have shewn to be impossible. Hence the complement of a set of
the first category is not of the 'first category.
A set of the second category has the cardinal number c of the continuum.
This is obvious in case the limiting set G, of a sequence GI, G,... G,...
of non-dense closed sets, each of which is contained in the following one, has
any complementary intervals. We can therefore confine ourselves to the case
in which G. is everywhere-dense; in which case the greatest intervals
* The distinction between sets of the first and second category is due to Baire, Annali di
Mat. (3), vol. II, p. 65,




89]


The first and the second categories


115


1,,,... 6,... which are complementary to G1, G2,... Gn,... respectively,
form a sequence which converges to zero. Taking any complementary
interval A of G1, a number n, can be found such that A contains at least two
intervals complementary to Gn, in its interior, and these we denote by Ao, A1.
Again n2 > n,, can be found such that in each of the intervals A0, A1 are contained at least two intervals complementary to Gn2; those interior to A0 we
denote by A00, Ao,, and those interior to A1 by AJ0, A,0. Proceeding in this
way we obtain a sequence of intervals
\p, lpqv ^pqr *...
each of which contains the next in its interior, and p, q, r,... have definite
values each of which is either 0, or 1. The point interior to all the intervals
of this sequence is a point of Gj which does not belong to G,, unless, from
and after some fixed index, all the indices are alike 0, or alike 1. Therefore
those points of G' which do not belong to G. have a (1, 1) correspondence
with all those numbers between 0 and 1, expressed in the dyad scale, which
do not contain identical digits from and after some fixed one; and it thus
appears that the points complementary to G. form a set of cardinal number c.
Any two sets of the second category have in common a set of points which
is also of the second category.
If G, is the limit of Gn, and H, is the limit of H, where G,, Hn are the
nth sets of two ascending sequences of non-dense closed sets, then the set
M(Gn, H,,), which is also closed, has for its limit a set of the first category.
But the complement of this set is the set common to the two sets of the
second category which are complementary to G,, H.; and this common set
is itself of the second category.
The definitions of sets of the first and of the second category can be
extended to the case in which all the sets concerned are contained in a perfect
set H, which takes the place of the continuous interval (a, b).
If G1, G2,... Gn,... are closed sets all contained in H, and each one nondense in H, then the set Mf(G, G2,... G,,...) is said to be a set of the first
category relatively to the perfect set H, and its complement relatively to H
is said to be a set of the second category relatively to H.
The perfect set H may be non-dense in the continuum; or it may contain
continuous intervals, finite or indefinitely great in number.
It will appear, from the theory of order-types developed in the next
chapter, that the points of any perfect set can be made to correspond
uniquely with the points of a continuous interval (a, b), in such a manner
that the relative order of two points of the perfect set is the same as the
relative order of the corresponding points in the continuum, the end-points
of a complementary interval of the perfect set corresponding to one point of
8-2




116


Sets of Points


[CH. II


the continuum. To a closed set non-dense in H, there corresponds a closed
set non-dense in the continuum; and a set of the first or second category
relatively to H corresponds to a set of the first or the second category, respectively, in the continuum. It thus appears that the properties of sets of
the first and of the second categories in the continuum, which have been above
established, can be immediately extended to the case of sets of the first and
second categories relatively to any perfect set H.
This is a particular case of the general property of any perfect set H
considered as the domain in which sets of points are defined; viz. that II
plays the same part relatively to such sets, as a continuous interval does
relatively to sets of points defined in it.
EXAMPLES.
1. Let Pi, P2, P3,... P,,... be an enumerable set of points in an interval (a, b); the
set may be everywhere-dense in (a, b). The finite sets
(P1), (Pl, P2)(,      3) (Pl, Pl. ( P2,... Pn)...
are each closed, and the given set is the limiting set, which is therefore of the first category.
The remaining points of (a, b) form a set of the second category.
2. Denoting the points of the interval (0, 1), as in Ex. 5, ~ 75, by
x= — + t    -2..  l2.. In "
where cs< kn; let the fixed integers k, k,... k,,,... form a sequence which increases
without limit. If* al, a2,... a.,... is any sequence of positive integers which increase
without limit, let G, denote the set of those numbers x, which are such that the integers
cl, C2,... c,... are all < a. The sets G,, G2, G3'... G,... are a sequence of perfect sets,
each one of which contains the preceding ones; the set G, is then a set of the first category.
3. The numbers of the continuum (0, 1) may be divided into sets, of the first and the
second categories, in the following manner:-All the numbers in (0, 1) may be expressed
as endless decimals; the finite decimals being therefore not used. Let t the set H consist
of all those numbers in which the digit 9 occurs only a finite number of times, and of
those numbers also in which, from and after some place, all the figures are 9. The complementary set K consists of all those numbers in which 9 occurs an infinite number of
times, except those in which every figure is 9 from and after some place. The set H is
the limit of a sequence of non-dense closed sets H1, H2,... H,,... each of which is of
cardinal number c. For, let H, consist of the numbers of the form 'abc... k999..., in
which every figure is 9, after some fixed place, and in which none of the figures a, b, c,... k
is 9; together with those decimals in which no figure is 9. No number of the set H1 can
lie within the interval ('abc... k899..., 'abc... k999...) which is therefore a complementary
interval of the set. The set H, may be taken to consist of the numbers of the form
'abc... hk999..., in which k is not 9, and not more than n-l of the figures a, b, c,... A,
are 9; together with those decimals in which 9 does not occur. That each of the sets H,
is of cardinal number c, follows from the fact that it contains all the decimals in which 9
* Broden, Math. Annalen, vol. LI.
t See Schonflies, Bericht fiber die Mengenlehre, p. 106.




89, 90]


Sequences of closed sets


117


does not occur; and these, if interpreted in the scale of 9, represent all the numbers of the
continuum (0, 1). The set H is everywhere-dense, since it contains that everywhere-dense
set of numbers in which every figure is 9, after some place. The set K, being of the second
category, is also everywhere-dense, and of cardinal number c.
4. The following method of dividing the continuum (0, 1) into two portions, each of
which is everywhere-dense, and of cardinal number c, has been given by Broden*':-Let
10 +    1+2+... + l,+... denote a divergent series of positive numbers, such that the limit
of I,, as n is indefinitely increased, is zero. Let a be a positive number <1, and let
nh, n2,... n,,... be a sequence of increasing positive integers. It is possible to choose the
divergent series so that each of the ratios In2/l,, 1,3/1n,... is <a: if this be done, the
series  In1, is convergent, its sum being < 1-t. Each of the series obtained from
=l=                                1 -Go
Inc,, by leaving out a finite number of terms, is also convergent. The convergent series
t=1
so obtained, form an unenumerable set: for they are obtained by multiplying the terms of
the series 2 lI,, each either by 0, or by 1; and thus there is a series corresponding to each
L=1
fractional number expressed in the dyad scale. Corresponding to each convergent series,
there is a divergent series which consists of 10+ 1 +... + ++..., with the convergent series
removed from it. We obtain in this manner an unenumerable set of divergent series. The
convergent and divergent series, each of which consists of terms of 10+lj+-...- +  +-...,
may now be correlated with the numbers of the continuum (0, 1). Let these numbers be
expressed in the dyad scale, in the form 'a1a2a3..., where every a is 0, or 1, and the
case in which every figure is zero after some place, is excluded. To one of the series
p+lq+lr+"..., we may take that number in which ap, a,, a,.,... are all 1, and the
remaining digits 0. The points of (0, 1) are thus divided into two classes; one of these
consisting of all the numbers which correspond to convergent series, and the other of those
corresponding to divergent series.
DIMINISHING SEQUENCES OF CLOSED SETS.
90. A closed set may be either
(1) a non-dense closed set defined, as we have shewn, as the end-points
of an everywhere-dense set of non-overlapping intervals, together with the
limiting points of the end-points, or
(2) a finite number of non-abutting closed intervals, or
(3) the set obtained by adding to a non-dense closed set the internal
points of some of the complementary intervals.
The sets (2) may be regarded as the particular case of (3), which arises
when the non-dense closed set is a finite one.
The closed sets here considered will be taken to be of any one of the
types thus indicated.
Let P1, P2, P,... Pn,... be an unending sequence of closed sets, each one
of which contains the one which succeeds it; then it will be shewn that a


* Crelle's Journal, vol. cxvmII, p. 29.




118


Sets of Points


[CH. II


set P, exists the points of which are contained in every one of the closed
sets, and that this set P, = D (PI, P,... P,,...) is itself a closed set, which
may however contain only one point or a finite number of points.
To prove this theorem, suppose (a, b) divided into a finite number of parts;
then in one at least of these parts (a,, b,) there must exist points which
belong to Pn for every value of n; for otherwise the sequence would be a
finite one. Dividing (a,, b,) into a finite number of parts, in one (a,, b2) at
least of these there are points which belong to every Pn. Proceeding in
this manner, and choosing the mode of division so that (an, bn) converges to
the limit zero, the point which is in the interior of all the intervals
(a2, b1), (a2, b2),... (an, bn)...
is a point which belongs to every Pa, and is therefore a point of Pa,; thus
P, contains one point at least.
To shew that P, is a closed set, let p,, 2,... pr,... be a sequence of
points in it which has p for its limiting point.  Then p, p,,... p,.,... are all
points of Pn whatever n may be; and since Pn is closed, p is a point of Pn.
This holds for every value of n, hence p is a point of P,; which establishes
the result.
The theorem   is a generalization of the results of ~ 63. In fact, if P,,
P2,... Pn,... are taken to be the derivatives GM0, G(2),... GI,... of a set G,
the existence of the closed set G(w) follows from the theorem.
Again we may take P, = G (a, P2= G(a),... P, = G(a),... where ac, 2,...,)... is any sequence of numbers of the second class, of which /3 is the
limiting number. The existence of the closed set G(s) then follows from the
theorem.
91. Let us now suppose that, for the sequence P,,... P,,... of closed
sets, each one of which contains the next, a positive number C exists such
that the content In of Pn is, for every n, greater than C. It will then be
shewn* that the content 1. of P, is   0C.
In order to establish this theorem, the following lemma is required:If Gi, 02 be two closed sets with contents 7I, 12 respectively, then the set
D (GI, G2) of points common to Gi, G2 is a closed set of content I'; and the
set M (Gi, G2) of points belonging either to Gi or to G2, or to both, is a closed
set of content I", where ' + " = I1 + I2.
That D (Gi, G2) is closed, follows from the fact that any limiting point of
it must be a limiting point both of Gi and of G2, and therefore belongs to the
set. That M(Gi, G2) is closed, follows from the fact that any limiting point
of it must be a limiting point of one at least of the sets GI, G2. If the points
* This theorem was given by W. H. Young, in his paper on "Open sets and the theory
of content," Lond. Math. Soc. Proc. Ser. 2, vol. II, p. 25.




90-92]


Diminishing sequences of closed sets


119


of G, be removed from the set MJ(G0, G2), the remainder is a measurable set
of measure I" - I, in accordance with the theorem of ~ 82. But the set so
obtained could also be obtained by removing from G2 those points which
belong to D (G?, G0); hence the measure of the set is also I2-I'. Therefore
it follows that I + I" = I1 + I2.
Let us now suppose that, if possible, the set P, has its content I, less
than C. Now the set P, - P, is measurable, and has for its measure I1 - 1,;
hence P, - P, contains as component a closed set Q1 of content greater than
I - I, -, where e is arbitrarily small, and is taken < C- I,.
This closed set Q, has, in accordance with the lemma, in common with
P2, a closed component Q2 whose content is > 21 + I1-  -  - J, where J
is the content of the set of all points belonging to the closed component Q1
of P1 only, or to P, only, or to both of these; and it is clear that J k I'1.
Therefore the component Q2 of P2 has its content greater than the positive
number I2-I,, - e, and is itself contained in the component Q1 of P1, of
which the content is greater than I - 1 - e. Proceeding in a similar
manner, we obtain closed components Q,, Q4,... of P3, P4,... P,..., each
of which contains the next, and none of which contains points of P,. Now
these closed sets Q1, Q2,... Qn,... have, in accordance with the theorem of
~ 90, at least one point in common; hence Pi, P2,... have a point in common
which does not belong to P,; and this is contrary to the definition of P0.
THE COMMON POINTS OF A SYSTEM OF OPEN SETS.
92. If*, G,  -G2,...... be a sequence of sets of points, such that each
set Gn contains the next G,,+,, and if the interior measure of each set is
greater than some fixed number C, whatever n may be, then the set Go of points
common to all the sets has an interior measure _ C.
It will be observed that the sets are not assumed to be measurable.
Let m, (G0,) denote the interior measure of a set G,; then, by ~ 81, closed
components P,, Q2, Q,...,... of the sets G,, G2,... can be found such
that the content of P1,
1
1
I(Pi) >   (m,(?)-(,1
I (Q2) > m, (G2) - I,
I(n)>m()..........................
1n


* W. H. Young, loc. cit. p, 25.




120


Sets of Points


[CH. II


where e is an arbitrarily small positive number. The set Q2 has a closed
component P2 which is also a component of P1, of content
I (P2) = I (Q2) + I (P1)O-I 7{M (Pi, Q2)} - I (PI) + I (Q2)-m, (G,)
1
>  (Q2)-2  e
2
> m, (AG) - (2 +)
Next take that closed component of Q2 which is also a component of P2;
it can be shewn as before that
(P3)> m, (G3)- (I+    +) 
and so on. We have now a sequence of closed sets P1, P2,... Pn,... each of
which contains the next, and such that the content of each of them is
> C-e; therefore the set P, of points common to all these, has its content
> C - C, and P, is a component of G,. It follows, since e is arbitrarily small,
that the inner measure of G, cannot be less than C.
93. We are now in a position to establish the following general
theorem:If*, G, G,... Gn,... is a sequence of sets of points, each of which sets is
a component of a closed set of finite content 1, and if the interior measure of
each of the sets G1, G2,... G,,... is greater than a fixed number C, then there
exists a set of points of interior measure? C, and of the power of the continuum, such that each point of the set belongs to an infinite number of the
given sets.
The conditions of the theorem are satisfied if all the sets lie in the
same finite interval of length l; also the sets are not assumed to be
measurable ones.
Choose a closed component of each of the given sets, of content > C;
let these components be Q, Q2,... Qu,.... Choose an integer m such that
mC - I < (m + 1) C, and let us consider the first n (> m + 1) of the sets
Q1, Q2.... The points common to any pair of these closed sets form a closed
set, and the set which contains all the points which belong to at least two of
the n closed sets is also a closed set Q1,n of content I,. Those points of Q1,n
which belong to Q, form a closed set of content - In, hence there is a set of
points of Q1, of measure  I (Q1) - I,, which do not belong to any of the sets
Q2, Q3,... Qn; and the measure of this set is > C - In. Similarly each of the
sets Q2, Q3,... Qn has a component of measure > C - I, consisting of points
which do not belong to any of the other sets, or to Q1. The measure of all


* W. H. Young, loc. cit. p. 25.




92, 93]


Common points of open sets


121


these sets added together is > n (C - Ihn); and it must be less than 1, since
the sets do not contain any points common to two of them, and they are all
enclosed in a set of measure 1. Hence
n (CG-In)< (m + 1) C, or,, > (1      + ) C.
It has thus been shewn that the closed set Ql,, has the power of the continuum, since its content is proved to be positive; and this holds for every
value of n which is > mz + 1. Considering now the next n sets Qn+1, Qn+2)...
Q2,n there is a closed set of content > (1 -  l + ) C, consisting of points each
of which belongs to two at least of the sets; and a similar result holds for
each system of n sets Qrn+l, Qrn+2,... Q(r+l)n.
We have now an infinite sequence of closed sets Qi,n, Q2,n, Q3,n  Qr,...,...
each of which has content > (1- m +  ), and the points of each of them
belong to two at least of the given sets. By applying similar reasoning, and
taking n' sets at a time, we see that there are an infinite number of sets each
of content > (1-  -+  ) (1- m+   ) C, and such that each point of any one
of them  belongs to four at least of the given sets. Proceeding in this
manner we obtain sets of points, each of content
> nml (I -   I (                    +) C+,
n            n           n ')  n(  )    ( 8, /
and such that each point of each set belongs to at least 2S+1 of the given
sets. Now let n, n',... n() be so chosen, that
m+1     1   m+l    1     m+1      1
n     2'       < 4        (s) <2s+l;
then the content of each of the sets which contains points belonging to 2S+1
at least of the given sets is
>C (    2 )(-I     )   (-        ) > C... +  2 +)}
>   ( l-e).
The process can be carried on without limit; and we see that the set
which consists of all points belonging to 2s+1 at least of the given sets contains closed components of content > C(1 - ). Considering the sequence
PI, P2,... of sets such that P1 contains all points that belong to two at least
of the given sets, P2 contains all points that belong to 22 at least of the
given sets, and so on, it is clear that Pi contains P2, and P2 contains P3, etc.
But the interior measure of each set is > 0(1 - e); hence, in accordance with
the theorem of ~ 92, there exists a set of points common to all the sets




122


Sets of Points


[CH. II


P1, P2,... of interior measure _ C(1 - e). This set consists of points which
belong each to an infinite number of the given sets; and its interior measure
is _ C, since e is arbitrarily small. The set has the power of the continuum,
since it contains closed components of content greater than zero.
The theorem that has been now established is of considerable importance
on account of the applications of it which can be made in various parts of
the theory of functions; it is due* to W. H. Young. That particular case
of the theorem in which the sets are all measurable was first statedti, without
proof; by Borel.
An important case of the theorem arises if we suppose each of the sets
to consist of a finite, or an enumerably infinite, set of closed intervals; in
which case the sets are all measurable. The theorem may then be stated as
follows:If there be given an infinite number of sets of intervals, in a finite
segment, each set consisting of a finite, or enumerably infinite, number of nonoverlapping intervals, and if the measure of each set of intervals is greater
than some fixed positive number C, then there exists a set of points having the
power of the continuum, and of interior measure _ C, such that each point of
the set belongs to an infinite number of the given sets of intervals.
This theorem contains the completion, and generalization, of a theorem due
to Arzela' which is stated by him as follows:Let y, be a limiting point of any set of numbers (y), and let
Go = (yi, y2/2, *)
be a sequence of numbers of (y) which converges to the limit yo. Assuming
the variables to be orthogonal coordinates of a point in a plane, let the set of
straight lines y= y, y  y2, y =  3,..., be drawn, and let a set of intervals
be taken on the portion of each of these straight lines which is in the interval
(a, b) of x. Suppose that each set of intervals is finite in number, and that
this number is variable from one straight line to another, but increases
indefinitely as the index in ys increases indefinitely. Let the sum of the
intervals 81,s, 2,s, *...* 8, on the line y =  s, be ds. If for every value of s,
ds is greater than C, a determinate positive number, there necessarily exists
at least one point x0 in the interval (a, b), such that the straight line x = xo
intersects an infinite number of the intervals 8.
Arzela subsequently removed the condition that each set of intervals is a
finite one.
* Proc. Lond. Math. Soc. Ser. 2, vol. ii, p. 26.
t Comptes Rendus, December 1903.
Rend. dell' Ac. dei Lincei (4) 1, (1885), p. 637; a second proof, which is however not rigorous,
has been given by Arzela in the Memorie della R. Acc. d. Sc. di Bologna, Ser. 5, vol. vii, 1899.




93, 94]


Sets in General


123


THE ANALYSIS OF SETS IN GENERAL.
94. It has been proved that a closed set can always be analysed into an
enumerable set and a perfect one, either of which may be absent; we now
proceed to consider the case of a set which is not necessarily closed. Before
doing so, it is necessary to classify the points of a set, according to the cardinal
number of those points of the set which are contained in the immediate
neighbourhoods of such points*.
An isolated point of a set G is such that in a sufficiently small neighbourhood of the point there are no other points of G. For this reason an isolated
point may be said to be of degree zero in the set.
A point P, which is a limiting point of G, and is such that in a sufficiently
small neighbourhood of P there is an enumerable set of points of G, is
said to be a point of enumerable degree in the set, or of degree a in
the set.
If a limiting point P of the set G is such that in any neighbourhood of
P, however small, there is an unenumerable set of points of G, the point P is
said to be of unenumerable degree in the set G.
In case the point P is such that, in every neighbourhood of it, the cardinal
number of the points of G contained in such neighbourhoods is c, the point
is said to be of degree c in the set.
It is not definitely known whether cardinal numbers exist which lie
between a and c; but if any such cardinal number x exists, a point would be
of degree x in the set, if a neighbourhood of P exists such that in that
neighbourhood, and in every smaller neighbourhood, there is contained a
part of the set which is of cardinal number x.
The points of unenumerable degree consist of all the points whose degrees
in the set are greater than a.
If a set G contains no point which is of unenumerable degree in the set, the
set is enumerable or finite.
If P be any point of the set, an interval which contains P can be found,
such that the part of G contained in this interval is enumerable; and the same
holds for any point Q of G which is not contained in the interval round P.
In this manner we can proceed until we have a non-overlapping set of
intervals which contain all the points of G. Since these intervals form an
enumerable or a finite set, and in each of them there is a finite or enumerable
part of G, it follows that G is an enumerable set.
* The analysis here carried out, of sets in general, was given by Cantor, Acta Math. vol. VII.
A more elementary presentation of the subject has been given by W. H. Young, Quart. Journ. of
Math. vol. xxxv, 1903.




124


Sets of Points


[CH. II


It is clear that, if G contains even one point of unenumerable degree in
the set, then the set is unenumerable.
The part of a set G which consists of points of unenumnerable degree is
itself unenumerable, and forms a set which is dense-in-itself
For the points of G which are of degrees 0 or a form an enumerable
set; this being seen to be the case if we remove from G the points of unenumerable degree in the set. If the points of unenumerable degree could
form a finite or enumerable set, then the whole set G would be enumerable,
which cannot be the case.
To prove that the set of points P, of unenumerable degree in G, is densein-itself, we have to prove that in every arbitrarily small neighbourhood of P
there exist points of the same kind. In an arbitrarily small neighbourhood
of P, there is an unenumerable set of points of G, and these cannot all be
points of degree 0 or a in the set, for there are only an enumerable set of such
points altogether. Hence, in any arbitrarily small neighbourhood of P, there
are points which are themselves of unenumerable degree in G; therefore P
is a limiting point of the set of those points which are of unenumerable
degree in G.
It follows from this theorem that every set G which is unenumerable
contains a component which is dense-in-itself. An enumerable set may or
may not contain a component which is dense-in-itself; but if it does contain
such a component, its derivative contains a perfect set, and thus the enumerable set is irreducible.
The more general theorem can be established that, if G is an unenumerable set, those points of G which are of the same degree x ( a) in G
form a set of which the cardinal number cannot exceed x, and which is densein-itself
If Pz be such a point, we can include Px in an interval such that x is the
cardinal number of the points of G in the interval. Doing the same with any
other point QZ of the same kind which is not in the interval round P, and
proceeding until all such points are in intervals, we have a finite, or
an enumerable, set of non-overlapping intervals, in each of which is a set of
points of G of cardinal number x, and all the points of degree x are included
in these intervals. It will appear from the general theory of cardinal numbers,
that the cardinal number of all the points of G included in this enumerable
set of intervals cannot exceed x; hence the set of points Px cannot have a
cardinal number greater than x.
To shew that the set is dense-in-itself, we observe that, in a sufficiently
small neighbourhood (a, /) of P, there is a set of points of G, of cardinal
number x; and that none of these points can be of degree in G higher than x.
For if there were such a point Q of degree higher than x, in some interval




94, 95]


Sets in General


125


(a',,/') contained in (a, 3) and containing Q in its interior, there would be
a set of points of G of cardinal number higher than x; but this is impossible,
as all these points would be in (a, /). Again, the points of G in (a, /) cannot
be all of degree lower than x; for if they were so, their cardinal number
would be lower than x, which is contrary to the hypothesis. Moreover,
since in any arbitrarily small neighbourhood of Px there are points of the
same degree x in G, PR is a limiting point for such points. Therefore the
set of points such as Px is dense-in-itself.
The set of points of degree c is dense-in-itself, and of cardinal number c.
95. Any set G consists of isolated points which form an enumerable
set called the adherence of G, and of limiting points which form a set called
the coherence of G. Denoting the adherence and the coherence of G by Ga,
Gc respectively, we have G = Ga + Gc.
The set Gc can in a similar manner be split up into its adherence and its
coherence, which we denote by Gca and Gc2 respectively; thus
Gc = Gca + Gc2.
The set Gca is an isolated set, and therefore enumerable; and if we proceed
to resolve Gc2 in a similar manner into its adherence Gc2a, and its coherence
Gc3, and then to resolve Gc3, it is clear that the process may be continued
any number n of times. We thus obtain
G = Ga + Gca + Gc2a +... + Gc' —la + Gcn.
The set Gcn-la may be named the adherence of G of order n, and Gc' may be
denominated the coherence of G of order n.
It may happen that, for some value of n, Gcn vanishes; in that case G
has been split up into a finite number of enumerable sets, and is consequently
itself enumerable. If this be not the case, the process may be continued
indefinitely, and Gcn then exists for every value of n. We then define
D (G, Gc, Gc2,... Gc',...),
the set of points common to all the coherences of G, to be the coherence of
order o, and denote it by Gc*. It is clear that every point of G which does
not belong to one of the sets Gcn-la, belongs to Gcw, hence we have
G =  Gc-la + Gc,
the summation being taken for all values of n.
We now split up Gc" into its adherence Gcwa, and its coherence Gc%+l, and
proceed further to obtain the adherences and coherences of G of the orders
of the various numbers of the second class. If ai, a,,... an,... is a sequence
of numbers of the second class, which has /3 for its limit, the coherence of
order / is defined by
GcO = D (Gca, Gca,,... Gcan,...).




126


Sets of Points


[CH. II


We now obtain a resolution of G of the form
G =  GcPa + Gcy,
where y is any number of the first or second class, and the summation refers
to all values of p which are less than 7. Each adherence GcPa is an isolated
set, and therefore enumerable; and if G contains a component which is densein-itself, this component is contained in Gco.
First suppose G to be an enumerable set; the process of analysis must
then cease for some number y of the first or second class. For if GcPa existed
for every number ry of the second class, we should have obtained an unenumerable set of adherences containing no points in common, and all belonging
to G: thus G could not be enumerable.
The cessation of the process may take place in two different manners:(1) if for some number ry of the first or second class, Gc = 0, G has
been resolved into an enumerable set of adherences, and it contains no component which is dense-in-itself:
(2) if for some number ry,
Gc a = 0,
in which case Gc =- Gc+~1l the set GcY then contains no adherence, and
every point of it is a limiting point, and Gc' is therefore dense-in-itself. The
set G has consequently been resolved into an enumerable component which
contains no part that is dense-in-itself, and into a set which is enumerable
and dense-in-itself.
Next, let us suppose that G is an unenumerable set. Then it has been
shewn that those points of G which are of unenumerable degree in G form a set
that is dense-in-itself; and those points which belong to the adherences of all
orders are points of zero, or of enumerable, degree, and thus form an enumerable set. It follows, since all points that do not belong to that part of G which
is dense-in-itself belong to the adherences, that the number of adherences
must be enumerable; and thus that, for some number 7 of the first or second
class, GcY is dense-in-itself. The set GcY may consist of an enumerable set
dense-in-itself, and of sets of higher cardinal numbers dense-in-themselves.
It has thus been shewn that any set G may be represented by
G= U+ Va,+    v+ VY,
where U is an enumerable set which contains no component that is dense-initself, Va, is an enumerable set of points of degree a dense-in-itself, Vc is a set
of cardinal number c consisting of points of degree c dense-in-itself, Vx is a
set dense-in-itself consisting of points of degree x, where a.< x < c.




95, 96]                 Inner limiting sets                        127
If, as is probable, no cardinal numbers exist between a and c, the sets VT
can be omitted. A set such as Va, Vl, Vc is denominated a homogeneous set
of degree a, x, c, in the set G.
If G is a closed set, then as has been shewn in ~ 73, Vc is perfect, and
YV cannot exist.
INNER LIMITING SETS.
96. Let us suppose that each rational point p in the interval (0, 1) is
enclosed inhere                                has the interval  lue for all
the points. In this manner the rational points are enclosed in a set of over2             1
lapping intervals, whose sum is less than X2 (q- 1) 2, or than 2X2 -, which
can be made as small as we please by choosing X small enough. The
equivalent set of non-overlapping intervals defines, by means of the endpoints and their limits, a closed set {q1}, such that for any point of the set
q - ql -3, for all points P.
4q  *    q3               q
Now consider the set of points defined by
a,   a2    a3        a,n
x =    + +1  +... +  I- +...,
where each a is _9, and the a are such that an infinite number of them are
different from zero. It has been shewn by Liouville that these numbers x are
transcendental. Let
p   _+ a,  a+       a,, thus
q   10  102! +'    10- '   s 
then             -p     n+   -.. < +    a(,,    ) 
q   10 (n+')!  ~n     q     )    n
It follows that, if x is one of the above transcendental numbers, whatever
value X may have, it is interior to an interval (P —.,  + -). For suppose
q    3q  q
p        lx                    1 x   1
q = 10n!; then  P - x < - < X provided X; and, however small
q       q    q      -      - 10(n_3) n! a
X may be, values of n can be found for which this inequality is satisfied. Therefore rational points - can be found however small X may be, such that x lies
within the intervals (_3 -    + -)    It thus appears that, besides the
original points - enclosing which the intervals are drawn, there are other
q




128


Sets of Points


[CH. II


points which lie inside the intervals for all values of X, when X is diminished
indefinitely.
This example, which is due to Borel*, shews that if each point x of a set
be enclosed in a series of intervals 81 (x), 82 (x),... 3n (x),... assigned according
to some prescribed law, and such that the upper limiting value of,,(x) for
all the points of {x} has the limit zero when n is increased indefinitely, the
magnitude and position of S (x) being assigned for each n and each x, then,
in general, there are points which do not belong to the given set {x} remaining
in the interior of the set of intervals {8n (x)}, however great n may be.  It
is clear that every point x' not belonging to {x}, which is in the interior of an
interval of the set [{8 (x)}, for every value of n, must be a limiting point of the
set {x}. For if p is a point which is not a limiting point of {x}, the distances
of p from the points of the set have a definite finite minimum     c; hence,
when n is so great that the upper limit of &3 (x), for a fixed value of n, and
for all points of {x}, is less than c, the point p is exterior to all the intervals
of the set {S (x)}.
The points not belonging to the unclosed set {xj which are interior to the
set {8 (x)) for every value of n, are among the limiting points of the set {x}:
and it will appear that some or all of the limiting points of {x} may have this
property, according to the law of choice of the intervals of the set.
If all the intervals {n (x)} be taken of equal length 2cn, with the x in the
centre of its interval, where Cn has the limit zero when n is indefinitely
increased, then every limiting point of {xn} lies within the set {n (x)}.  For,
however small c,, may be, there are points of the set whose distance from a
limiting point p is less than c,.
If the points of a set G be enclosed in a series of sets of intervals {n (x)},
which are subject to the condition that the maximum of the lengths 6n (x) for all
points of the set has the limit zero, when n is increased indefinitely, then the
set G, together with those points, if any, of the derivative G', not being points of
G, which are within the intervals 1{n (x)} for every value of n, is said to be the
inner limiting sett for the sequence {,n (x)} of sets of intervals.
If G is a given set of points, and it is possible so to choose the sequence of
sets {n (x)}, that no points which do not belong to G remain in the interior
of the intervals [8, (x)} for every value of n, then the set G is said to be an inner
limiting set of points.
It has been shewn above that every closed set is an inner limiting set.
* Legons sur la theorie des fonctions, p. 44.
f This term is due to W. H. Young, who has investigated the properties of such sets,
see Leipziger Berichte, August 1903, " Zur Lehre der nicht abgeschlossenen Mengen." For further
properties see also Proc. Lond. Math. Soc. Ser. 2, vol. I, p. 262.




96, 97]                  Inner limiting sets                       129
If a point which does not belong to an inner limiting set is interior to one
or more of the intervals {8_ (x)}, but is not interior to any of the intervals
{8n (x)}, then that point will be said to be shed from the sequence of sets of
intervals at the index n.
In accordance with the theorem of ~ 66, the set {8n (x)} may be replaced
by a set of non-overlapping intervals {A}, which have the same internal
points as {8n (x)}; and the points which are not interior to {An} or to {8n (x)}
form a closed set. It thus appears that every inner limiting set is complementary to a set of points which is the limit of a sequence Gn of closed sets, such that
Gn is contained in OGn+.  Conversely, every set which is complementary to
G,, the limit of an ascending sequence of closed sets, is an inner limiting set,
the intervals complementary to GO being taken as {AJ}.
In case the closed sets Gn are all non-dense, the set G, is a set of the first
category, and the complementary set is of the second category. Therefore
it follows that every set of the second category is an inner limiting set.
Every enumerable set is a set of the first category, for it may be exhibited
as the limit of a sequence of finite sets; hence the complementary set is of
the second category, and is therefore an inner limiting set.
In the case of any enumerable set {P}, those of its limiting points which do
not belong to {P} form an inner limiting set.
To prove this, let Q be the set of those limiting points of the set
P,, P2,,.n,... which do not belong to that set; then the points of Q
can be enclosed in intervals {81} which do not contain P,; and in the interior
of the intervals {81} a set {82} may be chosen enclosing the set Q, and excluding
the point P,, and so on. Then the sequence of sets {8n} has for its inner
limiting set {Q}; and the only limiting points of {Q} which do not belong to
{Q} are, or may be, the points P1, P2,... Pl,... which have each been shed at
a definite index.
It can easily be seen that an inner limiting set remains such, if a finite
number of points be added to, or subtracted from the set. Also the sum of a
finite number of inner limiting sets is itself an inner limiting set; but this is
not in general true of the sum of an indefinitely great number of inner
limiting sets.
97. It will now be shewn that every inner limiting set is either enumerable or else of the power of the continuum.
Let an inner limiting set P be defined by means of a sequence of sets
{An(), A(2),... An(),...}, each of which consists of non-overlapping intervals.
The set {A4} is measurable, and its measure is
L {A(l) + A,('2) +,,. + A,() +...} = mn,
r=oo


H.


9




130


Sets of Points


[OH. II


where mn, diminishes as n increases. If m, has a limit, when n is indefinitely increased, which is greater than zero, say C, then ran> C, for
every value of n; and thus, in accordance with the theorem of ~ 92, the
inner limiting set has a measure _ C, and therefore contains closed components of positive content; therefore in this case the inner limiting set has
the power c of the continuum.
There remains now   to consider the case in which mn has the limit
zero, when n is increased indefinitely. It is clear that in this case no
interval of {An} can also be an interval of all the sets {An+i}, An+2},...; for, if
it were so, the measure of all these sets would exceed the length of the
particular interval, which is contrary to the hypothesis Lmn = O. Let us
first suppose that the sets {An} are all everywhere-dense in the interval in
which they are all contained; then any particular interval An(r), since it
cannot be an interval of all the following sets, must contain at least two
intervals of one of the following sets a,,,}. Let us denote these two intervals
by do, d1. Applying the same argument to each of the intervals do, d1, each
must contain two intervals of some following set; and thus we have four
intervals doo, do,, dlo, d1l, all contained in An(). Proceeding in this manner,
we have intervals d with indices consisting of every permutation of the digits
1 and 0; and if we consider any sequence, such as dol, do,, cdo0, d,00,o...,
the indices form a sequence of radix fractions expressed in the dyad scale,
each interval containing the next in its interior; for it is clear that at each
stage of the process the intervals in a particular interval may be so chosen
that they have no end-point in common with it. Since a sequence of intervals
can thus be found which corresponds to any irrational fraction expressed in
the dyad scale, and since there must be a point of P in the interior of all the
intervals of such sequence, it appears that in Al(r) there is a set of points of
P which has the same power as the set of irrational numbers between
0 and 1; and that power is c.
Next, let us suppose that the sets of intervals {An} are not all of them
everywhere-dense in their domain; and suppose that the inner limiting set
P contains a part Q which is dense-in-itself, so that the derivative Q' is
perfect. The perfect set Q' may be placed into correspondence with all the
points of the continuum (0, 1) so that the order of corresponding points in
the two sets is the same; and to each point in the second continuum there
corresponds a single point of Q', except that the end-points of an interval
complementary to Q' correspond to a single point in the second continuum.
The points of Q correspond to points of a set Q1 everywhere-dense in the
second continuum; and those intervals of the set {An} which contain points
of Q, correspond to intervals of a set {An'}, which is everywhere-dense in the
second continuum.
Those points of the second continuum which are interior to all the sets




97, 98]


Inner limiting sets


131


{A'/}, form a set of power c, as has been shewn above; it therefore follows
that the set Q has also the power of the continuum. For the points interior
to all the sets {A'} are all either points of Q1, or else points which correspond to the end-points of intervals complementary to Q', and these latter
form at most an enumerable set.
The following theorem has now been established:An inner limiting set has the power of the continuum if it contains a
component which is dense-in-itself, and if it contains no such component it must
be enumerable. Its measure is the lower limit of the measures of the nonoverlapping set of intervals by which it is defined.
For the only sets which contain no component dense-in-itself are enumerable; and it has been shewn that an inner limiting set which contains such
component has the power of the continuum.
It thus appears that an enumerable set, which contains a component
which is dense-in-itself; cannot be an inner limiting set.
98. It will now be shewn that every enumerable set, which contains no
component that is dense-in-itself, is an inner limiting set.
Let P be an enumerable set, and let us first suppose that the derivative
P' is also enumerable; then in this case P contains no component dense-initself, for the derivative of such a component would be perfect, and would be
a component of P', which is impossible when P' is enumerable. Divide P
into two parts PI and P,; and of these let PI consist of those points which are
not limiting points of the set P' -D (P, P'), composed of those points of P'
which do not belong to P; while the other part P2 consists of those points which
are limiting points of P'- D (P, P'). Since all the limiting points of the
enumerable set P' - D (P, P') which do not belong to the set itself belong
to P,, the set P2 is, as has been shewn in ~ 96, an inner limiting set. The
points of P1 not being limiting points of P' - D (P, P'), each point of PI can
be enclosed in an interval which contains no points of P' - D (P, P'); and the
set of intervals thus obtained can be taken as the set {18} of intervals
enclosing Pi. It follows that, since the points of P'-D (P, P') are not
contained in a properly chosen sequence of sets of intervals enclosing the
points of PI, and are each shed at a definite index from a properly chosen
sequence of intervals enclosing the points of P2, the set PI + P2 or P is an
inner limiting set. We have now shewn* that:Every reducible set is an inner limiting set.
Next let us suppose that P', the derivative of the enumerable set P, has
the power of the continuum. If P' contained all the points of any interval
(a, p), P could not be an inner limiting set; for the points of P in (a, f) would
* See Hobson, Proc. Lond, Math. Soc. Ser, 2, vol. II.
9-2




132


Sets of Points


[CH. II


be everywhere-dense in this interval, and would form a set dense-in-itself, which
has been shewn to be impossible. Since P' does not contain all the points in
any interval, and is closed, it can be resolved into the sum of a perfect set G,
and an enumerable set L1, consisting of points interior to the intervals complementary to GI. The set P may be divided into two parts P1 and Q1, where P,
consists of those points which are interior to the complementary intervals of G1,
and Q1 consists of those points which belong to G,: it may happen that Q1 does
not exist. It can be shewn that P1 is an inner limiting set, whether Q1
exists or not. For P1 consists of a series of sets PPn, P12,... P,,... interior to
the complementary intervals (al, bl), (a2, b2),... (an, be,),... of G1; but the set Pyn
in (an, bn) has all its limiting points in that interval, and those belonging to
ti are enumerable; and therefore, in view of what has been proved above, Pn
is an inner limiting set.
The sequence of sets of intervals which enclose the points of P1n may be
so chosen that all the intervals of every set are interior to (an, bn); thus no
limiting points of P not belonging to P1, except those belonging to Pn', are
ever interior to any interval of the sequence assigned to P,,; and as this holds
for every n, it follows that P1 is an inner limiting set, and its points are such
that they can be enclosed in a sequence of sets of intervals which from the
beginning contain no points of G,.
The set Q1 consists of points which belong to G(, and therefore Q, has no
limiting points in L1. If every point of G? were a limiting point of Q1, the set Q1
being dense in G6, would be dense-in-itself; were it so, Q, could not be an inner
limiting set. It follows that Q1 is not dense in GI, and thus Q1' does not contain
all the points of GI. Let Q1' be resolved into an enumerable set L2 and a
perfect set G2: the latter may be absent. The set Q, may then be resolved
into a component P2 contained in the intervals complementary to G2, and a
component Q2 contained in G,; thus P = P1 + P2 + Q2. The same argument
applied to P2, as was applied to Pi, shews that P2 is an inner limiting set;
and the intervals of the sequence which encloses its points may be taken
to be all interior to the complementary intervals of G0. The set Q2 in G2
may be treated as Q1 in G1 was treated, and we thus have Q2=P+ Q3,
where P3 is an inner limiting set, and Q3 is contained in a perfect set G3.
Proceeding in this manner, it may happen that for some integer n, Qn does
not exist, and then P is expressed as the sum of a finite number n of inner
limiting sets, and is itself therefore an inner limiting set. If no integer n
exists for which this happens, we consider the set M (P1, P2,... P,n,...),
where n has every integral value. It may happen that this set contains
every point of P; but if not, we take the set
P-M(PiP2,...Pn...)
and resolve it as before into an inner limiting set P,, and a set Q, contained
in a perfect set G,, but which cannot be dense in G,, since it cannot be dense



98]


Inner limiting sets


133


in-itself. We then proceed to resolve QO into P.+1 and a set Q,,+ contained in
a perfect set G+,,. We proceed further, and may obtain in this manner sets
whose index is any transfinite ordinal number of the second class; and thus
P is resolved into
P + P +... + P,+ P.++... + P + Q,
where / is a non-limiting number of the second class, or else into
P + P2 +... + P+... +P +...
with no last term. Since P is enumerable, this process must come to
an end at, or before, some definite number a of the second class; and the end
can only come, either when there is no component Qa in Ga, or when there
is no Ga.
It has thus been shewn that, when P contains no component that is densein-itself, it can be resolved into a finite, or enumerably infinite, set of inner
limiting sets, of which there may, or may not, be a last set. Let Py be one
of the components into which P has been resolved, y denoting a number of
the first or second class. We now fix on a sequence of sets of intervals enclosing
the points of Py, such that all the intervals are interior to the intervals
complementary to G,; then the set Py+, + P+, +..., which is contained in G,
has no limiting points in any of the intervals which enclose the points of Py,
for all its limiting points must be in Go. The sequence of sets of intervals
having thus been fixed for every P,, we can now shew that each limiting point
p of P, which does not belong to P, is shed from the whole sequence of sets of
intervals, at a definite index. The point p is either a limiting point of P,,
belonging to L1, or is contained in G1. In the former case it is shed from the
intervals enclosing P, at a definite index; and, not being a limiting point of
P2 + P3 +..., it is shed from the intervals enclosing the points of that set, at a
definite index; consequently it is shed from the intervals enclosing P, at a
definite index, the greater of the two former ones. In the latter case, unless p
is in G2 or in P2', it is not a limiting point of P, + P +..., and never comes
into any of the intervals enclosing the points of P1; it is therefore shed at
a definite index. If p belongs to G1, G2,... and to every G before Ga, but is
not in Ga, it may be a point of Pa. In that case it is not a limiting point
of the set Pa+i + Pa+2 +..., and does not come into the interior of any of the
intervals which enclose the points of P1, P2,..., or any P with index less
than a. It is therefore shed, at a definite index, from the sequence of sets
of intervals enclosing the points of P. It has thus been established that:The necessary and sufficient condition that an enumerable set may be
an inner limiting set is that it contains no component which is dense-initself
A corollary to the above proof is that every enumerable set is the sum
of an inner limiting set, and of a set which is dense-in-itself.




134


Sets oj Points


[CH. II


99. Any unenumerable set can, in accordance with the result of ~ 95,
be expressed in the form P = U+ + a +  Vy + Vc; and we observe that if Vc
is absent, the necessary and sufficient conditions that P may be an inner
limiting set are that Va and F- V  should both be absent; this follows from
the preceding results.
If Vc exists, we observe that no point of U + Va, +, VTo can be a limiting
point of YC; for any limiting point of V, must be a point of degree c in the
set P. If Vc is everywhere dense in (a, b) it follows that U + Va + Y2 V  is
absent. The set V, may be non-dense in (a, b), or it may be dense in some
parts of (a, b) and non-dense in other parts.
It will be shewn that Vc is in general made up of a part which is nondense in (a, b) and of a finite, or indefinitely great, number of parts each of which
is everywhere-dense in a particular interval in which it lies. Suppose that an
interval (a, /) can be found in which Vc is everywhere-dense; and let x be a
point in (a, b) such that x  13. Then those values of x for which Vc is everywhere-dense in (a, x), together with those values for which this is not the case,
define a section of all the numbers of the continuum (/, b); and this section
defines a number /3  /3. Similarly we may assign a number a1 < a, so that
(al, /3) is the greatest interval containing (a, /8) which is such that V, is
everywhere-dense in it. If, in the parts of (a, b) external to (a,, /3i), the set
V, is dense in any interval, then we proceed to fix the greatest interval for
which it is everywhere-dense. In this manner we obtain a finite, or enumerably infinite, set of detached intervals contained in (a, b), in each of which Vc
is everywhere-dense; and the remainder of (a, b) may consist of a set of
detached intervals and of a set of points. In this remainder the points of Vc
form a non-dense set.
No point of U + Va + E Vx can be in an interval (al, 8/3) in which Vc is
everywhere-dense. If Vc is the part of Vc which is non-dense in (a, b), every
point of U  V+ Va +  V, must lie in one of the intervals complementary to the
perfect set V,'. It is to be observed that in Vc are included the end-points of
the intervals (a1, /3), in case those end-points belong to Vc.
In order that P may be an inner limiting set, it is necessary that the
part of U + Va + Z V,, which is in each interval complementary to V', should
be an inner limiting set; and this cannot be the case unless Va and  V, are
absent.
It has thus been shewn that:
In order that an unenumerable set of points may be an inner limiting set, it
is necessary that the set should contain no points whose degrees in the set are
other than 0, a, or c, and that it should contain no component which is densein-itself, and of which the points are of degree a in the set.




99, 100]                Inner limiting sets                       135
The determination of the necessary and sufficient conditions that any given
unenumerable set of points, however defined, may be an inner limiting set has
now been reduced to the problem of determining the criteria for the case of
a set which is dense-in-itself and all the points of which are of degree c in
the set. The case in which the latter set is non-dense in its domain may be
reduced, by the method of correspondence, to that in which it is everywhere
dense; and the problem is therefore reducible to that of determining the
conditions under which a given everywhere-dense set of points all of degree c
in the set may be a set of the second category. No investigation of all the
possible types of such sets has yet been carried out, and therefore the problem
remains as yet unsolved. A set which is everywhere-dense in (a, b), and of
which the points in every sub-interval have the power of the continuum,
may be of the first category, and thus not be an inner limiting set; or it
may be of the second category, and therefore be an inner limiting set. The
question has been raised by Schonflies* whether every such set is necessarily
either of the first or of the second category; this question must certainly
be answered in the negative. For, if we divide (a, b) into any finite number of
parts, and place in them alternately inner limiting sets which are dense and
of power c, and dense sets of the first category and of power c, it is clear
that the whole set so constituted cannot be either of the first or of the second
category. The outstanding question as to the criteria that such sets may be
of the second category, is of considerable importance in relation to the Theory
of Functions.
NON-LINEAR SETS OF POINTS.
100. Most of the properties of linear sets of points can be extended
without essential modification to the case of sets of points in two, three, or
more dimensions; and those respects in which sets of points in more than one
dimension differ, as regards the formulation of their properties, from linear
sets are sufficiently exemplified by the case of plane sets. It will therefore be
sufficient, for the purpose of indicating the principal properties of non-linear
sets, to confine our account to the case of plane sets.
Each point (x, y) of a plane set is defined by the two numbers x, y
which are the rectangular Cartesian coordinates of a point. A set which
extends over the whole plane may be made to correspond with the points of
a set which lies in a finite rectangle; this correspondence may be made by
7rX         VrY
means of the relations x= tan 2, y = tan —, when X, Y are each
restricted to have values between + 1 and -1. We shall consequently
assume that the plane sets under consideration consist of points lying in a
finite rectangle whose sides are parallel to the axes of coordinates.
* See SchSnflies, Gottinger Nachrichten, 1899, p. 282, also Bericht fiber die Mengenlehre, p. 81.




136


Sets of Points


[CH. II


In the case of plane sets, a rectangular area whose sides are parallel to
the coordinate axes, plays the same part as a linear interval in the case of
linear sets. A set contained in such a rectangle is said to be bounded.
Corresponding to the fundamental principle that a series of intervals,
each of which contains the subsequent ones, has one point interior to all the
intervals, provided that the lengths of the intervals converge to zero, we have
the principle that the points interior to a set 1,, $.,... 8,,... of rectangles
each of which contains the next, consist of a single point or of a linear
interval, according as both, or only one, of the pairs of sides of the rectangles
have the limit zero, when n is indefinitely increased.
The theorem, that every infinite bounded plane set has at least one limiting
point, is then proved by dividing the rectangle, in which the set is contained,
into a finite number of parts by means of lines parallel to the axes. At least
one of the resulting rectangles must contain an infinite number of points of
the set either in its interior or on its boundary; choosing such a rectangle, we
proceed to divide it as before into a finite number of parts, and continually
apply the same argument; in all these rectangles, there is at least one point
which must be a limiting point of the given set, since we may choose the
mode of subdivision so that both pairs of sides of the rectangles have their
limit zero. In any rectangular area whatever, which has a limiting point P
of the set in its interior, there are an infinite number of points of the set.
A plane set is everywhere-dense when points of the set lie within every
rectangle, with sides parallel to the axes, which can be drawn in that rectangle
in which the set lies.
A plane set is non-dense when in every such rectangle another can be found
which contains no points of the set.
The definition of the successive derivatives of a plane set, and the proof
that all these derivatives are closed sets, is on exactly the same lines as in the
case of linear sets.
101. The frontier of a set of points G in plane space or space of any
number of dimensions, being defined, as in ~ 84, to be the set of points each of
which belongs to one of the sets G, C (G), and is a limiting point of the other
set, it will be shewn that*:If the complementary set C (G) exists, then the frontier of G and C (G)
always exists, and is a closed set.
Let P be any point of G, and P' a point of the complementary set C (G),
and consider those points of G which are on the straight segment PP', i.e.
those points of which the coordinates are  +, '  +     where (x, y) and
* Jordan, Cours d'Analyse, vol. i, p. 20.




100-102]


Non-linear sets


137


(x', y') are the coordinates of P and P' respectively, and k denotes a positive
number (including zero). The linear set of points of G on PP' has, in accordance with the theorem of ~ 46, an upper boundary Q. This point Q which
may coincide with P, is a point of the frontier of G and C (G); for if Q is a
point of G it is also a limiting point of C (G), and if it is a point of C(G), it
is a limiting point of G. Therefore, if C(G) exists, there is always a frontier
of G and C(G). Again let Q1, Q2... Q,... be an infinite set of points of
the frontier; this set has at least one limiting point Q. Such a point Q is
itself a point of the frontier; for, in the set {Qn, there is an infinite number
of points all of which belong to G, or all to C(G), of which Q is the limiting
point. If these points all belong to G' and to C(G), then Q belongs to
G' and to C{(G)}'; if they belong to G and to {C(G)}', then Q belongs to G'
and to IC(G)}'. In either case Q is a point of the frontier; and thus, since
every limiting point of the frontier belongs to it, the frontier is a closed set.
If all points of the plane belong to the frontier of G and C(G), then G has
no interior points. If every point of C(G) belongs to the frontier, then there
are no points exterior to G.
102. If (x, y) and (x', y') are two points P, P', then the positive number
{(x -  )2 + (y - y')2} is said to measure the distance* of P from P'.
If P is a point of a set G,, and P' a point of another set G2, then the
distance PP' has either a lower limit or a lower extreme value, for all pairs
of points of the sets GU, G2. In case this lower limit, or lower extreme, is a
positive number A (> 0), the sets G1 and G2 are said to be detached from one
another.
If two bounded and closed sets G1, G2 are detached from one another, they
contain at least one pair of points P, P' such that their distance from one
another is measured by A.
For let e, 62,... en,... be a sequence of decreasing positive numbers
converging to the limit zero. A pair of points P1, P/' of G1, G2 can be
determined, such that PP?'2 < A2 + e1; again a pair P2, P2' can be determined,
such that P2P22< A2 + e2, and, in general, a pair Pn,,Pn' of points can be
determined, for which PnPn'2 < A2 + en. If (x,, Yn) and (xn', Yn') are the
coordinates of Pn, Pn', the coordinates (Xn, y,, xn, yn') determine a point p,
in the four-dimensional continuum. The set of points p,,,... pn,... has at
least one limiting point (x, y, x', y'); let P, P' denote the two points (x, y),
(x', y') in the two-dimensional domain. It will be shewn that P, P' belong to
G1, G2 respectively, and that PP' is measured by A. A number m, can be
found such that x- x,, y - yn, a' -  y', y'-yn' are all numerically less than
an arbitrarily chosen positive number s7, provided n _ m; it follows that P is
* Instead of the distance so defined, Jordan employs, in this connection, the " cart," defined
as Ix-x'+y I -y'.




138


Sets of Points


[CH. II


a limiting point of the set P1, P2,... P,..., and that P' is a limiting point
of the set P1', P2',... P,'.... Since these sets belong to the closed sets G2, G2
respectively, it follows that P belongs to G1, and P2 to G2. We have, further,
0 -X |     - I -n | + I|n-n I  + I x n - $  2rX- +  [| n-n | n-,
and similarly, l y - y' < 2r + yn-Yn'l, for n _ nm. From these inequalities,
we see that (x- x')2 + (y- y')2< 8r12+ Ar +PnP,'2, where A is some fixed
number; hence PP'2 < 8?2 + A] + C,+ + A2, and since v, en are both arbitrarily
small, it follows that pp'2 _ A2; and thus PP', which is certainly not less
than A, must be equal to A. The theorem has thus been established.
103. A bounded and closed set of points is said to be connex or singlesheeted (d'un seul tenant), when it cannot be decomposed into two or more
detached closed sets.
If P, P' are any two points of a connex closed set G, then if e is any positive
number whatever, points pi, p2, p,... pn can be determined, all of which belong
to the set, and are such that the distances PpI, p1p2, P2ps,... ppP' are all - e;
and conversely, if this condition is satisfied, then G is connex.
The condition stated in the theorem is sufficient to ensure the connexity
of the set G. For if G can be divided into two separated closed sets G6, G2,
such that A is the lower limit, or the lower extreme, of the distances of pairs
of points of G,, G2, we may choose e to be <A. If P is a point of G,, and p, is
a point such that Ppi < e, the point p, belongs to GI; again if p2 is a point such
that pIp2 < e, P2 also belongs to G?, and so on. Since Pn belongs to G0, whatever finite value in may have, it is impossible that pP' < e, because pP' > A.
Again the condition is a necessary one. For let us suppose that, for some value
of e, the condition is not satisfied for every pair of points. If P be a point
belonging to such a pair, the set G may be divided into two parts G, and G2,
where G, is such that, for each point P' belonging to it, a definite set of points
of G1, viz. p1, p2,... Pn, exists such that Pp1, pP2,... pnP' are all i e, and G2
is such that for each point of it this condition is not satisfied. The two sets
GI, G2 are closed, and are such that the lower limit, or the lower extreme, of
the distance between pairs of points in them is > e. For if p is a limiting
point of G0, it belongs either to G1 or to G2; and since there are points pn of
G0, such that ppn < e, the point p clearly belongs to G,; therefore G, is a
closed set. Again if q is a limiting point of GQ, it cannot belong to G1; for a
point P' of G2 can be found such that qP'< e, hence if q' belonged to GI, so
also would P'. It is clear that no pair of points of G,, G2 can exist, of which
the distance is < e, hence for these sets A > e. It has thus been shewn that,
if for any e the condition is not satisfied, G can be divided into two detached
closed sets, and it is therefore not connex.
A  connex closed set, which does not consist of a single point, is a
perfect set.




102-104]


Non-linear sets


139


For an isolated point of the set could be considered as a set detached
from the set which consists of all the remaining points, and hence, if such an
isolated point existed, the set could not be connex.
It will be observed that a connex closed one-dimensional set can only
consist of a single interval.
The theory of plane sets and of sets of three or more dimensions is of
great importance in relation to its application to the Analysis Situs. Jordan*,
having given an arithmetical definition of a simple closed curve, has established the fundamental theorem that such a curve divides the plane into
two parts, respectively external and internal to the curve. The subject has
been further developed by Schonfliesjt, from the point of view of the theory
of sets of points.
104. The mode in which a non-dense plane set is determined by means
of areas, free in their interiors from points of the set, is not in all respects
similar to the mode in which a non-dense linear set is determined by means
of the complementary intervals. In the latter case each point P which does
not belong to the set is enclosed in an interval which contains no points of
the set, and this interval has a maximum length in both directions from the
point P, the end-points of such maximum interval 8 being points of the
closed and non-dense linear set, and this maximum interval is identical with
8, for all points P interior to 8. But in the case of a plane set, if we confine
ourselves to areas of given shape, such as rectangles, and these take the
place of the linear intervals 8, it is not the case that a closed set is defined
as the set of boundary points, together with their limits, of a unique system
of such rectangles.
If P be a point which does not belong to a given non-dense plane closed
set, and if we draw through P a straight line parallel to the line whose
equation is y = mx, then those points of the given set which lie on this straight
line are easily seen to form a closed set, and the point P must be interior to
a, complementary interval A, (P), of this closed set. If on one side of P
there are, in this straight line through P, no points of the given set, then on
this side the extremity of the interval Am (P) may be regarded as the point
in which the straight line intersects a side of the rectangle in which the
plane set is contained. The interval Am (P) exists for every value of m, and
the extremities of the intervals A, (P), for a fixed P, are in general points
of the plane set; the region of plane space Ap, in which these intervals
Am (P) lie, is free in its interior from points of the plane set; and such a
region is the true analogue, for plane sets, of the complementary interval of a
* See the (ours d'Analyse, vol. i, pp. 90-100.
t Gottinger Nachrichten, 1899, also Math. Annalen, vols. LVITI and LIX. The subject has also
been treated by Veblen, Trans. of the American Math. Soc. vol. vI.
+ See Schinflies, Gottinger Nachrichten, 1899, p. 282, also Bericht iiber die Mengenlehre, p. 81.




140                       Sets of Points                     [CH. II
linear set. The plane closed set consists of points on the boundaries of a
system of such regions Ap which do not overlap, and of the limiting points
of these points on the boundaries; and every point which does not belong to
the set is interior to one of the regions Ap. In this sense there is for each
point P of the complementary set a single region Ap which is the maximum
free region containing P in its interior; and all points interior to Ap have
their maximum free regions identical with Ap.
If, however, we work only with rectangular areas, which are usually the
most convenient in view of applications of the theory to the theory of integration, there exists in general no rectangular area corresponding to a point
P which has analogous properties to the region Ap. If we describe a square
of sides 2p parallel to the axes of coordinates and with its centre at P, then,
for any point P of the complementary set, when p is small enough there are
no points of the given set interior to or on the boundary of the square; and
p may be increased until one of the sides of the square contains a point of the
set, or is coincident with a side of the rectangle in which the whole set is
contained. When either of these things happens, we may keep this particular
side fixed in position, letting the other three increase their distances from P
by the same amounts; if a corner of the square comes to be a point of the set,
then both the sides intersecting at that corner are kept fixed; the square now
becomes a rectangle, and ultimately another side will either contain a point
of the set, or will fall on a boundary of the space in which the set exists.
Proceed in this way until we have a rectangle such that each of its sides
contains one or more points of the plane set, or else falls upon a boundary of
the domain of the set; we have then a definite rectangle corresponding to the
point P. But if we take a point Q inside this rectangle, and construct the
corresponding rectangle for Q, this need not coincide with the rectangle constructed for P; because a side of the rectangles, drawn with Q as centre, may
come into a fixed position, by meeting a point of the given set, before it has
reached the final position of the corresponding side of the rectangles constructed for P; and the maximum free rectangle for a point P, does not then,
in general coincide with the maximum free rectangles for points inside the
first.
105. It is however possible, for a given closed non-dense plane set G, to
construct an enumerable set of rectangles which is everywhere-dense, and
such that every point of G lies on the boundary of a rectangle, or is a limiting
point of points which lie on the boundaries of such rectangles. Let us
denote by S the rectangle in which the whole set G lies, and let s be the
rectangle constructed as above for a point P of the set G. Produce the sides
of 8, when necessary, until they cut the sides of S, thus dividing S into at
most nine different rectangles, of which one is 8, and the others may be
denoted by S,, where r = 1, 2,... 8. In each rectangle Sr take any point P,




104-106]


Non-linear sets


141


which does not belong to G, and construct for P,. the maximum free rectangle
8,. as before; let the sides of 3,. be produced when necessary until they meet
the sides of Sr, then Sr is divided into at most nine rectangles, which consist
of A3 and at most eight rectangles SXs when s = 1, 2,... 8.
Proceeding in this manner we obtain a set of rectangles
S, Sr, Srs, Sst... 
and in them a set of rectangles 8, Sr, 8rs, 3rst.., each of which contains no
points of G in its interior, each of the numbers r, s, t,... being one of the digits
1, 2, 3,... 8. If p be a point of G which is not on a boundary of any rectangle 3n, it must be in the interior of each of an unending series of rectangles
Sr, Srs, Srst,..., where r, s, t,... have definite values; and this set of rectangles must converge either (1) to a point in the interior of all of them, or
(2) to a linear interval, or (3) to a definite rectangle S, in the interior of all
of them. In case (1), the point to which the rectangles converge is a limiting
point of those points of G which lie on the boundaries of the definite
sequence of rectangles S,, 3rs, 83t,.... In case (2), there must be, on the
limiting linear interval, at least one point which is a limiting point of G: for,
if not, the whole interval could be enclosed in a rectangle which contains no
points of G; and this is impossible. In case (3), we start with the rectangle
S., and take a point P, not belonging to G inside it, construct the maximum
free rectangle 8,, produce its sides as before to meet those of SX and proceed
as before to construct Sst... and,,st.... This process can be continued
until an index is reached which may be any number of the second class, but
the point p must be reached before some definite number of the second class
appears as index; this following from the fact that the number of non-overlapping regions which are contained in a given space must be enumerable.
Thus the point p is reached after an enumerable set of steps of the process.
It has therefore been shewn that:If G is a non-dense closed plane set of points, an everywhere-dense
enumrerable set of rectangles can be determined, such that every point of G is
on a boundary of one or more of the rectangles, or is a limiting point of such
points, or lies in a linear interval which is the limit of a sequence of the
rectangles.
In case the set G is perfect, the rectangles of the set must either not abut
onr one another, or every common side must contain either no points of G, or
else a perfect set of points of G.
106. That a perfect plane set G has the power of the continuum* may be
proved by projecting the set on a straight line which we may take to be
a side of the rectangle in which the set is contained. The set of points
* See Bendixson, Bib. Svensk. Vet. Handl. vol. ix (1884), where the first proof of this theorem
was given.




142


Sets of Points


[CH. II


which are the projections of points of G is a closed set. For, if P be one of
the limiting points of the set of projects, let pp' be an arbitrarily small neighbourhood of P, of which P is the centre; draw straight lines PQ, pq, p'q'
perpendicular to pp' to the side qq' of the containing rectangle. Then in
the rectangle pqq'p' there are an infinite number of points of G; and if we
divide this rectangle into (2n + 1)2 equal parts by means of straight lines
parallel to pp' and to PQ, then in one of these parts at least there are an
infinite number of points of G in the interior or on one of the boundaries
parallel to pp'. Also one such rectangular part, at least, exists with its centre
on PQ; for otherwise P could not be a limiting point of the projection of G.
Divide this rectangle into (2n + 1)2 equal parts as before, then in one of
these at least with its centre on PQ, there must be an infinite number of
points of G; proceeding in this manner we shew that there is one point
at least on PQ which is a limiting point of G, and this point therefore
belongs to G; thus the projection of G is a closed set. An isolated point
P of the projected set must be such that there is a perfect component of
P on the straight line PQ. If the projected set is perfect, then it has the
power c of the continuum; and if it contains isolated points these must be
the projections of perfect linear components of G; therefore in either case
G has the power of the continuum.
It is clear that this method can be extended to the case of a set in any
number of dimensions; and we shew that the power of an n-dimensional
perfect set is c, if that of an n -  dimensional perfect set is c.
107. The content of a closed plane set may be defined in a manner
strictly analogous to Harnack's definition of the content of a linear closed
set. If the rectangle in which the set is contained be divided into rectangular portions, by drawing a finite number of straight lines parallel to the
sides of the rectangle, and the sum of the areas of those rectangular portions
be taken which contain in their interiors, or on their boundaries, points of
the closed set, then the content of the set is the limit of the sum when the
number of the rectangular portions is increased indefinitely in such a manner
that the greatest of the sides of all the rectangles has the limit zero. That
the content so defined has a definite value independent of the mode in which
the successive subdivisions of the original rectangular area are carried out,
provided only that the greatest of all the sides of the rectangular areas
diminishes indefinitely as the number of the rectangles is increased indefinitely, may be proved in precisely the same manner as in the case of
linear sets.
For plane sets in general, the exterior and interior measure may be
defined as in the case of linear sets.
The exterior measure nm (G) of a set G is the lower limit of the sum of the
areas of a finite, or indefinitely great, number of rectangles which enclose all




106-108]                  Non-linear sets                         143
the points of G in their interiors, when every possible such system of rectangles
is taken account of. The interior measure mi (G) is the excess of the area of
the rectangle in which G is contained over the exterior measure of the set
complementary to G; or we may take the equivalent definition, that the interior
measure is the uzpper limit of the contents of the closed components of G.
A plane set is measurable when the exterior and interior measures have
identical values.
All the theorems which have been proved in ~ 81-84 relating to the
measures of linear sets hold also for plane sets, and for sets in any number of
dimensions.
In the case of a set of points on a straight line, the content, or the
measure of the set, considered as a set in two dimensions, is always zero,
whatever value the content, or the measure of the set, considered as a linear
set, may have. The latter may be spoken of as the linear content, or the
linear measure of the set.
108. If G be any closed set of points in the rectangle ABCD, and through
the points P of AB straight lines PP' are drawn perpendicular to AB, and
D....Ptp  Q'                  P'
6~
A            P, Q                    P             B
FIG. 4.
if f(P) denote the linear content of the linear component of G which is
on the straight line PP', then the set of points P on AB, which is such that
f (P) _ ar, is a closed set, a- denoting any positive number.
Let P1 be a limiting point of the set; and if possible, let the linear
content of that component of G which is on PiP,' be < a; we can then find
a finite number of intervals 81, 82,.*. 8. on PIP,' whose sum is > AD - a,
and which are free in their interiors and at their ends from points of G. On
each of these intervals 8 we can describe a rectangle which contains no points
of G in its interior or on its boundaries: this may be done on either side of
PIP,'; for each point of 8 can be enclosed in a rectangle free from points of
G, and by the Heine-Borel theorem, a finite number of these rectangles,
enclosing all the points of 8, exists. Take a point Q belonging to the set of
points for which f(Q)  o a, and let PiQ be less than the breadth of all the




144


Sets of Points


[CH. II


rectangles described on the intervals {3} on one side of P,P,'. On QQ' there
is a finite number of intervals free from points of G, whose sum is > A D- a-,
by the assumption as to PIP'; hence the linear content of the component
of G which is on QQ' must be < ar, which is contrary to the hypothesis. It
follows that f(PI)  a-; hence the set of points on AB is closed.
It will now be shewn that, for a closed set of points G, if for every
position of P on A B, the linear content of the component of G upon PP' is
< a, then the content of G is < a-. AB.


D r; _
A             q   P    P                          B
FIG. 5.
Taking any point P of AB, on PP' a finite number of intervals, whose
sum is > AD - ac, can be found which are free from points of G; and on each
of these intervals a rectangle can be drawn on each side of PP' containing
no points of G in its interior or on its boundary. We can now draw two
straight lines pp', qq', one on each side of P, so that each of them passes
through the interiors of all the rectangles so described. We have now found
an interval pq containing P, such that in pqq'p' there is an area
>pq (AD - a)
free from points of G. Corresponding to each point P of AB such an interval
pq can be found; and, in accordance with the Heine-Borel theorem, a finite
number of these intervals can be selected such that every point of AB is in
the interior of one at least of them. The end-points of these intervals
divide AB into a finite number of parts such that, above any one part of
length a, there is an area > a (AD - a) free from points of G; and hence
there is altogether an area > AB(AD - a) free from points of G. It follows
therefore that the content of G is < AB. o-.
We shall now establish the following theorem, which is of importance in
the theory of double integration:If G be a closed set, and if the linear content of the set of points P on AB
for which the linear content of that component of G, which lies on PP', is > a,
have the value zero for every positive value of a, then the set G is of zero content.




108]


Non-linear sets


145


The points on AB, for which f(P) - a, can be enclosed in a finite number
of intervals whose sum is < e, where e is an arbitrarily small number; and in
each of the remaining parts of AB, the value of f(P) is < a; hence by the
foregoing theorem   the content of G is < a (AB - e) + e. AD; and since this
holds for arbitrarily small values of ar and e, it follows that the content of G
must be zero.
Conversely, it may be shewn that:If G be a closed set of zero plane content, the set of points P on AB, for
which the linear content of the component of G on PP' is > a-, has, for every
positive value of a, content zero.
Let I denote the linear content of the set (P) for which f(P) - a-; divide
AB into n equal parts, and AD also into n equal parts, and through the endpoints of these parts draw straight lines dividing the rectangle into equal parts
1
each of area 1. AB. AC. Then the sum of those parts of AB which contain
n2 
points of the set (P), is always greater than I; and in each such part there is
at least one point P, such that the sum of the parts of PP' which contain
points of G is > a. It follows that the sum of those rectangular portions
which contain points of G is > aI, however great n may be; and hence that
the content of G is      a-I.  Therefore it follows that G    cannot have zero
content unless I is zero.
EXAMPLES.
1. Let a set of points (x, y) in the rectangle for which 0  x  1, 0  y  1, be defined
as follows":-The numbers x, y are expressed in the dyad scale, and only those values of x
and y are taken which are expressed by terminating radix-fractions, the number of digits
being the same for x as for y. If x' denotes a terminating radix-fraction, there are only a
finite number of points (x', y') of the set on the straight line x=x'; similarly if y' denotes
a terminating radix-fraction, there are only a finite number of points of the set on the
straight line y=y'. The two-dimensional set is however everywhere-dense; for, considering
a straight line y=x+a, where a is a positive or negative radix-fraction with a finite
number of digits, we see that, corresponding to any number x expressed by a finite number
of digits greater than the number of digits by which a is expressed, there is a point (x, y)
on the straight line belonging to the set. The component of the set on the straight line
y=x +a, being everywhere-dense, and the values of a being everywhere-dense in the
interval (-1, 1), it follows that the set is everywhere-dense in the rectangle.
This example shews that an everywhere-dense two-dimensional set may be linearly
non-dense on each straight line belonging to two parallel sets. It also shews that a twodimensional set may exist which is extended, but is unextended on straight lines belonging
to either of two parallel sets.
2. Let a crosst formed by two pairs of straight lines parallel to the pairs of sides of
a square be constructed, and so that the remainder of the square consists of four equal
~ Pringsheim, Sitzingsberichte d. Mifilnchener. Akad. vol. xxix, p. 48.
t Veltmann, Schlomilch's Zeitsch. vol. xxvII, pp. 178, 314.


H.


10




146


Sets of Points


[CH. 11


squares at the corners. Let the interior points of the cross be removed from the square,
and then let a similar cross be removed from each of the remaining four squares. Proceeding in this manner, let the crosses be so chosen that the area of each square after the
-p
mth stage of the process is ab 2m times the area of each square after the preceding stage.
The sum of the areas of the squares which remain after the mth stage is
(4a),  b  (22 2 1+  2+)Q =(4a)'n (b? Q
where Q is the area of the original square. A non-dense closed set of points is defined
as the points which remain when this process is carried on indefinitely. The limit of the
sum of the crosses is that of
[l- (4a)mb   ('   )] Q;
and this is Q or < Q, according as a< '; it follows that the closed set has zero content if
a <, but if a=f, the content is b- Q.
SETS OF SEQUENCES OF INTEGERS.
109.  A  theory of sets of sequences of integers, of which the formal
character is similar to the theory of sets of points in any number of
dimensions, has been developed by Baire*, with a view to application to
the Theory of Functions.
A group of integers (a,, a2,... a), of order p, consists of a system  of p
positive integers arranged in a definite order.
The group (a,, a,,... ar), of order p, is said to be contained in each of the
groups (a,), (a,, a,), (a,, a, a3)... (a,, a,,... aO_-) of orders 1, 2, 3,... p - 1,
respectively.
A sequence of integers (al, a2,... ap,...) consists of an infinite number of
integers, defined in any manner, and arranged in an order similar to the
sequence 1, 2,3,....  This sequence is said to be contained in each of the
groups (), (ai, a2),... (al, a,... ap),....  Let P be a set of such sequences of
integers, and let A be any other sequence of integers; then if, for every n,
there are sequences in P, other than A itself, which are contained in the same
group of order n as A itself is contained in, i.e. sequences having their first
n integers the same as the first n integers in A, then the sequence A   is said
to be a limit of the set of sequences P.    The sequence A may or may not
itself belong to P.
The set P is said to be closed, in case all its limits belong to it. The set
is said to be perfect when it is closed, and also every sequence in the set is a
limit of the set.
A set E of groups of integers is said to be complete if, when g is any group
of order p belonging to E, the groups of orders 1, 2, 3,... p - 1, which contain
g, also belong to E.
* Comptes Rendus, vol. cxxix, 1899, p. 946.




108, 109]


Sets of sequences of integers


147


A complete set E of groups of integers is said to be closed, if every group
g belonging to E contains at least one group of higher order than itself, which
is also contained in E.
Having given a complete set of groups E, a sequence A may exist such
that all the groups containing A belong to E. The set F of all sequences
such as A, is said to be determined by the set of groups E. The set F, if it
exists, is closed.
Every closed set of groups E determines a closed set of sequences F, and
conversely, every closed set of sequences F is determined by a unique closed
set of groups E. In case F is perfect, E is also said to be perfect. In order
that E may be perfect, it is necessary and sufficient that every group belonging
to E should contain at least two groups of one and the same order superior to
its own order, and belonging to E.
If P is a set of sequences, then the set P' of those sequences which are
limits of the set P is said to be the derived set of P, and may be denoted by
P'. The derived set P' is closed.
The successive derivatives P", P"'I,... P(),... Pa), of finite or transfinite
orders, are then defined as in the theory of sets of points. If P is a closed
set of sequences, there exists a number a of the first or the second class, such
that P(a) = P(a+l). Unless P is an enumerable set, it can be resolved into the
sum of an enumerable set and a perfect set.
Let us consider a perfect set of groups E determining a perfect set of
sequences F. A set P of sequences all belonging to F is said to be non-dense
in F or in E, provided that every group of E contains at least one group of E
which contains no sequence of P.
A set of sequences P all belonging to F is said to be of the first category,
relative to F, if there exists an enumerable sequence of sets P,, P,... P,...,
each of which is non-dense in F, and such that each sequence of P is part of
one at least of the sets P,, P,... P,.... The set obtained by removing the
set P from F is said to be of the second category relative to F. The same
generic distinction between sets of the first and of the second category holds,
as in the theory of sets of points.


10-2




CHAPTER III.


TRANSFINITE NUMBERS AND ORDER-TYPES.
110. A    PRELIMINARY account has been given in Chapter II, of the
theory of transfinite ordinal and cardinal numbers; it was shewn that the
introduction of such numbers was suggested by the exigencies of the theory
of linear sets of points, and that, in particular, the necessity for the use of
transfinite ordinal numbers arises whenever a convergent sequence of points is
transcended by adjoining to the points of the sequence their limiting point and
any further points which it may be desirable to regard as belonging to the same
set as the points of the sequence. The fundamental discovery of G. Cantor,
that the rational points of an interval form an enumerable set, whereas the
set of points of the continuum is unenumerable, by establishing the existence
of a distinction between the characters of two infinite sets, suggests the
development of a general theory of cardinal numbers of infinite aggregates.
The procedure we adopted, of introducing the fundamental notions of transfinite ordinal and cardinal numbers in connection with the theory of sets of
points, is in accord with the historical order in which the whole theory of
transfinite numbers and order-types was developed. The account of the
theory of transfinite numbers given in Chapter II, is in general agreement
with Cantor's earlier presentation* of his ideas; his latert and more abstract
treatment of the subject is the one upon which the account given in the
present Chapter is founded.
In order that the reader may be put into a position to form his own
conclusions as to the validity of a scheme which must be regarded as still,
to some extent at least, in the controversial stage, it has been thought best
to postpone any discussion of the difficulties of the theory, until after the
conclusion of the detailed account of the theory in its constructive aspect.
* See his "Grundlagen einer allgemeinen Mannigfaltigkeitslehre," Leipzig, 1883, or Math.
Annalen, vol. xxI; see also Zeitschrift filt Phil. und phil. Kritik, vbls. LXXXVIII, xci and xcII.
Cantor's ideas were foreshadowed in a paradoxical form by Bolzano in his "Paradoxien des
Unendlichen," Leipzig, 1851; and although infinite numbers had been discussed by earlier
writers, Bolzano is the only real predecessor of Cantor in this department of thought.
t This is contained in the two articles " Beitrage zur Begriindung der transfiniten Mengenlehre," in the Math. Annalen, vol. XLVI (1895), and vol. XLIX (1897).




I1O, 111]


Cardinal numbers


149


In the last part of the Chapter, some critical remarks upon the logical
basis of the theory will be made; these must necessarily be of an incomplete
character, partly from considerations of space, and also because any complete
criticism of such a scheme as Cantor's theory of transfinite numbers would
involve the consideration of questions of an epistemological character which
for obvious reasons cannot be adequately dealt with in a work of a professedly mathematical complexion. Objections which may be urged against
some parts of the theory, will however be fully stated. Some consideration
will also be given to the question, whether, and how far, the theory is indispensable as a logical basis of continuous Analysis.
THE CARDINAL NUMBER OF AN AGGREGATE.
111. A collection* of definite distinct objects which is regarded as a single
whole is called an aggregate.
An aggregate may be denoted symbolically by a large letter M, the
elements of the aggregate by small letters m; and the constitution of
the aggregate may be denoted by the equation M= {m}.
The consideration of questions which arise in connection with this
definition, as to the mode in which the objects of the aggregate must be
specified, in order that the aggregate may be adequately defined, and as
regards the conditions, if any, which must be satisfied in order that a
collection may be regarded as a whole, or aggregate, of such a character
that it can be an object of mathematical thought, will be postponed. For
the present, it is sufficient to remark, that an adequate definition of any
particular aggregate, which is not necessarily finite, must contain, as a
minimum, a set of rules or specifications by means of which it is theoretically
determinate, in respect of any object whatever, whether such object does or
does not belong to the aggregate. The set of prime numbers, for example,
is regarded as an aggregate, although when a particular number is presented
to us, we may be practically unable to decide whether that number is prime
or not. In this case however, a finite number of processes will suffice to
decide the question.  If however, we take the case of the algebraical
numbers, the state of things is different; for we are not in possession of any
general method which enables us to decide whether a given number is
algebraic or not. Nevertheless, the question being regarded as having a
definite answer, the algebraical numbers are regarded as forming an aggregate, in the sense here employed.
* This definition is given by Cantor, Math. Annalen, vol. XLVI, p. 481, as follows:-" Unter
einer 'Menge' verstehen wir jede Zusammenfassung M von bestimmten wohl unterschiedenen
Objecten m unserer Anschauung oder unseres Denkens (welche die 'Elemente' von M genannt
werden) zu einem Ganzen."




150


Transfnite numbers and order-types


[CH. III


An aggregate does not depend, for its validity as a mathematical entity,
upon the possibility of producing all its members, successively or otherwise,
but upon the sufficiency of the rules by which its elements are to be
distinguished, as belonging to it, in that particular kind of objects to which
they belong; that is, upon the sufficiency, in this direction, of its definition
of membership.
Two aggregates M, N are said to be equivalent to one another when
they are such that a law of correspondence can be established between the
elements of one aggregate and those of the other, such that to each element
of one of the aggregates, there corresponds one and only one element of the
other aggregate.
This relation of equivalence between two aggregates M and V, may be
expressed symbolically by M  N7, or N ~M.
It is clear that, if each of two aggregates is equivalent to a third, the two
aggregates are equivalent to one another.
Aggregates which are equivalent to one another are said to have the same
power or cardinal number.
A cardinal number is accordingly characteristic of a class of equivalent
aggregates.
The question whether two defined aggregates have or have not the same
cardinal number, is thus equivalent to the question whether it is, or is not,
possible to establish a systematic (1, 1) correspondence between the elements
of the two aggregates, in accordance with the above definition of equivalence.
A particular aggregate can ordinarily be shewn to be equivalent to itself.
The law of correspondence between an element and another element which
can be set up, is in general of a character which admits of a certain
arbitrariness. The cardinal number is accordingly regarded as independent
of the notion of order in the aggregate.
The power or cardinal number of an aggregate M has been defined by
Cantor as the concept which is obtained by abstraction when the nature
of the elements of M, and the order in which they are given, are entirely
disregarded.
Cantor regards the fact, that equivalent aggregates have the same
cardinal number, as a deduction from this definition. Some critical remarks
upon the definition of the cardinal number of an aggregate will be made
in ~ 155.
The cardinal number of M is a characteristic of M which may be denoted
by M, to indicate that both the order of the elements, and their precise
individual nature, are irrelevant as regards the cardinal number.




111-113]


Cardinal numbers


151


The relation of equivalence M~ N, between two aggregates, implies the
equality M= N; and this equation expresses the necessary and sufficient condition for the equivalence of M and N.
Since Cantor regards the cardinal number of M as independent of the
precise nature of the elements of M, we may in accordance with this view,
substitute for each element the number unity. We have thus a new
aggregate which is a collection of elements each of which is the number 1,
and is equivalent to M; and this new aggregate is regarded by Cantor
as a symbolical representation of the cardinal number M.
THE RELATIVE ORDER OF CARDINAL NUMBERS.
112. Every aggregate MA, which is such that all its elements are also
elements of M, is called a part or sub-aggregate of M.
If Mi is a part of 1M,, and Ml is a part of M, then M, is a part of M.
A finite aggregate cannot be equivalent to any of its sub-aggregates; but,
as will be seen in detail further on, an infinite aggregate always possesses
sub-aggregates which are equivalent to itself. This is the characteristic
distinction between finite and infinite aggregates, and has in fact been
employed by Dedekind and others to define an infinite aggregate, as one
which is equivalent to one of its parts.
If two aggregates M, N with the cardinal numbers a _ M, / - N, are
such that, (1) there exists no part of M which is equivalent to N, and
(2) there exists a part N1 of N which is equivalent to M, it is clear that the
corresponding conditions are satisfied for any two aggregates which are
equivalent to M, N respectively; and thus the two conditions characterise
a relation between the cardinal numbers a, / of the two aggregates. When
the above conditions are satisfied we say that a is less than 3, and that / is
greater than a; which is expressed symbolically by a < /, /3 > a. This is the
definition of inequality for two cardinal numbers, and of the relations greater
and less in the purely ordinal sense in which they are here used.
The condition contained in the definition is inconsistent with the relation
of equality between a and / being satisfied. For if a =- /, then Ma N, hence
since N1A M, we have N1x N: therefore, since M~ N, there must be a part
of M, say M,, such that M, M, which would involve M1~ N; but this is
contrary to one of the conditions contained in the definition of inequality.
It is easily seen that if a < /, and / < y, than a < ry.
113. It has been seen that the three relations a = /, a < /, /3 > a are
mutually exclusive; but the question arises whether any two cardinal numbers
a, /3 whatever must satisfy one of these relations. An affirmative answer to
this question would be required before it could be maintained that all cardinal




152


Transfinite numbers and order-types


[CH. III


numbers can be regarded as being alike capable of having relative rank
assigned to them, in a single ordered aggregate.
Two aggregates M, N of which we may denote parts by M1, N, must
satisfy one and only one of the following four conditions:(1) M, N have parts Mi, N,, such that M1  N, and N,, M.
(2) M has a part M,, such that M,1  N; but no part of N exists which
is equivalent to M.
(3) There is no M1 which is equivalent to N; but there is an N1 which
is equivalent to M.
(4) There exists no M, equivalent to N; and also no N1 equivalent to M.
It will be proved that, if the condition (1) is satisfied, then M = N. The
condition (2) expresses the relation defined as M > N. The condition (3)
expresses the relation defined as M < N.
It has not yet been proved that the relation (4) is an impossible one;
except that, in the case of finite aggregates, it may be easily seen that it
involves M= N. Until this point is cleared up, it cannot be maintained as
an established fact that the cardinal numbers a, / of any two aggregates
whatever satisfy one of the three relations a = /, a > /, a < /.
Two aggregates which are such that their cardinal numbers a, / stand to
one another in one of the relations a = /, a > 3, or a < /, may be said to be
comparable with one another. Otherwise they are incomparable with one
another.
THE ADDITION AND MULTIPLICATION OF CARDINAL NUMBERS.
114. If M, N are two aggregates which have no element in common,
then the aggregate which has for its elements all those of 111 and all those of
N is called the sum of the two aggregates M, N, and may be denoted by
(M, N). A similar definition applies to the case of the sum of any number
of aggregates no two of which have an element in common.
If M', VN' are two other aggregates with no element in common, such
that M     M', NV N', it is clear that (Ml, N)  (M', N'); and thus the
cardinal number of (M, N) depends only on those of M and N.
If M1= a, N =/3, we define the result of the operation of addition oJ a and /
to be (M, N).
From the independence of cardinal numbers of the order of elements, we
deduce
a+/3=/3+a, a+(/3+r)=(a+/3)+ry;
thus the operation of addition of cardinal numbers obeys the commutative
and associative laws.




113-116]


Cardinal numbers


153


115. If an element m of M be associated with an element n of r, so
as to form a new element (m, n), the aggregate of all possible elements
which can be formed in this way is called the product of M and N, and may
be denoted by (M. N).
If M     M', N~ N', it is clear that to each element (m, n) of (M. N),
there is a corresponding element of (M'. N'), hence (M. N)'(M'. N'), and
thus (M.N) depends only on M and N.
The cardinal number of the product-aggregate (M. N) is defined to be the
product of the cardinal numbers of M and N.
The product of M and N may also be defined as the cardinal number of the
aggregate which is obtained by substituting for each element of N, an aggregate
which is equivalent to M.
It is seen on reflection that this definition is equivalent to the first one.
Since, as can be shewn from the definition,
(M. N)   (N. M), (M. (N. R)) ~ ((M.    N.R)
and                 (M. (N, R))  ((M. N), (M. R)),
we see that cardinal numbers satisfy the relations
a. / = /3.  a, a (3.) = (a. /) 7,  (/3 + 7) = a/p  + a~/.
It has thus been shewn that the multiplication of cardinal numbers obeys
the commutative, associative, and distributive laws.
The definition of multiplication may be extended * to the case in which
the number of factors is not necessarily finite. Let us consider a class of
aggregates M, where the class contains either a finite or an infinite number
of aggregates, and suppose no two of the aggregates have an element in
common. Let there be chosen from each of the aggregates in the class, one
element, and conceive that this is done in every possible way; we have now
a new aggregate, each element of which consists of an association of elements,
one from each of the aggregates of the given class. The new aggregate is
said to be the product-aggregate of the given class of aggregates, and its
cardinal number is defined to be the product of the cardinal numbers of all
the aggregates of the given class.
CARDINAL NUMBERS AS EXPONENTS.
116. If we have two finite aggregates M, N containing x and y elements
respectively, we may suppose that to each of the y elements of N, one
element of M is made to correspond, so that the same element of M may
be use0.,ny number of times; any particular such correspondence we call


*:i ' t7Whitehead, American Journal of Math. vol. xxiv, where the theory of cardinal numbers
is treaeid.. b-i  'he Peano-Russell symbolical method.




154


Transfinite numbers and order-types


[OH. III


a covering (Belegung) of N by M. The total number of ways of covering N
by M is xy. To put the matter in a concrete form, the total number of ways
of distributing y things among x persons, where any number of the y things
may be given to one person, is xy; any particular mode of distribution is
what we have called a mode of covering the aggregate of y things by the
aggregate of x persons.
The definition of covering an aggregate N  by an aggregate M   is
immediately extensible to the case of infinite aggregates. As before, the
covering denotes any system by which to each element of N is made to
correspond a particular element of M, the same element of M   being
employed any number of times, or not at all.  Denoting by N/M  each
particular mode of covering N by M, we thus form the new aggregate (N/M)
which contains as its elements all such coverings.
It is seen at once that, if M   M', N~ N', then (N/M) (N'/M'). Thus
the cardinal number of (N/l) depends only on the cardinal numbers of M
and N.
The cardinal number of the aggregate (N/M), each element of which is
a covering of N by M, and in which every possible mode of such covering
occurs as an element, is denoted by the symbol ag, where a =l M,  - N; thus
ap (N/IM).
It is easy to shew that
((N/M). (R/M)) ~((N, R)/M)
((R/Ml). (BRIN)) (R/(M. N))
(R/(N/M))   ((R. N)IM).
Hence if M=a, N=/3, R= ry, we see that, in accordance with the
above definition of exponentials,
C.C a = a+y, av. /Y = (a. /3), (aP)v = a *Y;
and thus the same laws hold as for exponents in which only finite cardinal
numbers are involved.
THE SMALLEST TRANSFINITE CARDINAL NUMBER.
117. The cardinal number of the aggregate of all the finite integers
1, 2, 3,......... is called Alef-zero, and is denoted by,0; thus N0= {=}.
The number No is identical with the number which has previously been
denoted by a.
If we add to {n} a new element e, we obtain the sum-aggregate ({t}, e),
and this is equivalent to {na, for we may make e in the first of these
aggregates correspond to 1 in the second, and in general n to n + 1; and thus
({N}, e) {nj. From this, we obtain 0o + 1 =  o0, a relation which differentiates
No from all the finite cardinal numbers.




116-118]


Cardinal numbers


155


The cardinal number No is greater than all the finite cardinal numbers,
and it is less than any other transfinite cardinal number.
Since the finite aggregate (1, 2, 3,... k) is a part of {n}, but no part of
the finite aggregate is equivalent to {n}, by the definition of inequality we
have No > k.
To prove that, if a is any transfinite number, say that of an aggregate M,
which is not equivalent to {n}, then N0 < a, we have to shew that M  contains
a part which is equivalent to {n}, and that there exists no part of {n}
which is equivalent to M. A first, second, third,... nth element can be
chosen from the elements of M in any manner, and this process can be continued without limit; thus M always contains a part which is equivalent to
{n}. Any part of the aggregate [n} which is not finite, consists of finite
numbers chosen from 1, 2, 3,... n,...; and of these there must be one which
is smallest: denote it by e,. Then the next greater can be denoted by e,
and so on. Thus this part of {n} is (e1, e2, e3,...), which is equivalent to
in}; and therefore the theorem is established.
118. It has been shewn that No + 1 = No: a similar proof would shew that
No + n = No, where n is any finite integer.
In accordance with the definition of addition, No + N  is the cardinal
number of the aggregate (1, 3, 5,... 2, 4, 6,...), for NR is the cardinal number
of each of the aggregates (1, 3, 5,...) (2, 4, 6,...); hence, since the cardinal
number of (1, 3, 5,... 2, 4, 6,...) is the same as that of {n}, we have
N, + RN=R, which we may write as
No. 2 = 2. No = No.
From this relation, by repeated addition of N. to both sides of the identity,
we find
No. n = n. N = No.
In order to express the product No. N, we form the aggregate {(n, n')} ot
which the elements (n, n') consist of every pair of finite cardinal numbers.
Let nn + n'= s, then s has the values 2, 3, 4,...; and for any fixed value of s
the numbers n, n' have a definite number of sets of values. Let s = 2, we
then have one element (1, 1): let s = 3, we then have two elements (1, 2),
(2, 1): for s= 4, we have (1, 3), (2, 2), (3, 1), and so on. The elements of
{(n, n')} may thus be arranged in order so that the element (n, i') is at the
pth place, when p = n + ( + n' -1)(n   ' - 2); thus the aggregate {(n, n')}
2
is equivalent to {p}, which has the cardinal number NR.
It has now been proved that N. No = No, or K,2 = N; and from this the
theorem  0'" = N, follows by repeated multiplication by N0.




156


Transfinite numbers and order-types


[CH. III


The theorems n. No=,, NO2= No, express in a symbolical form the
results which have been proved in ~ 54, that a finite, or enumerably infinite,
number of enumerable aggregates makes an enumerable aggregate.
THE EQUIVALENCE THEOREM.
119. The proof referred to in ~ 113, will now be given, that if M, N are
any two aggregates such that M contains a part M1 which is equivalent to
N, and N contains a part N, equivalent to M, then M= = N. This theorem,
which may be called the equivalence theorem, was first proved by Schroder *
and independently by Bernstein t; but the form in which the proof is here
given is due to Zermelo+.
Lemma I. If a cardinal number a remains unaltered by the addition of
any one of the enumerable set of cardinal numbers pi, p2,... p,,..., it remains
unaltered if all these cardinal numbers p are added to it at once.
If M, PI, P2,... P,,... be aggregates of which the cardinal numbers are
a, pI, 2,... pn,...; and such that PI, P2,... Pn,... are all parts of -Mi.
We have then, M = (PI, MI) = (P,, M,) =... = (P=, Mn)...; where MI, M2,...
are all parts of M, and in virtue of the hypothesis made in the statement of
the theorem,
M= MI    12 = =... =,....
We may denote the (1, 1) correspondence which can be set up (see 111)
between M and Mn, by M,, = b,,M; and this for every n. Now it is clear
that this relation of correspondence is such that
OM =(pPi, pM) = (P,, M2)=....
Hence              M =(P,, M),
MI =  1M= (lP,, 1M,) = (P', M,),
where P2, M' are those aggregates which correspond to P2, Mi respectively
in the correspondence denoted by ~bi.
Also, with a similar notation,
X/= 01c<2Mi= (Q1bP2, Sb012 M3) = (P% M3),............'........................................,..
',.-i = ('  2... -,_,M= ('N0...,-_ P,, ' 02...._l 1,M)= (PI', M,').
From these results we deduce
M = (iP, ~P, Pi",... M",., M,');
and no two of the parts PF, P',, P3',... P,' of M, have elements in common.
* See Jahresbericht d. Deutsch. Math. Verg. vol. v, p. 81 (1896); also Nova Acta Leop. vol. LXXI,
p. 303 (1898).
t See Borel's Lefons sur la theorie desfonctions, p. 103.: Gittinger Nachrichten, 1901, p. 34, "Ueber die Addition transfiniter Cardinalzahlen."




118, 119]            The equivalence theorem                      157
This process of division of M can be continued indefinitely; and we then have
l = (P1, P2', P3,...  / ),
where P,.' for every r is included, and M.' consists of those elements which
belong to MA' for every value of r. From this we see that
a =p +2+p3 +... + a';
where a' is the cardinal number of M,'.
Let us now consider the special case of the lemma which arises when
Pi, P2,... are all equal, say to p. In this case, we see that, from the
hypothesis a =p + a, the result a = Nop + a' follows, where Mo denotes the
cardinal number of the series of finite integers.
Now since No = 2o,, we have  op + a' = 2Nop + a'= Nop + a; it has thus
been shewn that if a = a + p, then a = a + Nop.
Returning to the general case, we have
a = a + Ropl = a  + Nop2=...;
it now follows that   a = NOpi + Nop2 +... + a",
where a" is the value which a' takes when NopI, Mop2,... are substituted for
PI, P2,.-..
We now have
a = 2No (pl + p2 +...)+  = a + No (p + p2 +...)
= (No + 1) (pi +p2 +...) + a"= a +pI +p2 +...:
and therefore the Lemma has been established in an extended form.
Lemma II. If the sum of two cardinal numbers p and q when added to
a leaves a unaltered, then a is unaltered by the addition of either p or q.
For if                   a = a +p + q,
we have seen that         a = a + No (p + q);
hence                   a =a + (No + 1)p + Noq,
and also                 a      = a +  + (No + 1) q;
from these equalities we have
a=a+p, and a=a+q.
We are now in a position to prove the equivalence theorem. If
a=,3+p, and 3 =a+q,
we have                      a = a +p+q,
and hence, by Lemma II,   a=a+p=a+ q=/3;
therefore if M has a part equivalent to N, and N has a part equivalent to M,
it follows that M = a = / - N.




158


Transfinite numbers and order-types


[CH. III


In case the condition a=/3+p, holds, but there is no corresponding
condition 3 = a + q, we have in accordance with the definition in ~ 112, a > 3.
It follows that the sum of two or more cardinal numbers is greater than, or
equal to, any one of the cardinal numbers.
The following theorem may be established:If the cardinal number a is unaltered by the addition of p, then if /3  a,
the cardinal number /3 is unaltered by the addition of q, where q < p.
For let /3=a+y, p=q+r; then from a=a+p=a+q+r, we deduce
that a = a + q. It then follows that
=a+7=a+q + y=a +r +7=          + q =    + r.
120. A proof has been given by Cantor* that if an aggregate exists of
which the cardinal number is a, then an aggregate always exists of which the
cardinal number is greater than a.
The proof is a generalization of the second proof given in ~ 56, that the
cardinal number c of the continuum is greater than that of the rational
numbers. The proof may be put into the following form:
Suppose M = nm} to be an aggregate of cardinal number a; which aggregate
M may be supposed to be simply ordered in any manner. In M let each
element m be replaced either by A or by B, where A, B are two given
objects; then M is replaced by a similar aggregate (see ~ 122), in which each
element is either A or B. An infinity of such aggregates will be obtained
differing from one another in respect of whether A or B has been put in the
place of each element of M; denoting the aggregate of all such possible
aggregates MAB, by {MAB}, it will be shewn that the cardinal number of
{MAB} is greater than that of M. ' In the first place, it can be seen that the
cardinal number of {MAB} is equal to, or greater than, that of {m}; for, taking
any one element m0 of {m}, replace it by A, and all the other elements by B;
we have then an element of {MAB}, and there is such an element corresponding to each element mn0 of {m}; thus those elements of {MABI, in which
there is only one A, form an aggregate of cardinal number equal to that of
{m}. Next, let us assume that, if possible, all the elements of {MAB} are
placed into (1, 1) correspondence with those of {m}; it will then be shewn that
an MAB can always be found which is not included in the correspondence. Each
M~AB in {MABR now corresponds to a definite mnO in {m}; form a new aggregate
M'AB in the following manner:-For each element OM~A in {MAB}, in which A
takes the place of mo in {m}, write B; and for each element M0AB in {MAB}, in
which B takes the place of mn, in {m}, write A; in this manner we form an
aggregate M'AB in which each element is either A or B, which is similar to
{m}, and which is not identical with any MAB that occurs in the correspondence
* See Jahresbericht d. Deutsch. Math. Vereinigung, 1897.




119-121]


Cardinal numbers


159


between {MABI and {fn}. It has thus been shewn that the cardinal number of
the aggregate of all the MAB is greater than that of M. If M is the aggregate
a, a, a2, a,... a,,,... which is similar to the aggregate of integral numbers, and if
for A and B we write 0 and 1, then the aggregate {M0o} may be interpreted as the
aggregate of all the rational and irrational binary fractions; and this aggregate
is thus shewn to be unenumerable. Instead of replacing the elements of
{m} by two letters A, B, we might have taken any finite number of
letters without altering the principle of the proof. In ~ 56, the ten digits
0, 1,... 9, were taken instead of A and B. It will be observed that, even if
{m} is normally ordered (see ~ 130), the new aggregate [M} is not given as a
normally ordered aggregate; and in default of proof it cannot be assumed
that it is capable of being arranged in normal order.
To replace all the elements of an aggregate either by A or by B, is
equivalent to taking a part * of the given aggregate. The theorem has thus
been established that, the cardinal number of the aggregate, each element of
which is a part of a given aggregate, is greater than the cardinal number of the
given aggregate, all possible parts being contained in the new aggregate.
DIVISION OF CARDINAL NUMBERS BY FINITE NUMBERS.
121. If two aggregates have the same cardinal number, and if each of the
two aggregates be divided into the same finite number n of parts, such that
the n parts of the first aggregate all have the same cardinal number, and
also the n parts of the second all have the same cardinal number, then it
can be proved that the cardinal number of one of the parts of the first
aggregate is the same as that of one of the parts of the second aggregate.
Symbolically, the theorem may be stated in the form:-if a, / are cardinal
numbers such that na = n/, then a = /.
This theorem has been proved by Bernsteiut. It will be sufficient to
give the detailed proof in the case n = 2, as the proof in the general case is
obtained by generalization of that employed in the particular case.
Since an aggregate is equivalent to itself, any special mode of exhibiting
such equivalence, by which each element is made to correspond to a definite
other element, is called a transformation of the system into itself. As regards
all such possible transformations the following propositions may be seen to
hold:(13:) The transformations of an aggregate M into itself form a group Om.
(2) Let 1 X, %X, % 3,... denote a sequence of transformations of Mh into
itself, 1 denoting the identical transformation, and let this sequence form a
group which is necessarily a sub-group of OM; then the condition that the
~ See Borel, Lecons siur la theorie des fonctions, p. 108.
t Inaugural Dissertation, "Untersuchungen aus der Mengenlehre," Halle, 1901. This is
reproduced in Math. Annalen, vol. LXI.




160


Transfinite numbers and order-types


[cCH. III


sequence forms a group is that, corresponding to any two integers m, n, there
is a third r, such that %mn = %,.. Further, let us suppose that to every X,
there corresponds a definite 'n, such that XXn = 1. If m be an element of
M such that m * X% (m), for n = 1, 2, 3,..., then %X (m) $ X, (m), where n and n'
are any unequal integers.
(3) If m and m' are any two distinct elements of M, and if m = Xn (m'),
for n = 1, 2, 3,..., then X, (m) = Xn' (m'). For if %X (m) = x', (m'), we should
deduce that m = X'x% (m) = X'%L (m) = X",, (m'), which is contrary to the
hypothesis made.
(4) If TI, T2,... are parts of M, such that each one T has no element in
common with another T, we may say that the T's form a system of separate
parts of M.
If T={t} is a part of M, and if t +n(t'), for n= 1, 2, 3,..., then the
equivalent aggregates T, X, (T), X2 (T),... form a system of separate parts of M.
(5) If T is a part of M which satisfies the condition stated in (4), then
M = M + T
For T, %1(T), X%(T),... are all parts of M having the cardinal number T;
and if R is the part of M which remains when all these separate parts are
removed, we have
M=R + No. T;
hence                    M + T= R+ (M0 + 1) T
=R+    o.T= M.
To proceed to the proof of the theorem:-Let
(a)                      = x, + x =  3 + 4,
(b)                     1 = X2,
(C)                    $3 = x4;
then it is required to shew that a = 23, which involves 2 = 4
The three equations (a), (b), (c) may be regarded as denoting that there
are three reversible transformations of the aggregate M into itself, which
may be denoted by Oa, Ob,,c respectively; the reversibility of these transformations is expressed by a2 =  b = 5c2= 1.
The transformation  a, involves xi = (xi3, x14), where xs are those elemeint:
of xl which are transformed into elements of x3, and  1,i4those:'whieh' are
transformed into elements of x4; on the whole we have
(X1 = (x13, X14),
X2 = (X23, 524)  where   
(6)                      -                where ~ik = =i.
23 =3 (X31, 232),
k4 =  (a41, X42),




Cardinal numbers


161


If T, is any part of xI, and T2 an equivalent part of x2, we may denote by
x1X, x2 the aggregates obtained by interchanging those elements of x, which
belong to T1 with those of x2 which belong to T2; we have then a similar set
of equations to (6) for the new starred aggregates, and =*= =, x* =2,
3C= X3, 4* = 4. If then the theorem be proved for the starred aggregates,
it holds for the original ones.
We have to shew that, after suitable transformations, a system of division
of the aggregates into parts, of the form in (6), can be found, such that
X13 +  414 =  4, and  31 + w32 = 32.  For from  these equations we deduce
X= =  = X14, X3 = 4 = X23, and then the aggregates x2, x4 are such that each
has a part which is equivalent to the other; and consequently, in accordance
with the equivalence theorem, x2, x4 are equivalent to one another; or c2 = 4.
It has in fact to be shewn that x,1 can be so chosen, that it is negligible
with respect to cardinal number, in comparison both with x14 and with x23.
We form the systems of transformation
Ob= X3'    c = X4, bcSbb = X6,...,cX3,       = X5, Oc bb=% c eX7, **;
each transformation x in this system has one inverse, given by the scheme
X4%nX4n+ = 1, X4n+2X4n+2 = 1, %4n+3X4n+3 = 1; thus the transformations x form a
group of reversible transformations of M = (xc, x2) into itself.
An element el3 of x,1 is either, (i) transformed into an element of x24 by a
transformation x with finite index, or else, (ii) e13 is not transformed into an
element of x,4 by any of the transformations x. Suppose, then, that for every
element e1, of x13 the second of these cases arises, then Xs2, X2r+1 transform
the elements of x,1 into aggregates which are respectively in x23 and x14, and
in them these aggregates form an enumerable system of separate parts of
each. For, in the case contemplated, X2 transforms xC3 into a part of x23; by
X4, the elements of x13 become elements of x23 or x24, consequently in accordance
with (ii), X4 (xc,) is a part of x23. In this manner it is seen, that x13 is transformed, by every,,, into a part of x23, and by every X%2s+1 into a part of '14.
It then follows, by Lemma (5), that $13 + X14 = 14, ~t + x23 = 21, and the
theorem is then completely established. The remainder of the proof consists
in shewing that, by an exchange of elements of xC with elements of x24, it is
possible to arrange so that the case just considered always arises.
Suppose x,,' are those elements of x,1 which are transformed by X2 into
elements of x24; let x,13 denote those elements different from x,,' which are
transformed by 3X into elements of x24 which were not affected by X2, and so
on; we have then the scheme
x13'            X2 (X13/)                    in X24,
X13' f+ 13/     X2 (x13) + X3 (xC13)         in 24,
x c13  13" + x13"  X2 (x13') * %X (X13 ) + %4 (X13")  in X4,
+ ** 3  X2 (x13 ): X3 (X131).. #+ X1 (x3 (n)) in x24.
H.                                                        11




162


Transfinite numbers and order-types


[CH. III


We take now the equivalent sums
00               00
[x13] =  X,, (), and [x24] = I Xn+l (X8,(n)),
Zn= 1             n=1
and we carry out an exchange of [X13] with [x24]; we then have
13 = [X13] + [(X13)], X24 = [X24] + [(24)].
When the exchange has been made of the elements of [x,3] with those of
[x24], we denote the new aggregates by starring the original ones; we have
then, in accordance with the formulae (6), expressions for x,*, x2*, x3*, x4,
and we can, as has been shewn above, attend to these, instead of to the
original x1, x2, x3, x4. Now no element of x13* is transformed into an element
of xc* by any of the transformations X, it being understood that the
transformations % are not to affect the substituted elements; and thus by the
reasoning which has been given above for the case in which no element of
x3 is transformed into an element of X24, the theorem is established. Bernstein
has also proved that if 2a = a + /, where a, / are cardinal numbers, then oa /3.
THE ORDER-TYPE OF SIMPLY ORDERED AGGREGATES.
122. An aggregate M is said to be a simply ordered aggregate when each
element mn has a definite rank relatively to the other elements of M, so that, of
any two elements m, mn' whatever, it is known which has the higher and which
has the lower rank.
If m has a lower rank than m', the fact is denoted symbolically by m < mn';
and if a higher rank, by in > m'.
If an aggregate is given at first unordered, it may be possible to order
the aggregate in a variety of essentially distinct ways. If the aggregate is
finite, the ordering of it may be accomplished by arbitrarily assigning to each
element its rank relatively to the others. In case the aggregate is an infinite
one, the ordering of it consists in the setting up of some general rule which
suffices logically to assign the relative order of any two elements.
Besides simply ordered aggregates there exist also doubly or trebly
ordered aggregates, or also aggregates with higher degrees of multiplicity
of order. Each element of such an aggregate possesses two, three, or more
distinct characteristics of an ordinal character. Simply ordered aggregates
only will be here considered.
Two simply ordered aggregates M, N   are said to be similar, when a
(1, 1) correspondence can be established, in accordance with some law, such
that to any two definite elements in, mn' of M  there correspond two definite
elements n, n' of N, in such a manner that the relative order of m, n' in M,
is the same as that of the corresponding elements n, n' in N.
This relation of similarity may be represented symbolically by M- T.




121-123]            Simply ordered aggregates                   163
Every simply ordered aggregate is similar to itself.
Two simply ordered aggregates which are similar to a third are similar to
one another.
All simply ordered aggregates which are similar to one another are said to
have the same order-type.
An order-type is accordingly characteristic of a class of similar aggregates.
The order-type of a simply ordered aggregate M is defined by Cantor
as the concept which is obtained by abstraction when the nature of the
elements of M is disregarded, their order being alone retained. The ordertype of M is then denoted by M. This definition will be further discussed
in ~ 155. That similar aggregates have the same order-type is regarded
by Cantor as a deduction from this definition.
If in M, we further disregard the order of the elements, we obtain MA, the
cardinal number of AM.
The order-type of M is, from Cantor's point of view, regarded as a simply
ordered aggregate similar to M, such that each element is the number 1. If any
order-type be denoted by a, the corresponding cardinal number is denoted by a.
Corresponding to any given transfinite cardinal number, there is a
multiplicity of order-types, which form a class of order-types; each such class
of order-types is characterised by the common cardinal number of all the
order-types of the class.
The order-types which belong to thi. class corresponding to a cardinal
number a, form an aggregate which has a cardinal number a'. It will appear
that a' is always greater than a.
If the order of every pair of elements in a simply ordered aggregate M be
reversed, the aggregate in the new order is denoted by *M.
If M- a, then the order-type *M is denoted by *a.
The order-type of the aggregate of all the finite integers in their natural
order (1, 2, 3,...), is denoted by w. This is therefore the order-type of
every aggregate (a,, a2,... an...) which is similar to (1, 2, 3,...).
The aggregate (... a... a3, a2, a,) has the order-type *w.
THE ADDITION AND MULTIPLICATION OF ORDER-TYPES.
123. If M, N denote two simply ordered aggregates, and if the aggregate
(M, N) be formed, in which all the elements of both M and N occur, and
which is such that any two elements of M have the same relative order as in
M, and that any two elements of N have the same relative order as in N,
and further that each element of M has a lower rank than all the elements
of N, then the new simply ordered aggregate (Ml, N) is said to be the sum
of the two simply ordered aggregates M and N. It is clear that if M = M',
11-2




164


Transfinite numbers and order-types


[CH. III


N   N', then (M, N)   (M', N'), and thus that the order-type of (M, N)
depends only on the order-types of M and N.
If M= a, N = 3, the sum a +/3 is defined to be the order-type of the sum
(M, N) of the two simply ordered aggregates, as defined above.
This defines the operation of addition of order-types. It will be seen
that the addition of order-types does not obey the commutative law.
For if a = M, /3 = 1V, then a +  /3= (M, N): but / + a = (N, M); and the two
order-types (M, N), (VN, M) are in general different from one another.
If n denotes a finite integer, w + n is the order-type of the ordered
aggregate (el, e2, e,... f, f2,... fn), whereas n + co is the order-type of
(fl, f2,... fk, el, e2, e,...). It is clear that the first of these aggregates is
not similar to (g,, g2, g*...), but if we let fi,f,... fn correspond to gl, g, g... gn,
then el to gn+l, e2 to gn+2,... and in general e,, to gn+mn, it is seen that the
second of the above order-types is similar to (gl, gg, 3...). It thus appears
that n + o = o, but co + n  wo.
124. In the simply ordered aggregate N, let us suppose that in the place of
each element is substituted a simply ordered aggregate similar to M, whereby
a new simply ordered aggregate is formed; this may be denoted by M. N.
It is clear that if MI  Mi', Ni   N', then M. N= M'. N', thus the order-type
of M. N depends only on the order-types of M and N.
If a = M, /3 = N, the product a. 3 is defined to be M. N, the order-type of
M. N, as just defined.
It will be seen that the product a. /3 is in general different from /3. a, and
thus that the multiplication of order-types does not obey the corlmurMt.tive
law. For example co. 2, is the order-type of the aggregate formed by substituting in (a,, a,) for each of the two elements an aggregate of type o;
co. 2 is therefore the order-type of (b1, b2, b3,... cl, c2, C3...), in which there is
no last element, and no element immediately preceding cl. On the other
hand, 2. co is the order-type obtained by substituting for each element in
(a,, a2, a....), an aggregate consisting of two elements; and 2. co is thus
the order-type of the enumerable aggregate (an,, a1, a21, a22, a31, a2...), which
is similar to (b,, b2, b,...), as may be seen by making a, correspond to b2n-1
and Ca12 to b2n. It has thus been shewn that 2. c = co, but co. 2: co.
THE STRUCTURE OF SIMPLY ORDERED AGGREGATES.
125. An examination of the structure of a simply ordered aggregate M
can, in general, only be attempted by considering the nature of those
aggregates which are its parts, and in each of which parts the order of
the elements is the same as that of the same elements in the whole
aggregate. The simplest transfinite part of an ordered aggregate is that
which has one of the types co, *c. Such parts we speak of as ascending
sequences, and descending sequences, respectively, contained in M.




123-126]


Simply ordered aggregates


165


Two ascending sequences {an], {al'}, contained in M, are said to be related
to one another, provided that, corresponding to any element a,, of the first,
there are elements a',n of the second, such that an < a',,; and provided also
that, corresponding to any element a,' of the second, there are elements a,,
of the first sequence, such that an' < aCn,.
Two descending sequences {bn}, {bn'} contained in M, are said to be related
to one another, provided that, corresponding to any element bn of the first
sequence, there are elements b'wn of the second, such that bn > b',; and
provided also that, corresponding to any element b,' of the second, there
are elements bn/ of the first sequence, such that b' > b,,,.
An ascending sequence {an], and a descending sequence {bn}, contained in
M, are said to be related to one another, if an < b,,,, for every n and n'; and
further, provided there exists in M no element, or only one element m, which
is such that an < m < bn, for every n.
Two sequences contained in an ordered aggregate, which are both related
to a third sequence, are related to one another.
Two sequences in an ordered aggregate, which are both ascending, or
both descending, and of which one is a part of the other, are related to one
another.
126. Suppose that in an ordered aggregate M, there is an element mo
which satisfies the following conditions, with respect to an ascending sequence
contained in AM:
(1) for every n, an < mno;
(2) for every element in of M which is < mo, there exists a number n
such that an, al+1, an+2,... are all > m; then the element m0 is said to be
the limiting element, or limit of {an} in M; and m, is said to be a principal
element of M.
Similarly, if we suppose that in M, there is an element mn, which satisfies
with reference to a descending sequence {an} contained in M, the following
conditions:
(1) for every n, an > mo;
(2) for every element mi of M which is > nto, there exists a number n
such that a,,, a,,,+, an+2,... are all < mi; then the element m, is said to be a
limiting element, or limit of {anc in M; and n,, is said to be a principal
element of M.
A sequence contained in M can never have more than one limiting
element in M.
If a sequence in M  has a limiting element m, in M, then m, is the
limiting element of every sequence in M which is related to the first one.




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Two sequences which have the same limiting element in M, must be
related to one another.
It is clear that, if M, M' are similar ordered aggregates, an ascending or
a descending sequence in M corresponds to a sequence of- the same kind in
M'. To every principal element in M, there corresponds a principal element
in M'.
An ordered aggregate which is such that every element is a principal
element is said to be dense-in-itself.
If, in an ordered aggregate, every sequence which is contained therein has
a limiting element in the aggregate, then the ordered aggregate is said to be a
closed aggregate.
An ordered aggregate which is dense-in-itself, and also closed, is said to be
perfect.
An ordered aggregate which is such that between any two whatever of its
elements, there are other elements of the aggregate, is said to be everywheredense.
The properties of an ordered aggregate thus defined, are also properties of
any similar aggregate; hence the terms may be applied to the order-types
which are symbolised by replacing the elements of the ordered aggregates
by 1; there can exist therefore an order-type which is dense-in-itself, or
closed, or perfect, or everywhere-dense.
The terms which have been here employed for the purpose of describing
certain peculiarities which may exist in an ordered aggregate, or in the
corresponding order-type, are identical with those which we have employed
in analogous senses in Chapter II, in the case of sets of points or numbers.
There is however a distinction which must be noticed between the use of the
terms in the two cases. To illustrate this distinction, let (Pi, P2, P... Pn...)
be a sequence of points on a straight line, which sequence has a limiting
point P, on the right of the points Pn; then if Q be any point of the
straight line on the right of P,, the two ordered aggregates (P1, P2, P3, 
Pn,... P(), and (P1, P2, P3,... P,,... Q), are similar, and have the same
order-type o + 1. In the first of these aggregates, P,( is the limiting element
of the sequence (Pi, P2,... P,...); and, in the second aggregate, Q is the
limiting element of the same sequence; and therefore both the ordered
aggregates are closed, in the sense explained above.  The first,of these
aggregates forms a closed set of points, in the sense of the term defined in
Chapter II; but the second does not, since Q is not a limiting point of the
set of points {Pn}. The distinction rests upon the different use of the terms
limiting element and limiting point, in the two cases of an ordered aggregate
of elements in general, and that of a set of points in the continuum. The
question whether an element is a limiting element of an aggregate to which




126, 127]


Simply ordered aggregates


167


it belongs, or not, in the sense defined above, is answered by examining the
structure of the ordered aggregate itself. In the case of a set of points in
the continuum, a particular point may be a limiting element of the aggregate
of points considered merely as an aggregate of elements with a particular
order-type; but the question as to whether the same point is a limiting
point of the set of points, considered as chosen out of the continuum, can
only be answered after an examination of the ordinal relation of the point
to other points of the continuum which do not belong to the set; in fact,
the set must be regarded, for this purpose, as an aggregate which is only a
part of another aggregate, the continuum. It is now clear that a set of
points considered solely as an ordered aggregate of elements, without
reference to the fact that it is essentially a part of the continuum, may be
closed, or perfect; and yet that the same set of points need be neither closed
nor perfect, in the sense of the terms employed in the theory of sets of points,
which has been dealt with in Chapter II.
THE ORDER-TYPES 7,,, 7t.
127. Certain order-types which are of special importance will be now
examined.
The first of these is the order-type q, of the set R of rational numbers
between 0 and 1 (both exclusive), in their order as defined in Chapter I.
It will be shewn that the order-type v, is exhaustively characterised by
the following properties:(1)  - = No= a.
(2) In r, there is no lowest and no highest element.
(3) r is everywhere-dense.
In fact, every simply ordered aggregate M, which has these three
characteristics, is similar to the aggregate R.
To prove this, we first observe that, on account of the condition (1), the
order of the elements in both M  and R can be so altered that each of
them is reduced to the order-type o. Let this be done; and denote by Mo,
RB the new ordered aggregates
Mo = (m91, 91m2, m,...),
Ro = (ri, r2, r,...).
We have to shew that M= R; and to do this we have to shew how to
establish the requisite correspondence between the elements i of M, and
r of R. Let n1i be made to correspond to r,; then there are an indefinitely
great number of elements of M, which have the same relation, as regards
order, to nlh, as r2 has, in R, relatively to r1; of all these elements choose that
one vm%, which has the smallest index as it appears in M,; and let mn,2 be




168


Transfinite numbers and order-types


[CH. III


made to correspond to r2. Of all the elements of M, which are related to
rnm and m 2, in the same manner, as regards order in M, as r3 is related to
r, and 2,, as regards order in R, choose that one m 3, which has the smallest
index as it appears in M,; and make mre, correspond to r3.
Proceeding in this manner, we make the elements r?, r2, r3... r, of R,
correspond to the elements in,, in2, mi... mm, of A2; and so far as these
elements are concerned the relations of rank are preserved in the correspondence: we proceed then to choose, in the same manner as before, the
element mg,+, which is to be made to correspond to r~+1; and thus we obtain,
for every r1,, the corresponding me,. It must however be shewn that this
process exhausts all the elements m of M, that is to say, that in the
sequence i, e2, e3,... El,,... every integral number p occurs in some definite
place. This can be proved by the method of induction. Let us assume
that the elements mi,,  mn,,... m,, all occur in the correspondence that
has been set up between the whole of R and at least a part of M, then
we shall prove that mn+, also occurs. Upon this assumption, let X be so
great that among the elements min, vnz,, ine,... imn, all the elements mn,, m,,
ml3,... mn, occur. Then if rn,,+ is not also among those elements, choose out
of rx+l, rx+2, rA+3,... that element rx^+ with the smallest index which has the
same relation to rl, r,,... rx, as regards order in R, that min,+ has relatively
to mi, nbel, m2,... m n, as regards order in M. Then the element mn,+ has
the same relation to in,, mi,, n 2,... meA+sl, as regards order in M, as rA+S
has to r,, r2,... rx+s_,, as regards order in R. It thus appears that mrn+, is
the element with the smallest index as it appears in M,, which has, in M,
the same relation as regards order to min, mn,... m,+s  that,+s has
relatively to r,, r2,... r^+s-, in R; hence min+ - mn,1; that is, the element
mn+1 occurs in the correspondence which has been established between M
and R. It has now been shewn that il and R are similarly ordered aggregates.
Examples of the order-type q are the following:(1) The aggregate of all negative and positive rational numbers including zero, in their natural order.
(2) The aggregate of all rational numbers which are greater than a, and
less than b, where a, b are two real numbers such that a < b.
(3) The aggregate of all real algebraical numbers in their natural order
in the continuum, or of all such of these numbers as lie between two real
numbers a, b.
(4)  The aggregate of a set of non-abutting linear intervals which are
such that their end-points and the limiting points of these end-points form
a non-dense perfect set of points in a linear interval.
The rational numbers of the interval (0, 1), including 0 and 1, form
an aggregate of the order-type 1 + q + 1.




127, 128]


Simply ordered aggregates


169


128. We now proceed to the consideration of the order-type 0, of points
forming a linear continuum.
It will be shewn that any simply ordered aggregate M is similar to the
aggregate X of all real numbers of the continuum (0, 1), in their natural
order, provided (1) M is perfect, and (2) in Il, an aggregate S, with the
cardinal number i0, is contained, which is so related to M, that, between any
two elements m,, mi of M, there are elements of S.
If S has a lowest and a highest element, these can be removed without
affecting its relation to M; and thus we may suppose S to be of the type iv,
of the aggregate R of rational numbers which lie between 0 and 1, both
exclusive, in their natural order.
Since S  R, we may suppose the elements of S to be made to correspond
in order to the elements of R; and it will be shewn that this correspondence
enables us to establish a correspondence between the elements of M and of X.
We suppose that each element of M, which belongs to S, corresponds to
that element of X which belongs to R, just as in the correspondence of S
with R already established. Any element m of M, which does not belong to
S, is the limiting element of a sequence {in,} of elements of S. To this
sequence {mn}, there corresponds a sequence {rn} in X, all the elements of
which belong to R; and this sequence {rn} has a limiting element x in X
not belonging to R; we take therefore m in iM to correspond to x in X. If
we take a different sequence {xc'}, which has the same limiting element m as
before, in M, then there corresponds to it a sequence t{r'} in R, which has the
same limiting element x as before, in X. It will now be shewn that, in
the correspondence so established between the elements of M and of X, the
relative order of two elements of M is the same as that of the corresponding
elements of X. This clearly holds of any two elements of M which are also
elements of S. Consider next two elements mi and s, of M, the first of which
does not, and the second of which does, belong to S; and let x,, r be the corresponding elements of X. If r < x,, there exists an ascending sequence in R,
of which x, is the limiting element, such that all its elements are > r; then to
this sequence there corresponds an ascending sequence in S, all the elements
of which are > s, and of which m is the limiting element; hence s < n. If
r > x,, it can, in a similar manner, be shewn that s > m. The proof that,
corresponding to any two elements inz, in2 of M which do not belong to S,
the elements x,, x2 of X are such that mn > m2, according as  < x2, is of
a precisely similar character to that just given. It has thus been shewn that
M and X are similar aggregates, and that the type 0 is characterised by the
conditions (1) and (2).
The above characterisation* of the type 0 contains Cantor's ordinal theory
of the constitution of the linear continuum.
* See Russell, Principles of Mathematics, vol. i, p. 303, also Veblen, Trans. Amer. Math. Soc.,
vol. vi, and Huntington, Annals of Math., Ser. 2, vols. vi and vii.




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Transfinite numbers and order-types


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A non-dense perfect set of points in a linear interval has not the ordertype 0, but the set of complementary intervals together with the limiting
points of their end-points does form an aggregate of order-type 0, when the
elements consisting partly of points and partly of intervals are taken in the
order in which they occur in the continuum.
129. The order-type *w + co may be denoted by 7r, and is the order-type
of the negative and positive integers in their natural order. This order-type
has properties distinct from that of o. For example, n + wt has been shewn
to be identical with co, where n is a finite integer, but n + r is not identical
with 7r. From  either of the equations n +r=z +7r, or r + n =r+m,
there follows m = n, or more generally t:If n, n' are finite integers, 4 and t' other order-types, from the equation
n + r + 4 = n' + 7r + '", there follows n = n', 4= S'.
To prove this theorem, we observe that, if the two aggregates be placed
into similar correspondence, the lowest elements correspond to one another,
then the second, and so on; hence n = n' is proved at once: and we now have
7r +  = 7r + 4'.
When two simply ordered aggregates M, + Z, N, + Z' of order-types
7r+ 4, 7r + ' are placed in correspondence in order, either M, corresponds
to N,., or M, corresponds to a part of VN,, or else N, corresponds to a part
of M,. In the last two cases the order-type wr must be split up into
7r = 7Ir + 7r2, where 7r = 71, and 772 is some other order-type; but from the
definition r = co + co, it is clear that every mode of dividing 7r into two
parts without altering the relative order of the elements, leaves it in the
form *to + Co; hence it is impossible that 77 = 71r + 7r,, and 7r = wr1, and therefore MI, corresponds to N,,. Hence also Z corresponds to Z', or 4= 4'.
NORMALLY ORDERED AGGREGATES.
130. The order-type of a simply ordered aggregate is, as we have already
seen, such that the structure of the aggregate as revealed by an examination
of the sequences contained in it, may be of the most varied character; the
various sequences may be ascending or descending ones, and may or may not
have a limiting element within the aggregate.
Of all the possible order-types, those are of especial importance which
have been defined by Cantor as the order-types of normally ordered aggregates
(wohlgeordnete Mengen).
A  normally ordered aggregate M is one which satisfies the following
conditions:(1) M has an element m, of lower rank than all the other elements.


t Bernstein, loc. cit., p. 9.




128-130]


Normally ordered aggregates


171


(2) If M1 is any part of M, and if M contains one or more elements
which are of higher rank than all the elements of Mi, then there exists one
elemeent m' of M, which immediately follows the part-aggregate M,, so that
there are no elements of M which are intermediate in rank between, m' and all
the elements of M.
The special case of (2) which arises when Mi consists of one element,
shews that a normally ordered aggregate is such that each element has one
which immediately follows it, unless the element is the highest element of
M. It is however not necessarily the case that M has a highest element.
If e,, e2, e3,... e,,,... is an ascending sequence of elements contained in M,
and such that elements exist in M, which are of higher rank than every e,
then there exists an element e' of M which is higher than all the e,,, and
such that every element e" of M which is lower than e' is lower than e,, ee+,
e,,+2,... for some definite value of n.
Every part of a normally ordered aggregate has a lowest element.
Let M, be a part of M; if M1 contains mni the lowest element of M, then
m, is the lowest element of M,. If M, does not contain m,, consider that
part of M which contains all those elements every one of which is of lower
rank than all the elements of Ji];; this part of M  must have an element
which immediately follows it; and this element belongs to Mi, and is its
lowest element.
If a simply ordered aggregate M itself, and also every part of M, has a
lowest element, M is normally ordered.
The condition (1) is fulfilled. Let M, be a part of M such that M contains
elements which are higher than all those of M,; let these form the aggregate
M2, and let m be the lowest element of M,. Then m is the element which
immediately follows M,; and thus the condition (2) is satisfied.
This property of a normally ordered aggregate, that every part of it has a
lowest element, might be adopted as the definition of a normally ordered
aggregate.
A somewhat simpler property which might be employed to define a
normally ordered aggregate is the following:An aggregate M is normally ordered t if, and only if, it contains no part of
which the order-type is wo.
If M is not normally ordered at least one part of it must have no lowest
element, and this part contains a sequence whose order-type is *co. An
aggregate which has a lowest element, and is also such that each element
has one that immediately succeeds it, is not necessarily normally ordered,


t See Jourdain, Phil. Mag., Ser. 6, vol. vII, p. 65.




172


Transfinite numbers and order-types


[CH. III


even if each element has one immediately preceding it.  This can be seen
by considering an aggregate with the order-type o + *w.
131. The following properties of normally ordered aggregates can be
proved in a very simple manner:Every part-aggregate of a normally ordered aggregate is itself normally
ordered.
Every ordered aggregate which is similar to a normally ordered aggregate
is itself normally ordered.
If in a normally ordered aggregate M there be substituted for the
elements normally ordered aggregates, in such a manner that if M,]n, M,,
are the aggregates substituted for any two elements m, n', then Mm a  Mm',
according as n < mn', the resulting new aggregate is normally ordered.
132. The part of a normally ordered aggregate M zwhich consists of all
those elements which are of lower rank than an element m, of M, is called the
segment of M determined by the element nm.
The aggregate which remains when the segment of M, determined by
the element m, is removed from M, is called the remainder of M determined
by the element m. The element m is the lowest element of the remainder.
If S is the segment of M formed by n, and R is the remainder, then
M=(S, R).
Of two segments S, S' determined by the elements mn, In' of which n < m',
we say that S is the smaller and S' the larger segment, or S < S'.
It can easily be seen that, if M, M, are two similar normally ordered
aggregates, a segment of M  corresponds to a similar segment of M,, the
element by which the segment of M   is determined corresponding to the
element of Ml by which the segment of M1 is determined.
A normally ordered aggregate is not similar to any of its segments.
Assume that, if possible, S  M, and suppose the elements of S, M are
put into correspondence. To the segment S of M, there must correspond a
segment S, of S, so that S1 = M = S, where S1 < S. Since S, = M, we find in
a similar manner a segment S2 < S1, which is similar to M, and so on; and in
this way we obtain an unending sequence S > S, > S2... > S,... of segments
of M which are all similar to M. Let m, ml1, m2... mn... be the elements
which determine the segments S, S,, S,... Sn...; then mi > ml > mn2... > n,....
The aggregate (...... mn,... m,, vn,, m) would be a part of M which has no
lowest element, and is of type *w, which is impossible if M  is normally
ordered.
If M is an infinite normally ordered aggregate, it always has parts which
are similar to M, although such a part cannot be a segment.




130-132]


Normally ordered aggregates


173


A normally ordered aggregate cannot be similar to any part of one of its
segments.
Let us assume that, if possible, S' a part of a segment S, of M, is similar
to M. Since S' M, we can place the elements of S', M in correspondence,
then to the segment S of M there will correspond a segment S1 of S', where
S1- S; let then S, be determined by the element e, of S'. Since e, is also
an element of M, it determines a segment M1 of M, of which S is a part,
and which has a part similar to M. Proceeding in the same manner, we
determine a segment M2 of M, which has a part that is similar to M;
and in this way we obtain an unending sequence of segments of M, all
similar to Ms, so that M > M1 > M,...> MA.... The elements which determine
these sequences form a part of M which is of type o*, and this is contrary to
the hypothesis that M is normally ordered.
Two dfferent segments of a normally ordered aggregate cannot be similar.
For one of these segments is a segment of the other.
There is only one mode of putting the elements of two similar normally
ordered aggregates into correspondence, so that the relative orders of the
elements are unaltered in the correspondence.,
For if in two modes of placing the aggregates in correspondence two
elements f,f' of one aggregate M, correspond to one element e of the other
M', the segments of MA determined by f, f are each similar to the segment of
M' determined by e; but it has been shewn to be impossible that M can have
two different segments which are similar to one another.
A segment of one of two normally ordered aggregates has at most one
segment of the other aggregate which is similar to it.
If S, S' are similar segments of two normally ordered aggregates M, M',
then to every smaller segment S, < S, of M, there corresponds a similar
segment S' < 8', of M'.
If S8, S2 are two segments of the normally ordered aggregate M, and
81,,,' are two similar segments of a normally ordered aggregate M', then if
Sl < S, it follows that S,' < S.
If a segment S of M is not similar to any segment of another normally
ordered aggregate M', then no segment S' > S of M is similar to any segment
of M' nor to M' itself; and the same holds of M itself.
If M, iM', two normally ordered aggregates are so related that, to any
segment of either, there corresponds a similar segment of the other, then Ml- M'.
Any element e of M determines a segment of M which corresponds to a
similar segment of M'. Let this latter be determined by an element e' of
M'; we then take e to correspond to e'. To every element of M we therefore




174            Transfinite numbers and order-types              [cH. III
find a corresponding element of M', and it is seen by applying the foregoing
theorems that the relative order of the elements is preserved.
133. If two normally ordered aggregates M,'M' are so related that, (1) to
every segment S of M, there corresponds a similar segment S' of M', and (2)
at least one segment of M' exists to which there is no corresponding similar
segment of M; then there exists a definite segment S' of M' such that S1'  M.
Consider all those segments of M', which do not correspond to similar
segments of M. Among these, there must be one S1' which is the least of
all; this follows from the fact that the elements which determine these
segments of M' form an aggregate which has a lowest element, and this
lowest element determines the segment S/,. Every segment of M' which is
greater than S,', is such that there exists no corresponding similar segment
of M; but every segment of M' which is less than S,' has a corresponding
similar segment of M. Since to every segment of M there corresponds a
similar segment of SI', and to every segment of S/' there corresponds a similar
segment of M, it follows that Mi S/'.
If the normally ordered aggregate M' has at least one segment to which
there corresponds no similar segment of M, then to every segment of M there
corresponds a similar segment of M'.
Let S' be the smallest segment of M', to which there corresponds no
similar segment of M. If there existed segments of M to which no corresponding similar segments of M' exist, let Sx be the smallest of all such
segments of M. To every segment of S1 there corresponds a similar segment
of S/', and conversely; hence S, S,', which is contrary to the hypothesis
that there exists no segment of M which is similar to S/'.
If M, M' are any two normally ordered aggregates, then either (1) M and
M' are similar, or, (2) there exists a segment S' of M' which is similar to M,
or, (3) there exists a segment S of M, which is similar to M', and these
possibilities are mutually exclusive.
The following four possibilities may be contemplated, as regards the
relation of M to M':(1) To every segment of either M or M' there corresponds a similar
segment of the other aggregate.
(2) To every segment of M there exists a corresponding similar segment
of M'; but there is at least one segment of M' to which no similar segment of
M corresponds.
(3)  To every segment of M' there corresponds a similar segment of M;
but there is at least one segment of M to which no similar segment of M'
corresponds.




132 —134]


Ordinal numbers


175


(4) There is at least one segment of M to which no similar segment of
M' corresponds, and also at least one segment of M' to which no similar
segment of M corresponds.
It has been shewn that (4) is impossible. In the case (1), it has been
proved that M   M'. In the case (2), it has been shewn that a definite
segment S,' of M' exists, such that 8,'- M; and in the case (3), that there
is a definite segment S1 of M such that S = M'.
It is impossible that at the same time M= M', and also M-= S': for, in
that case, M' Si'; and it has been shewn to be impossible that M' is similar
to one of its own segments.
It is also impossible that M= S/,, and also M'  S,; for there must then
exist a segment of S,/, which is similar to S, and therefore to M'; but this
is contrary to the theorem that a normally ordered aggregate cannot be
similar to one of its segments.
If any part of M is such that that part is not similar to any segment of M,
then that part is similar to M itself.
Any part M, of M is normally ordered; if then M1 be similar neither to M
nor to any segment of M, there must exist a segment of M1' of M1 which is
similar to M; and  I1' is a part of that segment of M which is determined
by the same element that determines the segment M' of M,. Therefore MI1
would be similar to a part of one of its segments, which has been shewn to
be impossible.
THE THEORY OF ORDINAL NUMBERS.
134. The order-type M of a normally ordered aggregate M, is said to be
the ordinal number which belongs to M; all similar normally ordered
aggregates have consequently the same ordinal number.
If M, M' are two normally ordered aggregates such that M has a segment
which is similar to M', whilst M' has no segment which is similar to M, then
the ordinal number a _M, is said to be greater than the ordinal number
/3 _ M'; and this relation is denoted by a > /. If M has no segment simnilar
to M', but M' has a segment similar to M, the ordinal number a is said to be
less than 3, and the relation is denoted by a < 3.
It follows from these definitions in conjunction with the theorem of ~ 133
that if a, / are any two ordinal numbers whatever, they satisfy one, and one
only, of the relations a =/3, a > /, a</; and that if a > /, then / < a.
Further it is seen that if a < / and / < y, then a < y; hence the aggregate
of all ordinal numbers is a simply ordered aggregate, when arranged in such
a manner that any one a, which has been defined as less than another one /,
precedes it.




176


Transfinite numbers and order-types


[CH. III


The sum a + 3 of two ordinal numbers is, in accordance with the general
definition of the sum of two order-types, the order-type of the normally ordered
aggregate (M, N), where M, N are two normally ordered aggregates such that
a=M, 13=N.
Since M, N each contains no part of type *o, the same is true of (M, N).
Hence the aggregate (M, N) is normally ordered; and thus a +/3 is an ordinal
number.
Since M is a segment of (M, N), we see that a < a + /.
N is a remainder of (M, N) determined by the lowest element of N,
hence N may be similar to (M, N); or, if not, it is similar to a segment of
(M, N): thus either /=a +,, or /3< a +/.
The addition of ordinal numbers obeys the associative law, but not in
general the commutative law; thus (a + /) + y = a + (/3 + 7), but a + 3 is in
general  /3 + a.
135. The product a.,3 of two ordinal numbers is, in accordance with the
definition of ~ 124, the order-type of the aggregate obtained by substituting
for each element of an aggregate of order-type 1, an aggregate of order-type a.
In accordance with the theorem of ~ 131, the aggregate thus obtained is
normally ordered, and of type dependent only on a and /3.
In general a. 3 is not equal to 3. a.
It is easily seen that a.3 >a, provided /3>1; and that if a/3=ay,
then 3 = y.
If a, 13 are two ordinal numbers such that a < 3, there exists an ordinal
number y such that a + y = 3; and this n mnber y is defined to be 3 - a.
For if M= 3, there is a segment of M which may be denoted by M1, such
that M1= a; let then M= (MI, S), therefore M= M  + S, and  - a = S.
136. Let 13, 2,,3.../n,... denote a simple sequence of ordinal numbers,
and suppose Ml, M2,... I/,,... are aggregates of which the order-types are
respectively the numbers of the sequence. The aggregate (Mi, I,... Mn,...),
which is obtained by replacing each element of the normally ordered aggregate
(1, 1, 1,...) of type c, by a normally ordered aggregate, is, ih accordance with
the theorem of ~ 131, itself normally ordered; and its type defines the sum
31 +132+. +..  +. = 3. If an denotes the sumM32 +1, +... + /n, we see
that an = (Mi,,,... Mn); and it is clear that a4n+ > an: hence
131 =,  2 = a2-al,.., 1?n = an - Y,1.It will now be shewn (1) that / > an, for every value of n; and, (2) that, if 13' is
any ordinal number < 8, there is some definite value of n such that a,
an+l,... are all > 3'.




134-137]


Ordinal numbers


177


(1) follows from the fact that each as is the ordinal number of a segment
of (M1, M2,... M.,...) of which / is the ordinal number.
To prove (2), we observe that a segment of (M1, M1'2,... M,...) exists, of
which /' is the ordinal number, and therefore the element which determines
this segment must belong to one of the aggregates HM, M',..'. M),... say M1J.
It follows that the segment is also a segment of (Mi, M.2,... Ml,); and therefore, 3' < an,, or a,, > /', n being > n.
It has thus been proved that 8 is the ordinal number which immediately
follows all the ordinal numbers al, a2,... ac,...; and it may be spoken of as
the limit of the sequence a,, a2,... a,,....
Thus every ascending sequence al, a2,... a,x,... of ordinal numbers determines a limiting number / = L an, which immediately follows all the numbers
7n = 00
of the sequence.
THE ORDINAL NUMBERS OF THE SECOND CLASS.
137. Every finite ordered aggregate is normally ordered, and its ordertype is the ordinal number of the aggregate. The finite ordinal numbers
may be spoken of as the ordinal numbers of the first class; to each such
ordinal number there corresponds a single cardinal number, and the properties
of the finite ordinal numbers are identical with those of the finite cardinal
numbers, the terms ordinal and cardinal simply defining the two uses of the
same number.    In the case of transfinite aggregates there is no such
identity between ordinal and cardinal numbers; in fact the arithmetic of the
one kind of numbers is essentially different from that of the other kind.
Corresponding to a single transfinite cardinal number there is an infinity
of transfinite ordinal numbers; all those transfinite ordinal numbers which
correspond to aggregates that have one and the same cardinal number a are
said to form a class Z(a), the class of normal order-types which have the
cardinal number a.
The ordinal numbers of all those order-types which have the same
cardinal number 0, as the aggregate of finite numbers, are said to be of
the second class Z(NO).
The ordinal number co = L. n, and is the smallest number of the second
class.
If M denotes the aggregate (m, m,,... m,n,...), then M= o, and  = 0.
Any number /3 which is < o, must be the order-type of a segment of M, and
M has only segments (mo, m2,... mn) with finite ordinal numbers n; thus /
must be a finite number; and therefore the only ordinal numbers < w are
finite ones.


H.


12




178


Transfinite numbers and order-types


[cH. ITT


Every number a of the second class has a number a +       immediately
following it.
For if a = M, a = Ko, we have a + 1 = (M, e), where e is a new element; and
since M is a segment of (M, e), we have a + 1 > a. Also
a+~ 1    =a+ 1 =   + 1 =0.
It has thus been shewn that a+ 1 is a number of the second class. Every
number < a + 1, is the order-type of a segment of (M, e); and such segment
can only be M, or a segment of M; hence no number < a + 1 is > a: therefore
a +1 is the next number greater than a.
If a1, a2,... a,,... is any sequence of mnmbers of the second class, there is
a number L. a,, also of the second class, which is the smallest number that is
greater than every number a, of the sequence.
If, as in ~ 136, we write /,8 = al, 2 = a,2- a... /, =  a- a-an_1,... then if
Gn =,n we have Lant=(G,, G2,... G,,...); and this number La, has been
shewn to be the smallest number which is > a for every value of n. To
shew that this number LIa is of the second class, we have, since /n 3< No, for
every value of n,
L. an c 0,.,0  No; and since L. an is not finite it must therefore = -0.
Two sequences {an), {a'nL of numbers of the second class, have the same
limiting number, when, and only when, the sequences are related to one another,
in accordance with the definition of ~ 125.
Let /, y be the two limiting numbers, and first assume that the sequences
are related to one another.  If 3 < y, then for some value of n, a' > 3,
a',+i > /,..; and hence for some value of n', we must have a, > /3, an'+~l > /,...
which is inconsistent with / being the limit of the sequence {an}.
If we assume / = 7, then, since an < y, for some fixed number r we must
have a'. > a,,, a'l,,> a,...; and similarly, since a', </, for some fixed
number s we must have a. > a',, as+,,> a ',...; hence the two sequences are
related to one another.
If n is a finite ordinal number, and a a number of the second class, then
n + a = a, and hence a - n = a.
For                          + o = o,
since             n +  =(e,, e,... en,; i,,.,...)
= (g,   *, gnl, gn1 *.)
where        1 = e, g2= e,,... g= e,,, g,+ =f, g+2 =f)...
Further,     ni + a = n +  ~ + (a - o) = w + (a -  ) = a.
If v is a finite number, then no = o. This is seen, by taking an aggregate
of the type o, and replacing each element by n new elements; then it is clear
that the new aggregate is also of type o.




137-139]                  Ordinal numbers                           179
It can easily be proved that (a + n)  = caw, where a is of the second class,
and n of the first class.
138. If a is any number of the second class, then the numbers of the
first and second classes, which are less than a, form a normally ordered aggregate
of type a, when they are arranged in order as defined above.
If M is an aggregate such that M= a, and if a' is an ordinal number < a,
then there is a segment M' of M such that M' = a'; and, conversely, every
segment of M determines a number of the first or second class which is < a.
For, since M = N0, any segment M' must have either a finite cardinal number, or
else must have 0, for its cardinal number. If e, is the lowest element of M,
a segment M' is determined by an element e' > e,; and every element e', of /M,
determines a segment M'. If-e', e" are two elements of M, both > e,, and M',
M" the segments of M determined by these elements, and a', a" their ordertypes, then if e' < e", it follows by ~ 132, that M' < M", and hence a' < a".
If then M= (e,, M'), and to the element e' of M', we make the element a'
of {a'}, correspond, the two aggregates M' and {a'} are placed in the relation
of similarity. It has thus been shewn that {a'} = M'; now M'= a -   =a,
hence  a'} = a.
Since a = N0, we have {a'} = No; and therefore the following theorem is
established:
The aggregate {a'} of all those numbers a' of the first and second classes,
which are ordinally smaller than a number a of the second class, has the
cardinal number No.
139. Every number a of the second class is either (1) such that it is
obtained from a number of the same class immediately preceding it, by the
addition of unity, or else, (2) such that there exists a sequence {a1,\ of Enmbers
of the first or second class, having a for its limit.
Let a = M; then if M has a highest element e, M-(M', e) where M' is
the segment of M determined by e; in this case M= M' + 1, or a = (a - 1) + 1.
If M has no highest element, then the aggregate {a'} of all numbers < a,
which is similar to M, has no greatest number; and this aggregate ta'} being
of cardinal number No can be re-arranged as an aggregate [a',j of type co. In
this aggregate {a'j}, some of the numbers a,', a3,... will in general be less
than a', but others must be greater than a/'; for a,' cannot be greater than
all the other numbers of the aggregate, there being in {a'} no greatest
number. Let a'P2 be that number of {a',, with the smallest index, such that
a', > a/'; similarly let a, be that number with the smallest index such that
a     ' >, a, nd so on. We have now an infinite sequence.P, Cp  ap 1,,
12-2




180


Transfinite numbers and order-types


[OH. III


of numbers such that they are in ascending order, and such that their indices
are also in ascending order. Since n - pn, we have a'n  a'; hence for every
number a' which is less than a there exists a number a', which is > a'.
Since a is the number which follows next after all the numbers a', it is also
the number which follows next after all the numbers a,', a', a... which we
may write as a,, a, a3,... an,...; thus a = Lan.
It has thus been shewn that there are two kinds of numbers of the
second class, (1) those which have an immediate predecessor in the aggregate
of all such numbers arranged in ascending order, and (2) those which have
no such immediate predecessor, and are called limiting numbers.
A number of the first kind is obtained by means of the first principle of
generation, (see ~ 61), from the immediately preceding number.
A number of the second kind is obtained by the second principle of
generation, as the number a which next follows all the numbers at of any
sequence {an} of numbers of the second class.
THE CARDINAL NUMBER OF THE SECOND CLASS OF ORDINALS.
140. The totality of the numbers of the second class arranged in ascending
order forms a normally ordered aggregate.
If A. denotes the ordered aggregate of all those numbers of the second
class which are less than the given number a, then A, is normally ordered
and of type a - t. For the aggregate {a'] of numbers of the first and second
classes, which consists of {nj and Aa, has been shewn in ~ 138, to be normally
ordered, and thus
a'} = ({n}, A,),
hence                a'}:= In} + A., or A  = a - w.
Let M denote any part of the aggregate {a} of all the numbers of the
second class, such that in {a} there are numbers which are greater than all
the numbers in M; and let a, be one such number: then M is a part of Ao+l,,
which is such that all the numbers of M are less than at least one number a,
of A o+,. Since A+,,  is normally ordered, there must be a number a' of Ao+l,
being itself consequently a number of {a}, which is the next greater number
than all the numbers of M. Thus, since {a} has a lowest number w, the
conditions are satisfied that {a} is a normally ordered aggregate.
It follows by applying the results of ~ 130, that:Every part of the aggregate {a} of all numbers of the second class has a
least number.
Every such part, in order, is normally ordered.




139-141]


Aleph-numbers


181


It will now be shewn that the aggregate {a} of all the numbers of the
second class, has a cardinal number greater than  o.
If {a} = R0, the numbers of tc} could be arranged in the form
7Y, 'Y2,.  Yn,...
of type ac, in which of course the order would not be that of generation.
Starting from ry, let 7Py be the 7 with the smallest index which is such that
7p2 > y7; then let 7p3 be that 7 with the smallest index such that yp3 > 7p,;
and so on. We obtain in this manner a sequence
in ascending order, the indices 1, p,, p3,... being also in ascending order. In
accordance with ~ 137, there must be a definite number 8 of the second class,
namely   = Lyp,,, such that 8 > y,,, for every p,, and consequently such that
8 is greater than every yn; but this is impossible since {7y} contains every
number of the second class; hence {a} cannot equal No.
Every part of the aggregate {a} of all numbers of the second class has
either the cardinal number of {a}, or else the cardinal number Kn,,unless it is a
finite part.
Every such part, when the elements of it are in order of generation, being
part of the normally ordered aggregate {a}, is either similar to {a}, or else to
some segment A,, of {a}; hence the cardinal number is either that of {a} or
is AO =- o-c), and this last is either o0, or is finite.
The cardinal number of {a} is the cardinal number next greater than No.
If there existed a cardinal number less than {a}, and greater than N0, it
must be the cardinal number of some part of {a}; but it has been shewn that
every such part of {a} has either the cardinal number of {[a, or is N0, or is
finite.
The cardinal number of {a}, or of Z {t0} is denoted by Ml.
THE GENERAL THEORY OF ALEPH-NUMBERS.
141. It has now been shewn that the ordinal numbers of the second
class in their order of generation, form a normally ordered aggregate of which
the cardinal number is N1, the next greater cardinal number to Ko. The
ordinal type of the normally ordered aggregate {a} of all numbers of the
second class, is a number 2, which is the smallest number of the third class.
In analogy with the definition of the second class, and in accordance with
what Cantor has denominated the principle of limitation (Hemmungsprinzip),
the third class is taken to include all the ordinal types of normally ordered
aggregates, of which the cardinal number is Rx, and this class is consequently
denoted by Z(R1). The number f1, which is the order-type of all the numbers




182


Transfinite numbers and order-types


[CH. III


of the first and second classes, in the order of generation, and which
comes after all those numbers, is not the limiting element of any sequence
a,, a2,... an,... of numbers of the second class; for, as we have seen, every
such sequence has a limiting number within the second class. From the
point of view adopted by Cantor in his earlier writings,-and explained in
~ 61, in which the successive ordinal numbers are regarded as successively
generated, in accordance with postulated principles of generation, the number
12 must be regarded as generated by a third principle of generation, different
from the two principles of generation employed in the case of the numbers of
the first and second classes. This third principle of generation affirms that
every set of ordinal numbers similar to the aggregate of all the numbers of
the first and second classes, in their order of generation, is immediately
succeeded by a new number, ordinally greater than all the numbers of the
set, so that every number which is less than this new number is also less
than some of the numbers of the set.    When, proceeding from   2, the
numbers   2 + 1, 2 + 2,... 1 + 12,... are formed, all three principles of
generation will be required, in forming the numbers of the third class.
From the point of view adopted later by Cantor, and explained in the
present chapter, fl is simply defined to be the order-type of the totality of
the numbers of the first and second classes, in their normal order. The
numbers higher than f2 are then defined in the same manner, each one as
the order-type of the totality of the preceding numbers in normal order.
The existence of a whole series of classes of order-types of normally
ordered aggregates, i.e. of ordinal numbers, has been speculatively asserted
by Cantor*, who has however, up to the present time, in his published works,
confined his detailed investigations to numbers of the first and second classes.
To each of the successive classes of numbers, there corresponds a single
cardinal number, that of the totality of the ordinal numbers up to, and
including all the ordinal numbers of that class. The first ordinal number of
each class is the order-type of all the numbers of the preceding classes in
their order of generation. A new principle of generation is required for the
first number of each new class, since that number cannot be regarded as the
limiting number of any sequence of which the ordinal number is less than
that of the number in question. All the successive principles of generation
are however included in the one principle, that an aggregate of normally
ordered ordinal numbers has itself an order-type which is a new number;
and thus, from this point of view, all the principles of generation, from the
second, onwards, are replaced by this one principle.
142. In accordance with this theory, there exists an ordered aggregate
1,  2,  3,...,...,.....,... 7o, +....,   f. o... 7,


* See Math. Annalen, vol. xxI, pp. 587, 588, also vol. XLVI, p. 495.




141-143]


A le~ph-numnbers


183


which contains every ordinal number of every class; and there also exists a
similar aggregate
1,  2,  3,... I,... No,  N,  N2 A,...  N,,...  N,...  N,... 
of cardinal numbers, each element of which is the cardinal number of a single
class of numbers of the first aggregate.
That the first of these aggregates is normally ordered, may be seen by
remarking that if it contained any part, of the type *o), then such part would
also be part of the normally ordered aggregate formed by the numbers
1, 2, 3,... w,... a; where a is the highest number in the hypothetical part,
of type *o. This is impossible, and hence the first aggregate is normally
ordered.
Cantor has proved (see ~ 117) that N0 is less than or equal to the cardinal
number of any transfinite aggregate, and that N1 is the cardinal number next
greater than N,. A proof has been given by Jourdaint, that N2 is the next
greater cardinal number than Ni, who has also considered in some detail, the
ordinal numbers of the third class, and has given indications of extension to
the higher classes.
The question whether every transfinite cardinal number is necessarily an
Aleph-number, which is equivalent to asking whether every aggregate is
capable of being normally ordered, has engaged a considerable amount of
attention. That the answer should be an affirmative one, has been regarded
by Cantor as probable. Some discussion of attempts which have been made
to settle this matter, will be considered in ~ 161. A case of great importance
is that of the continuum, which is defined as a simply ordered, but not as a
normally ordered aggregate. No proof has yet been discovered, of the correctness of Cantor's view, that c = N,. In case c occurs at all in the aggregate
of Aleph-numbers, the continuum is capable of being normally ordered. The
possibility has also been contemplated that c may be greater than all the
Aleph-numbers.
THE ARITHMETIC OF ORDINAL NUMBERS OF THE SECOND CLASS.
143. The ordinal numbers of the second class have been defined as the
order-types of normally ordered, enumerably infinite, aggregates; and the
operations of addition and multiplication have been defined for these numbers,
in ~ 134 and 135. It now remains for us to define exponentials for numbers
of this class; and the definition is founded upon the following theorem:If | is a variable of which the domain consists of the numbers of the first
and second classes, including zero, and if y, 8 denote two constants belonging


t See Phil. Mag. for 1904, " On the transfinite cardinal numbers of number-classes in general."




184


Transfinite numbers and order-types


[CH. III


to the same domain, such that 8 > 0, y > 1, then there exists a single-valued
determinate function f(f), which satisfies the conditions
(1) f(0) = 
(2) If ~', 1" are any two values of:, such that:' < a", then f(') <f(").
(3) For every value of:, f (d + 1) =f(). 7.
(4) If {,} is a sequence of which: is the limiting number, then {f(j)}
is a sequence of which f(S) is the limiting number.
In the case 8= 1, the function f()) is denoted by yt; and then f(a),
satisfying the above conditions, defines the exponential function yt, for all
numbers y, f of the first and second classes.
To prove the theorem, in the first place we have
/(1) = 87, /(2)=   2,..., / (n) =  'ynz;
thus f(1) <f(2) <f(3),..., and the function is determined for every  < W.
Next assume that the function is determined for every:< a, a number of the
second class. If a is not a limiting number, f(a) =f(a - 1) y >f(a - 1);
and thus f(a) is determined. If a is a limiting number, and is preceded by
the sequence {an}, then {f(|n)} is a sequence, and f(a) = Lf(an). If {an'} is
another sequence such that a = Lan', then the two sequences {f(an)}, {f(a,')}
are related to one another, and therefore have the same limit; and thus f(a)
is uniquely determined. f (:) is now determined for every:: for if there were
values of a for which it were not determined, there must be a smallest of
such values; the theorem would then hold for   < a, but not for >   a;
which is contrary to what has been proved above.
144. If a, 3 are numbers of the first or second class, ya+P = ya. y7.
The function ( (I) = ya+, satisfies the conditions
(1)   (0) = a.
(2) If ' < A",  (') <  ~ (pf ).
(3) ~(f+1)=-(    f).7
(4) If {n} is a sequence such that L:, = A, then (b ( L) = L( (n).
It follows by ~ 143 that, if we take 6 = ya, then (b ( a) =/yayt; hence if
=/3, we have
7a+p = 7~a. 7Y.
Again, if a, / are two numbers of the first or second class
7"   (7y)P
If we putf() = 7ya, we find by applying the theorem of the last section,
that f() = (ya)t, where ya replaces 7.
7 being > 2, it can be proved that, for every 7f, 7y >. The theorem holds
for f:=0, f=l1; and if it be assumed to hold for all values of: which are
less than a given number a, then it holds also for I = a.




143-146]


Order-functions


185


For, first let a be not a limiting number: then, if a- l  ya-l, we have
(aq- 1) 'y -ya;
hence ya > ( - 1) + (a - 1  ) (: therefore since a - 1 and y - 1 are > 1,
and (a-1) + 1 = a, we have ya a. If a is a limiting number= La,,, then
since a,  ya,,, we have La,, Ly7n, or a _ 7ya. If there were values of: such
that 4 > 7t, there must be one of such values which is the least of all; and if
this were a, then: _ yt, if 4 < a, but a > 7y; which is contrary to what has
been proved above.
145. Of all the numbers of the second class, the smallest ones are those
which are algebraical functions of Co, of the form
n~. p1n + cI'1.. p.n-1i +.. - (.-pi + Po,
where Po, Pi,... Pn are finite numbers. If we write
01 = (0,t (0.2 = O), (03 = ()2,...
then we obtain the number e0 = Lwn. This number e0 is the smallest of a
species of numbers of the second class which are characterised by the property
e = e6, and which Cantor has designated e-numbers. Cantor has shewn that
the e-numbers form a normally ordered aggregate of type f, and therefore
similar to the whole second class of numbers. He has further shewn that
every number a of the second class is uniquely representable in the form
a = wao/o + (a)alK l+... -+ (at'/C,,
where a0, a,, 2,... a,. are numbers of the first or second class which satisfy
the conditions a0 > al > a2... > a,. 0, and Ko,  C2, K2,... K,), r + 1, are numbers
of the first class which are different from zero. For the detailed investigation
of the normal form, and for that of the special class of e-numbers, we must
refer to Cantor's original discussion*.
THE THEORY OF ORDER-FUNCTIONS.
146. A method of representation of any mode of ordering a given
aggregate M  has been given by Bernsteint. When the elements of the
aggregate are numbers, this method lends itself to a diagrammatic representation of the aggregate as ordered in any particular order-type.
An aggregate M is ordered, in the most general sense of the term, when
it is known as regards every pair of elements a, b, whether a < b; but a
particular mode of ordering the aggregate can be represented by means of
a function f(a, b) of the pairs of elements, which is defined by
f(a, b)= 1, if a < b; f(a, b)=-1, if a > b, and f(a, a)= 0.
M ath. Annalen, vol. XLIX, pp. 235-246.
t See his Dissertation, also W. H. Young, on " Closed sets of points and Cantor's numbers,"
Proc. Lond. Math. Soc., Ser. 2, vol. i.




186


Transfinite numbers and order-types


[CH. III


This function f(a, b) may be denominated an order-function of the aggregate I; and there is one order-function for each possible mode of ordering the
aggregate. The function must satisfy the condition that, if f(a, b)=f(b, c),
then each equals f(a, c).
Two order-functions fi (a, b), f2 (a, b) of a given aggregate Ml, represent
two methods of ordering the aggregate in one and the same order-type,
provided there exist a reversible transformation b, of the aggregate M into
itself, such that f, {I (a), b (b)} =f, (a, b).
All those order-functions of a given aggregate M, which correspond to an
arrangement of M in one and the same order-type, constitute a family of
order-functions; and there is one such family of order-functions corresponding
to each order-type in which the given aggregate M can be arranged.
It is clear that the order-functions of a family corresponding to M form
an aggregate with the same cardinal number as the group of transformations
of M into itself.
If, in particular, the aggregate M is that of the positive integers, then a
pair of elements (a, b) is represented by a cross-point of the rectangular trellis
formed in the positive quadrant by drawing all the straight lines, x = a, y=a,
for positive integral values of a, referred to rectangular Cartesian coordinates
x, y.
The natural order of the numbers 1, 2, 3,... will be represented by f(x, y),
defined for all the cross-points, so that f(x, y) = 1, when x < y, and f(x, y) =- 1,
when x > y, and also f(x, x)= O.
Any particular mode* of ordering the numbers 1, 2, 3,... will be represented by marking one set of cross-points + 1, and another set - 1, those on
the diagonal x = y, being marked zero.
It is, however, not every mode of so marking the cross-points that represents a possible ordering of the aggregate. That a mode of marking may
represent a possible order, two conditions must be satisfied. First, we must
have f(x, y) = -f(y, x); and thus points which are optical images relatively to
the diagonal y = x, must be marked with unities of opposite sign. Secondly,
the condition that if f(a, bi) =f(bx, c), then each =f(a, c) must be satisfied.
This condition may be expressed as follows:-Join every cross-point which is
marked + 1, or 0, with every other such cross-point, then the resulting figure
may be called the positive frame-work; join similarly all the pairs of points
marked - 1, 0; in this way we obtain the negative frame-work. Let lines
joining the two pairs of points (x,, y,), (x,, y,), and (xS, y,), (x,, y,) be called
conjugate lines. The condition which must be satisfied is that no side of the
positive frame-work can be conjugate to a side of the negative frame-work.
It can easily be seen that the condition so stated is necessary and sufficient.
* Some examples of order-types represented in this manner are given by W. H. Young,
Lond. Math. Soc. Proc., Ser. 2, vol. i, p. 244.




146, 147]


The Continuum


187


THE CARDINAL NUMBER OF THE CONTINUUM.
147. The arithmetic continuum has been defined as an aggregate of the
order-type 0 (see ~ 128), and it is thus not normally ordered. It has been
held by Cantor* that this aggregate, and perhaps every aggregate, is capable
of being arranged as a normally ordered aggregate; but no proof of the
correctness of this view has been obtained. If the continuum be capable of
arrangement as a normally ordered aggregate, its cardinal number c must be
identical with one of the Aleph-numbers; and in fact Cantor has believed
that c= R, the cardinal number of the aggregate of all the order-types of
normally ordered enumerable aggregates. As evidence of the probable truth
of this view, the facts may be cited that all the sets of points which have
actually been defined in connection with the theory of sets of points, have one
or other of the two cardinal numbers R0 and c, and that no such set of points
has been defined of which it is known that the cardinal number is > N0 and
< c. This negative evidence is however clearly insufficient to settle the
question whether every part of the continuum has one of the powers R0
or c, a question which has hitherto defied all attempts to obtain a conclusive
answer. As has already been pointed out, it cannot be assumed that every
two cardinal numbers are such as to be comparable with one another; but
a proof has been given by G. H. Hardyt that c, and presumably any cardinal
number whatever, must either be an Aleph-number, or else be greater than
all the Aleph-numbers. The mode of reasoning is a generalization of that
employed by Cantor in his proof (see ~ 117) that 0K is less than any other
transfinite cardinal number.
Because c _ Mi, it is possible to take elements from the number-continuum
corresponding to all the numbers of the first and second classes of ordinals.
For if this process came to an end, we should have c = N0, which has been
proved by Cantor not to be the case. It follows that a set can be selected
from the continuum equivalent to the aggregate of ordinal numbers of the
first and second classes. Now if a set could be selected from this aggregate
equivalent to the continuum, it would follow from the equivalence theorem,
proved in ~ 119, that c = N; and if no such set could be selected it follows
from the definition of inequality in ~ 112, that c > MN; thus it has been proved
that c _ 1. If now c > NK, a similar proof would shew that c _- N, and so
on. If c > PN for every finite n, then c - N,, and this process may be continued indefinitely through the Aleph series. A criticism of the validity
of this proof will be given in ~ 160.
It is known that every infinite closed linear set of points has one or other
of the two cardinal numbers N0, c; and if a set of points can exist of which
* Math. Annalen, vol. xxi, p. 550.
t Quarterly J. of Math. 1903, p. 87.




188           Transfinite numbers and order-types           [CH. III
the cardinal number has neither of these two values, it must be unclosed, and
may without loss of generality be taken as dense-in-itself. The difficulties of
dealing with open sets dense-in-themselves are so great, that attempts to find
a contradiction involved in the assumption of the existence of such a set,
possessing a cardinal number different from both M0 and c, have hitherto been
a complete failure.
148. A very remarkable relation has been given by Cantor between the
cardinal number of the continuum and that of the integral numbers. This
relation is expressed by c = 2:o, or more generally c = nso, where n is a finite
integer.
This theorem was applied by its discoverer to obtain a simple arithmetical
proof that the 0o-dimensional continuum has the same power as the onedimensional continuum.
In accordance with the definition of an exponent given in ~ 116, 2Ko is the
cardinal number of the proper fractions in the dyad scale,
b,  b,  b3
2   22  23  '
where every b is either 0 or 1. In this aggregate each number of the form
1-2 +< 1, where p and q are integers, occurs twice; hence
2q
2No = (S, X),
where X is the aggregate of real numbers between 0 and 1, and s is an
enumerable aggregate.
It follows from the above, that 2No = c + No. Now c + No = c + 2No, since
No = 2No; therefore c = c + No; whence we have 2Xo = c.
From this theorem we deduce c. c = 2No. 2No = 22Xo = 2No = c; and hence
by repeated multiplication by c, we find c" = c, where n is any finite integer.
Again            cXo = (20o)fo = 20o o = 2No = c,
and therefore the continuum of finite, or of enumerable dimensions, is
equivalent to the one-dimensional continuum.
The aggregate defined by all possible modes of covering the numbers of
the continuum by themselves, has the power cC =f, and this number f is
greater than c. This has been proved in ~ 120; for a part of the new aggregate
is equivalent to that obtained by replacing the numbers of the continuum
either by A or by B, and taking all possible aggregates which arise in this
way. Since this part has been shewn to have a cardinal number greater than
that of the original aggregate, it follows then thatf> c.
More generally, if a is any cardinal number, we have a" > a.
If the continuum be divided into any finite number n of parts, such that




147-149]


-The Continuumn


189


all the parts have the same cardinal number, then that cardinal number is the
same as that of the continuum*.
The parts may consist of sets of points of any kind.
The theorem may also be stated thus:-if na = c, then a = c.
To prove it, we have na = c=nc; and therefore by applying the theorem
of ~ 121, it follows that a = c.
149. The continuum is equivalent to the aggregate of all possible ordertypes of simply ordered aggregates of cardinal number No.
This theorem points the contrast between the aggregate of all ordertypes of simply ordered enumerable aggregates, which is of power 2:o, and
the aggregate of all the order-types of normally ordered enumerable aggregates, which is NR. The latter aggregate is, of course, a part of the former
one, and thus the theorem R ~ 20o, can be deduced.
The theorem may be also stated in the form: The total number of ways
of ordering the integral numbers 1, 2, 3,... is c.
If,/ is the order-type of an enumerable aggregate arranged as a simply
ordered aggregate, then a part of an aggregate of the type 7, considered in
~ 127, can always be found which is of the type 1/. To establish this, it can
be shewn that an aggregate of type /M can always be changed into one of
type 7, by insertion of new elements. If, between every pair of elements
nml, m2 of /t, there are other elements, then /t is of one of the types 7,
1+7, 7+1, 1 +  +1; so that FL is reduced to qR by the removal of the
lowest and the highest elements, when such elements exist. In any such
case, if we add to /u, aggregates of type 7, at the beginning and at the end,
we obtain an aggregate of type 7 + L + +7 which is of type q, whichever of
the types q, 1 +7,  +, +1,  +  +1 may be identical with u*. If pairs of
elements exist in the aggregate of type f/, such that there are no elements
between them, an aggregate of type v can be inserted between every such
pair, until a new aggregate of one of the types q, 1 +,  +,  +1, 1 ++ 1, is
obtained; then as before, by adding aggregates of type q, at the beginning
and end, we obtain an aggregate of type 7. It has thus been shewn that
any aggregate of type /u is a part of another aggregate of type 77. Since the
rational numbers in their totality naturally exist in the order-type 77, it follows
that an aggregate of any type /A can be made by taking a part of the aggregate of rational numbers, of type 7.  It follows that the aggregate of all
types /L has a cardinal number less than, or equal to, that of the aggregate
of all part-aggregates of the set of rational numbers arranged in type q7.
Now every aggregate (r1, r2,...), of which all the elements are rational
numbers, corresponds to a single point of a continuum of an enumerable number


* Bernstein, loc. cit., p. 31,




190


Transfinite numbers and order-types


[ci. III


of dimensions, of which the coordinates are x, =r, x2=r,.... Hence the
cardinal number of the aggregate of all part-aggregates-of the set of rational
numbers is less than, or equal to, the cardinal number of the M0-dimensional
continuum, that is, - c; and therefore the cardinal number of the aggregate
of all types p is _ c.
It will now be proved that c - the aggregate of all types tA. To every
real number between 0 and 1, there corresponds an infinite sequence b1b2b,...
where every b is either 0 or 1, expressing the number in the dyad scale.
After each b, insert an aggregate of type wr, and we then have an aggregate
brb7rb27rr..., of type v = b + r + b2 + 7r + b + 7r +.... Here, some of the b's
may be zero, and these may be simply omitted; thus r +0+7r = r+ 7r.
Hence to any real number x between 0 and 1, there corresponds the type
v = bl + r + b2 +....
It is now necessary to shew that the two order-types v, ', which correspond
to two different numbers, x, x', are necessarily distinct from one another.
If v = v', we can write the equality C + 7r + $  = C(' 4- + I'r, where C0, C0' are
each either 0 or 1; and from this we obtain, by means of the theorem of
~ 129, C0 = C[', and 1= ~'.  The last equation can be written
C2+ 7r+ 2 = C2 + 7r  2/,
and from this we conclude that C2= C2', 2= /'; and we can proceed onwards
in the same manner. From b, = b', b2 = b2',..., we conclude that x = x'. It
has thus been shewn that {x} - {v}; and from this, we conclude that c _ the
cardinal number of all order-types,u.
This part of the theorem  is due to Cantor*, and the first part to
Bernstein.  By combining the two results, the complete theorem is established.
This important result may also be expressed by saying that the totality
of all permutations of the sequence of positive integers has the power of the
continuum.
It may also be shewn that the totality of all parts of the sequence 1, 2, 3,...
has the power of the continuum.
For if we form a sequence by writing 0 for each of the numbers 1, 2, 3,...
which does not occur in a given part of (l, 2, 3,...); and 1 for each number
which does occur in the given part, then the sequence of O's and 1's thus
obtained, corresponds to a real number expressed in the dyad scale, and
therefore the numbers of the continuum are put into correspondence with
the parts of the sequence (1, 2, 3,...).
150. It can be shewn that the aggregate of all sets of points in the
n-dimensional continuum has a cardinal number greater than c.


* See Bernstein's Dissertation, p. 7.




149-151]


The C ontinuumn


191


For in the aggregate {P} of all points in an n-dimensional continuum,
we can substitute 0 for each point P which does not occur in a given set
of points of the continuum, and 1 for each point P which does occur; we
then obtain an aggregate consisting of 0's and l's: but it is known that
the totality of all such aggregates has the power 2c, which is >c.
On the other hand, the totality of all closed sets of points in the n-dimensional continuum has the same power c as the continuum.
Every closed set is the derivative of an enumerable set of points; and to
every enumerable set of points there corresponds a single closed set.
It follows that the cardinal number of the totality of closed sets is - the
cardinal number of the totality of enumerable sets of points chosen out of the
continuum. To shew that the latter is c, we observe that it is _ the aggregate of all combinations of points of the continuum in sets of N0 elements,
that is, ' co~, or - c.
Again, every single point of the N0-dimensional continuum corresponds
to a single point of the one-dimensional continuum, and this point is an
enumerable part of the continuum; hence the totality of enumerable sets
of points of the n-dimensional continuum is _ c. On combining this with
what has been proved above, we see that the totality of all enumerable sets
of points in the n-dimensional continuum is c; hence the totality of all
closed sets of points in the n-dimensional continuum is < c.
Again, the totality of all closed sets of points in the n.dimensional
continuum is _ c. For one such closed set can be taken in each of an
infinity of the domains x-= a, where xa is one of the n coordinates which
determine the position of a point in the n-dimensional continuum; and the
aggregate of all possible values of a has the power c.  We thus obtain
an aggregate of closed sets which has the power c; and it follows that the
aggregate of all closed sets in the n-dimensional continuum is _ c.
Since the totality of all closed sets of points in the n-dimensional
continuum is _ c, and at the same time is c c, it must have the cardinal
number c.
Since* every curve or surface in a continuum is formed by a closed set of
points, we see that every possible curve or surface corresponds uniquely to
a single definite real number.
151. A method of constructing a set of points of which the cardinal
number is N1, has been givent by G. H. Hardy.
If we start from the sequence
1, 2, 3, 4, 5,......(1)
* Bernstein, loc. cit., p. 43.
+ Quarterly Journal of Math., vol. xxxv, 1903, "A theorem concerning the infinite cardinal
numbers." A criticism of this construction will be given in ~ 162.




192


Transfinite numbers and order-types


[CH. III


of integral numbers, a new sequence
2, 3, 4, 5,...... (2)
is formed by omitting the first term.
Continuing this process, we form
3, 4, 5, 6,..... (3)
4, 5, 6, 7,...... (4)
5, 6, 7, 8,..... (5)
We now form a new sequence
1,  3,   5,  7,  9,...... (co)
by traversing the above infinite array of sequences diagonally. Then we form
3,  5,  7,   9, 11,...... (w+1)
5,  7,  9, 11, 13,...... (o+2)
7,  9, 11, 13, 15,...... (o+3)
9, 11, 13, 15, 17,...... (co+4).....................
1,  5,  9,13, 117,...... (co.2)
5,  9, 13, 17, 21,...... (co.2+1)
9, 13, 17, 21, 25,....... (co.2+3).....................
1,  9, 17, 25, 33,...... (0.3).......e.............
Thus sequences corresponding to all the numbers w. /, + v can be formed.
To form the sequences corresponding to (o2, we take the array of sequences
1,  3,  5,   7,   9,......(o)
1,  5,  9, 13,   17,...... (o.2)
1,  9, 17, 25,   33,....... (co.3)
1, 17, 33, 49,   65,...... (. 4)
1, 33, 65, 97, 129,...... (o.5)........................
and traverse it diagonally; we thus obtain
1,  5, 17, 49, 129,..... (co2).
Generally, if bi, b2, b3, b4,... is the sequence corresponding to t/, the
sequence b2, b3b, b5,,... corresponds to 8 + 1.    To obtain a sequence corresponding to a number 7 which is a limiting number of the second class,
we take the array of sequences corresponding to any ascending set of
numbers /, 8,,... of which the limit is y, and traverse it diagonally.  It is
clear that, in this manner, a sequence can be found for any given number of




The Continuum


193


the second class; but that the set of sequences so obtained is not unique. For
example, o might have been taken as the limit of 1, 3, 5, 7,..., or Co2 might
have been taken as the limit of o +1, w. 2  2,. 3 + 3,....
It will be shewn that the sequences bl, b2, b3,... can be so chosen that
in every case b, < b < b3,...; and that, if the sequences b1, b,... and bl', b2,...
correspond to any two numbers /, /3', where 3 < f', then there exists a
number Nr such that bn,'> b, for n _ N; and thus that the sequences are
distinct from one another.
Let us assume that sequences, corresponding to all numbers < y, have
been constructed in such a manner that this condition is satisfied. First,
let y be a non-limiting number, so that 7 = ' + 1. Then if 3 < y', there is
a number N such that a' > bn, for n _ N, where a', a2', a',... is the sequence
which corresponds to y'. But if a,, a2, as,... is the sequence which corresponds to y, we have an = a'n+i > n' > bn, for n _ N.
Hence, if the construction is possible for all numbers <, it is possible
for all numbers - fy, where 7 is a non-limiting number.
Next, let us suppose that y has no immediate predecessor, and that
= L. /3m; then also  y=L (/3, + v,), where the PI are finite numbers.
Now there is a number N1, such that b2,n> bl,n, for n _ N1, where bn,n
denotes the nth number in the sequence corresponding to /3. A fortiori, if
Y7m = /3m+ vm, we have c2,n = b,9+v^ > b2,, > b,,3, for n ' N1, where cm,n is the
nth number in the sequence corresponding to 7y. But if we take v2 > b1,N,-,
we have C2,n = b2, n+V _  + v2 > bl,N,-1 > bli, for n < N, and hence we have
c2,, > b,,,, for all values of n. Similarly Pv can be so chosen that 73 > 72, and
c3,, > c2,, for all values of n; and so on generally. If we write 7y for 1,
and c,,, for bl,,, we have a doubly infinite array
c1, 1,  c1, 2  C1,3   * * 
C2,1,  C2,2,  C2,3, '..
C3,1,  C3,2,  C3,3, *.
and we define the sequence corresponding to y, by traversing it diagonally,
so that c,= cn,n. If then / < 7, we can find m so that 8/ < 7y; then there
is a number K, such that c,,n > bn, for n  K.  But if n> m, we have
Cn = cn, > c,,,; and thus if n is greater than the greater of the two numbers
m, K, we have c, > b,. It has thus been shewn that if the construction is
possible for all numbers < 7, it is possible for all numbers < 7, whether 7 is
a limiting number or not.
In this manner a sequence is obtained which corresponds to any assigned
number y of the second class, and this sequence is distinct from those which
correspond to the numbers < 7, such sequences being also distinct from one
another.


H.


13




194


Transfinite numbers and order-types


[CH. III


The sequences may be correlated with points in the linear continuum
(0, 1). To correlate a sequence b1, b2, b3,..., we may take the binary radix
fraction in which the bth, b2th, b3th,... figures are all 1, and the remaining
figures all 0. In this manner a set of points is shewn to exist, such that
one point of the set corresponds to each number of the first or of the second
class. This amounts to the construction of a set of points of cardinal
number K. Just as an enumerable set of points is determinate, when the
point which corresponds to any assigned number n of the first class is
determinate, so the set of cardinal number N, is determinate, in the sense
that a definite point is determined corresponding to any assigned number
/ of the first or of the second class.
It may be remarked that a set of points of cardinal number NK, or of any
cardinal number >  o,, when arranged in normal order, cannot possibly be in
the order in which they occur in the continuum.
For if a set of points, in the order in which they occur in the continuum,
forms a normally ordered aggregate, each point and the next succeeding one
define a linear interval of which they are the end-points. We have thus a
set of intervals which must have the same cardinal number as the given
set of points. Each interval of the set abuts on the next one, and thus the
end-points together with their limiting points define an enumerable closed
set. Hence the given set must be enumerable.
GENERAL DISCUSSION OF THE THEORY.
152. An account having now been given of the abstract theory of aggregates, as developed by Cantor and others, the remainder of this chapter will
be devoted to a critical discussion* of the theory.
In accordance with Cantor's general theory of ordinal numbers, and of
aleph-numbers, there exist two aggregates
1,   2,... n,... a,  c+1l,... (,n+l,... 3,,...,
N),,...... M.,n+,... 0a,  *  Q 5 +  *** bi  * *R  *.,
the first, the aggregate of all ordinal numbers, and the second that of all
N cardinal numbers. These aggregates are both normally ordered, and are
similar to one another; and they contain, respectively, every ordinal number,
and every cardinal number which belongs to a normally ordered aggregate.
In accordance with the principle which is fundamental in the whole
theory, that every normally ordered aggregate has a definite order-type,
which is its ordinal number, and has also a definite cardinal number, it is
seen that the above aggregates have an ordinal number 7y, and a cardinal


* Most of the critical remarks here made have been published in the Proc. Lond. Math. Soc.,
ser. 2, vol. III.




151, 152]


General discussion of the theory


195


number R,. The ordinal number y must itself occur in the first aggregate,
and must therefore be the greatest ordinal number, i.e. the last element of
the aggregate; moreover M  must occur in the second aggregate, and must
be the last element of that aggregate.  There can, however, be no last
ordinal number; for, on the assumption of the existence of y, an aggregate of
ordinal number y + 1, can be formed. For example, by placing the first
element of either of the above aggregates after 7 or after KN respectively;
it can at once be shewn that there is no last ordinal number, and consequently no last aleph-number by. We have thus arrived at a contradiction.
Burali-Forti, who first pointed out this contradictiont, accounted for it
by denying the truth of the theorem, that any two distinct ordinal numbers
ai, a2 must necessarily satisfy one of the relations al > a2, a, < a, in accordance
with the definition which has been given in ~ 134, of the meaning of these
relations. However, Cantor's proof of this theorem (see ~ 133), does not
appear to be capable of refutation, and consequently the origin of the contradiction cannot be explained in the manner indicated.
B. Russell has suggested+ that the aggregates of all ordinal numbers
and of all aleph-numbers are not normally ordered, and therefore that these
aggregates have no ordinal number, and that their cardinal number is consequently not necessarily an aleph-number. He admits, however, that the
segments of either aggregate are normally ordered.  This explanation is
confuted by the argument that, if the above aggregates are not normally
ordered, then they must contain parts, of type *o; such a part would then
be a part also of a segment of one of the aggregates, and such segment
would not be normally ordered.
The contradiction has been explained by Jourdain~, by means of the
suggestion that there are ordered aggregates which have no order-type,
and no cardinal number; and that the above aggregates belong to such
class. To such aggregates he gives the name inconsistent aggregates, in
virtue of the fact that, of such an aggregate it is impossible to think,
without contradiction, as a " collection by the mind of definite and distinct
objects to a whole." It appears from a statement made by Jourdain 11, that
Cantor had himself, some years previously, arrived at the same conception
and name.
In accordance with this view of the matter, there exists an ordered
aggregate, viz. that'of all the ordinal numbers, every segment of which is
normally ordered, and has a cardinal number, and yet such that the aggregate
bt Rend. del. circolo mat. di Palermo, vol. xi, 1897, "Una questione sui numeri transfiniti."
+ The Principles of Mathematics, vol. I, p. 323.
~ Phil. Mag. 1904, " On the transfinite numbers of well-ordered aggregates."
I] Loc. cit. p. 67, note; see also Hilbert, Jahresbericht der deutsch. math. Vereinig. vol. viI, p. 184.
13-2




196


Transfinite numbers and order-types


[CH. III


itself, being "inconsistent," cannot, without contradiction, be thought of as
having a definite order-type. This amounts to a denial of the universal
validity of the fundamental principle that every ordered aggregate has a
definite order-type; and yet it is by means of this very principle that the
existence of the successive ordinal numbers is regarded as having been
established. Each successive ordinal number was defined to be the ordertype of the ordered aggregate of all the preceding ordinal numbers. The doubt
thus thrown upon the validity of the principle by means of which the existence of the complete series of ordinal numbers, and simultaneously, that of
the aleph-numbers, is established in Cantor's theory, naturally suggests that
a further scrutiny of the foundations of that theory is required.  It is not
clear, a priori, that an aggregate which is inconsistent, in the sense employed
above, may not be reached at an earlier stage of the process of forming the
successive classes of ordinal numbers, before the aggregate of all such
numbers, in the sense of Cantor's theory, is reached.  Moreover, it would
seem reasonable to expect, that so fundamental a distinction, as that involved in the notion of an inconsistent aggregate, should be indicated in the
general definition of an ordered aggregate, or in close connection therewith.
In any case, an explanation of the contradiction, on these lines, cannot be
regarded as satisfactory, until criteria have been obtained which shall suffice
to decide, in respect of any particular ordered aggregate, whether such
aggregate has an order-type and a cardinal number, or whether it is an
inconsistent aggregate.
153. Before proceeding to attempt the consideration of how far Cantor's
general theory of ordinal numbers and aleph-numbers can be accepted,
we shall examine the definition of an aggregate in general, with a view to
discovering whether it has, in the form given in ~111, the requisite degree
of precision. An attempt will then be made to decide what limitations
or qualifications must be imposed upon the nature of an aggregate, so
that, in the development of the theory, the possibility of being confronted
by such a contradiction as that which was pointed out by Burali-Forti, may
be removed at its source.
The term aggregate being taken as denoting a collection of distinct objects,
in the most general sense, the difficult question arises as to the conditions
under which the elements that form the aggregate can be regarded as
adequately defined. In the case of a finite aggregate, the elements may
be defined by means of individual specification, but this is not possible in
the case of a transfinite aggregate; individual specification must then, in
the latter case, be replaced by a law, or by a finite set of laws, forming the
norm by which the aggregate is defined. Prima facie the most general
definition of an aggregate which presents itself is that an aggregate consists
of all objects, such that each satisfies certain specified conditions. It is




152-154]


General discussion of the theory


197


however convenient to admit the case of two or more alternative sets of
conditions; thus an aggregate may contain all objects, each of which satisfies
either the conditions (A), or else one of the sets of conditions (B),... (K).
The conditions forming the norm by which the aggregate is defined must
be of a sufficiently precise character to make it logically determinate, as
regards any particular object whatever, whether such object does or does not
belong to the aggregate. As we have seen, for example, in the case of the
aggregate of algebraical numbers, the means at our disposal may not suffice
to render the actual determination possible, in any particular case; we therefore agree to fall back upon the logical determinacy as sufficient; thus it is
logically determinate as regards a number, defined in any particular manner,
whether that number is algebraic or not, and consequently we regard all the
algebraical numbers as forming an aggregate in accordance with the definition
of that notion. We shall accordingly define the term aggregate, as follows:All objects which are such as to satisfy a prescribed norm are said to belong
to an aggregate defined by that norm.  The norm consists of a set of specified
conditions, or of a specified set of alternative specified conditions; and this
norm must be sufficient to render it logically determinate, as regards any
particular object whatever, whether that object belongs to the aggregate or not.
It is clear that the elements of an aggregate, being subject to a common
norm, must have a certain community of nature which constitutes the ground
of the aggregation.
In the case of a finite aggregate, the norm may take the form of individual
specification of the objects which form the aggregate.
154. It is not clear that an aggregate defined in the above sense is
necessarily capable of being ordered at all. For example, it is difficult to
see that such an aggregate as that of "all propositions " could conceivably be
ordered; where it is assumed that the meaning of the word "proposition " is
taken as so definite, that this aggregate has a norm in accordance with the
definition above. Again, to take an example among aggregates of the kind
usually considered in Mathematical theory, we may consider the aggregate
obtained by covering the aggregate of real numbers by itself. This aggregate
which has the cardinal number f- cc, is equivalent to the aggregate of all
the functions of a real variable; it is difficult, if not impossible, to see how
order could be imposed upon this aggregate. If then, a transfinite aggregate
is to be given as an ordered aggregate, or is to have an order imposed upon
it, or rather discovered in it, it would appear to be necessary that the norm,
which constitutes the definition of the aggregate, should be of such a character,
that a principle of order is contained therein, or can at all events be adjoined
thereto; so that, when any two particular elements are considered, the conditions which they satisfy in virtue of their belonging to the aggregate, when
individualized for the particular elements, may be sufficient also to allow of




198


Transfinite numbers and order-types


[CH. III


relative rank being assigned to those elements in accordance with a principle
of order. This is in fact the case in such aggregates as those of the integral
numbers, the rational numbers, or the real numbers. In the case, for example,
of the positive rational numbers, the relative rank of any two particular elements
(p, q), (p',q') is assigned by the system of postulations, contained in ~ 11,
which defines the aggregate.  It may, of course, also be possible in other
cases, as in this one, to re-order the aggregate, in accordance with some
other law, extrinsically imposed upon the aggregate; but the nature of the
elements must be such that this is possible.
We can now state that:In order that a transfinite aggregate defined as in ~ 153, may be capable
of being ordered, a principle of order must be explicitly or implicitly contained
in the norm by which the aggregate is defined.
The relative order of any two elements chosen from an ordered aggregate
depends upon the individual characteristics of those elements, in accordance
with the principle of order.
In the definition of order-type given by Cantor (see ~ 122), according to
which the order-type of an aggregate is obtained by making abstraction of
the particular nature of the elements of the aggregate, it is assumed that
the aggregate is given as an ordered aggregate.  Again, in his definition
of cardinal number (see ~ 111), Cantor has assumed that the aggregate is
given as an ordered one; the cardinal number there appears as the result of
a double abstraction, viz. of the particular nature of the elements, and of the
order in which they are given.  The question however arises, whether the
definition of cardinal number should not be such as to be also applicable in
the case of aggregates which are not given as ordered aggregates. Cantor has
himself, in fact, in his theory of exponentials involving transfinite cardinal
numbers, contemplated certain aggregates as having cardinal numbers,
whilst such aggregates were not given as ordered aggregates, and primd
facie, at all events, are not capable of being ordered.
155. Taking the case of an aggregate defined as an ordered aggregate,
we now approach the consideration of the fundamental question, whether,
and under what conditions, if any, such an aggregate can be regarded as
having a definite order-type, and a definite cardinal number. This is equivalent to asking whether, or when, meanings can be given to those terms,
in accordance with general definitions, of such a character that they can
be treated as permanent objects for thought, or as mathematical entities
which may themselves be elements in aggregates.
With reference to Cantor's definition (see  111) of the cardinal number
of a transfinite aggregate, by abstraction, in accordance with which the
cardinal number is represented by replacing each element by an abstract
unity, it must be observed that such a substitution would replace the given




154, 155]


General discussion of the theory


199


aggregate by another one which had no longer any intelligible relation with
the norm by which the original aggregate is defined. The abstract unities
would be indistinguishable from one another, and the new aggregate would
be indistinguishable from any other non-finite aggregate of such unities. It
would be impossible to decide, as regards any particular abstract unity,
whether it belonged to the aggregate or not; in fact, to make complete
abstraction of the individual nature of the elements of an aggregate is to
destroy the aggregate. A definition by abstraction could be justified only
by the interpretation, that abstraction is made of those characteristics only,
in which the elements of the aggregate differ from the corresponding
elements of all possible equivalent aggregates.  Thus the existence of
aggregates equivalent to the given aggregate would appear to be essential,
if the latter is to be regarded as having a cardinal number to which any
definite meaning can be attached. On the grounds stated, the definition of
a cardinal number, as the characteristic or class-name, of a class of equivalent aggregates, is to be preferred to the definition given by Cantor.
Accordingly, an aggregate has a cardinal number, only when it is one of a
plurality of equivalent aggregates distinct from one another. The elements
of one of these aggregates must be essentially different from those of another
of them; it would not, for example, be admissible to consider two equivalent
normally ordered aggregates as essentially different from one another, when the
one can be obtained from the other by replacing the elements of some segment
by other elements, the remainder being left unaltered. In all cases the correspondence between equivalent aggregates must be definable by some norm.
We are thus led to the following statement containing a definition of
cardinal number:
The members of any particular class of equivalent aggregates have a
quality in common in virtue of their equivalence. The name of this quality of
mutual equivalence is the cardinal number, and may be regarded as characteristic of each aggregate of the particular class.
In Cantor's definition of the order-type of a simply ordered transfinite
aggregate (see ~ 122), abstraction is made of the nature of the elements,
their order in the aggregate being alone retained. The order-type is then
regarded* as represented by an aggregate of abstract unities, in the order
of the elements of the given aggregate. In any ordered aggregate, it is
however the individual characteristics of any two elements which determine
their relative order in the aggregate, in accordance with some principle of
order valid for the whole aggregate. If complete abstraction be made of
the characteristics of the various elements, order has then disappeared from
the aggregate. It must be supposed, that in Cantor's representation of the
order-type, there are attached to the abstract unities marks of some kind,
* See Math. Annalen, vol. XLVI, p. 497.




200


Transfinite numbers and order-types


[CH. III


which may in particular cases be marks indicating position in space or time,
by which the order of the various abstract unities is denoted; the given
aggregate is then really replaced by an aggregate of these marks, and the
abstract unities are superfluous. These marks, by which order is determined,
must also have been associated with the elements of the original aggregate.
It thus appears, that in a definition by abstraction, it can be only those
characteristics (if any) of the various elements which are irrelevant in
determining the order, of which abstraction is made: thus the aggregate is
really replaced by a similar one. On these grounds, that definition of an
order-type is to be preferred, in which the order-type is defined as the
characteristic, or class-name, of a class of similar aggregates. Accordingly,
in order that a given aggregate may have an order-type, to which a definite
meaning can be attached, it is necessary that the aggregate be one of a
plurality of similar aggregates.
We may accordingly state that:The members of any particular class of similar aggregates have a quality
in common, in virtue of their relation of similarity. This quality of mutual
similarity possessed by the aggregates is their order-type, and may be represented by a name or symbol, regarded as characteristic of each aggregate of
the particular class.
The considerations above adduced may be applied in the case of an
aggregate which is a segment of the hypothetical aggregate of all ordinal
numbers. In this case it is impossible to make abstraction of the nature
of the individual elements of the aggregate, without destroying the order,
because the elements are themselves nothing more than marks indicating
order. Hence it would appear, that the aggregate cannot in any intelligible
sense be considered as having an order-type, unless it be possible to define
an aggregate of objects of some other kind, which shall be similar to the
one under consideration.
156. We proceed to consider, from a somewhat different point of view,
those aggregates which consist of ordinal numbers in their order of generation.
There are two distinct methods of establishing the existence of a class
of mathematical entities.
(1) Their existence, as definite objects for thought, may be shewn to
follow as a logical consequence of the existence of other entities already
recognized as existent, or of principles already recognized as valid; so that
the existence of the new entities in question cannot be denied without
coming into contradiction with truths already known. This method may
be termed the genetic method.
(2) The existence of the entities may be postulated; and their mutual
relations, and their relations with other entities already known to exist, may
be defined by means of a complete system of definitions and postulations.




a y, 156]


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201


Accordingly, the objects in question are a relatively free creation of our
mental activity. The validity of the scheme thus set up is established when
it is shewn to be free from internal contradiction. Its utility is to be judged
by its applicability to the general purposes of the science, and by the light
it may throw upon the fundamental principles of that science, in virtue
of the scheme containing a generalization of what was previously known.
This method may be termed the method of postulation. It may, however,
be urged that the failure to discover contradictions within a scheme which
has been postulated is no proof that such contradictions do not exist, and that
such proof can only be supplied by the exhibition of a system of entities
already known to exist, such that the relations between them are in accordance with those postulated in the scheme in question.
Both these methods have been employed by Cantor in his theory of transfinite numbers and order-types.  In his earlier treatment of the subject, he
employed the second of the above methods.  The existence of the new
number o, and of the limiting numbers of the second class, was postulated,
in accordance with the second principle of generation.  Freedom  from
contradiction, and utility in connection with the theory of sets of points,
which suggested the postulations, were relied upon as the grounds upon
which the system of new numbers was to be justified. The first number 1Q,
of the third class, was introduced by a new postulation.
In his later and more abstract treatment of the subject, an account of
which has been given in the present chapter, Cantor applied the genetic
method. The existence of the number o is not directly postulated, but is
taken to follow from the existence of the aggregate {n}, of integral numbers;
co is defined to be the order-type of this aggregate, and it is assumed that
such order-type is a definite object which can itself be an element of an
aggregate. The existence, as definite entities, of the cardinal, numbers being
assumed, the successive ordinal numbers of the successive classes are obtained
by assuming as a general principle, that an ordered aggregate necessarily
possesses a definite order-type which can be regarded as itself an object, the
ordinal number coming immediately after all those that are the elements of
the aggregate of which it is the order-type.
It has been seen above, that the assumptions that an ordered aggregate
necessarily possesses a definite order-type, and that it also possesses a
definite cardinal number, both of which can be regarded as objects, lead to
the contradiction pointed out by Burali-Forti. It appears, therefore, that
the class of entities, which is constituted by the ordinal numbers of all classes,
and the similar aggregate of aleph-numbers, do not satisfy the condition of
being subject to a scheme of relations which is free from contradiction.
In fact, the principle, in accordance with which their existence is inferred,
conflicts with the definition of the aggregates as containing respectively




202


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[CH. III


every ordinal number, and every aleph-number.   It would then appear,
that the genetic process which led to the definition of the aggregates
of all ordinal numbers, and of all aleph-numbers, cannot be a valid one.
Thus the principle that every ordered aggregate has a definite order-type,
which may be regarded as a permanent object of thought, cannot be accepted as a universal principle to be used in a genetic mode of establishment
of the existence of a class of entities.  A denial of the validity of this
principle does not however preclude the less ambitious procedure of postulating the existence of definite ordinal numbers of a limited number of
classes, in accordance with Cantor's earlier method. So long as the postulation of the existence of ordinal numbers does not go beyond some definite
point, no contradiction will arise, and the validity of the scheme, for purposes
of representation, will suffice to justify the postulations which are made.
An attempt to examine the structure of such a class of ordinal numbers,
as that of the oth class, with cardinal number MR, or that of the f2th class,
with cardinal number No, will lead to the conviction that such conceptions
are unlikely to prove capable of useful application in any branch of Analysis
or of Geometry. Nevertheless, should inexorable logic compel us to contemplate the existence of such classes of objects, they would be a proper field
of exploration; we have however seen that there are grave doubts as to
whether this be in fact the case.
157. The genetic method being rejected on the ground that it leads to
the construction of a class of entities which in its entirety can have no
existence, we have to fall back upon the method of postulation. A consideration of the essential elements in the conceptions which lie at the base
of the scheme of finite integral numbers may afford guidance as to how far
we may properly proceed in the construction, by postulation, of transfinite
ordinal numbers of successive classes. The ordinal numbers of any one
particular class are those which belong to rearrangements of the elements of an
aggregate, of which aggregate the order-type is the lowest number of that class.
We may therefore consider primarily, the lowest numbers of the classes of
which the cardinal numbers are N0, N1, N... respectively. It was pointed
out in Chapter I., in the case of the finite numbers, that the existence of
an integral number does not follow as a mere logical consequence of the
existence of the preceding numbers, but that each ordinal, or each cardinal
number appears as the characteristic of the members of a family of similar,
or of equivalent, aggregates of objects, the number in question being then the
ordinal, or the cardinal, number of each member of the family. Thus the
notion of correspondence between the elements of different aggregates was
seen to be an essential element in the conception of either an ordinal, or
a cardinal, number as characteristic of a class of aggregates. In the genetic
method, as applied to the construction of the whole series of classes of




156, 157]        General discussion of the theory                 203
transfinite ordinal numbers, this notion of correspondence between the
elements of different aggregates having the same number plays no part;
and in fact, the existence of a number is constantly inferred from that of
a single unique ordered aggregate. For example, the existence of 12, and
of N, is inferred from the existence of the single aggregate of numbers
of the first and second classes.  Generally, in the whole scheme, the
existence of a new number is inferred from the existence of that unique
aggregate which contains the preceding ordinal numbers. That this procedure leads to contradiction has been already seen.    The transfinite
numbers must be regarded as obtained, or defined, in accordance with
the same principles as hold good in the case of the finite numbers, if
they are to be regarded as numbers, even in an extended sense of that
term. It seems then highly probable, that the neglect of the principle,
that correspondence between similar or between equivalent aggregates is
essential to our right to consider the numbers belonging to aggregates
as definite entities, may be the source of the contradiction which arises from
the thoroughgoing application of the genetic method that leads to Cantor's
complete series of ordinal numbers and aleph-numbers. In accordance with
this view of the nature of Number, finite or transfinite, the postulation
of the existence of a definite entity, which entity shall be entitled to be
regarded as a number, is only justified when it is shewn that other
aggregates exist besides the aggregate which consists of the preceding
ordinal numbers, of which other aggregates the postulated number is either
the characteristic ordinal or the cardinal number. Thus the postulation of
the existence of the numbers o, and  0, requires for its justification, the
exhibition of other aggregates besides {n}, that of all finite numbers; in this
case the requirement is satisfied by the definition of sets of points, or of other
geometrical objects, and thus there really exists a class of aggregates which
is similar to the ordered aggregate 1, 2, 3,... n...; and hence the postulated
order-type c, and the postulated cardinal number No, are really entitled to
rank as ordinal and cardinal numbers respectively. When we consider the
ordinal number 12, and the cardinal number RN, the state of the case is very
different. In order that the existence of 12 might be on a parity with that
of o, it would require to be shewn that it is possible to define a set of
objects, say points of the linear continuum, which should be such that, to
each prescribed ordinal number of the second class, there corresponds a
definite point of the continuum, i.e. to shew that a norm is possible which
would define a set of points of order-type 12. This has hitherto not been
accomplished, nor have aggregates having any of the cardinal numbers
2,, N3,... been defined by means of sets of rules. If it be urged that the
postulation of the order-type 12, and of the corresponding cardinal number
K1, does not of itself lead to contradiction, it may be replied that such postulation does not entitle 12 and N, to rank as numbers, in the sense in which




204


Transfinite numbers and order-types


[CH. III


co and N0 are numbers; for, in the latter case, the essential elements in the
original conceptions of ordinal and cardinal numbers are all present, whereas
this has not been shewn to be true of fQ and KN. Moreover, the postulation
of the existence of 12 and NK, if it does not of itself lead to contradiction, can
only be made by means of a principle which, when applied systematically,
certainly leads to contradiction. In accordance with the criterion laid down
above, N1, N,... cannot, at the present time, be regarded as definite entities,
and could not be regarded as in any true sense numbers, even if any meaning
could be assigned to them.
It may conceivably turn out, in the future, to be possible to justify the
postulation of the existence of certain of the numbers NK, N2,..., together
with the classes of ordinal numbers which would belong to them. It will,
however, certainly never be possible to do so for the whole class KNp}, where
/, is any ordinal number of the aggregate of all ordinal numbers, in accordance with Cantor's complete scheme, because such postulation leads to
unavoidable contradiction. The setting up of a scale of standards, to some
of which standards no aggregates not consisting of the preceding numbers
conform, involving, as it does, the employment of sphinx-like aggregates,
to each of which no other aggregates can be shewn to be similar, would
a priori appear to be an illegitimate extension of the notion of number,
an essential element having dropped out; and i posteriori it has been shewn
to lead to contradiction.
It may be urged that no contradiction would ensue if, in single instances,
the existence of order-types and powers, considered to be definite entities,
were postulated for aggregates of the unique character referred to above.
But if this were done, such order-types and powers would not be entitled to
rank as numbers; and such sporadic creations would be of no importance in
Mathematical theory. Systematic postulation of this character is just what
has been shewn to lead to a self-contradictory scheme of entities, and is
therefore illegitimate.
A cardinal number has been defined      by B. Russell, to be a class of
equivalent aggregates; it may then be urged that such class may contain
only one member, and that this is sufficient for the existence of the cardinal
number.   In fact, Russell inferst the existence of the number n+ 1, from
that of the numbers 0, 1, 2, 3,... n.
* Principles of Mathematics, vol. I, pp. 111-116.
+ Ibid. p. 497. Since Russell regards the activities of the mind as irrelevant in questions
of existence of entities, his view, and that here advocated, have no premisses in common. An
advantage claimed for the conception of the nature of number, here advocated, over that of Russell,
is that it does not lead to such a contradiction as that pointed out by Burali-Forti. Russell
objects (see p. 114) to the conception of a number as the common characteristic of a family
of equivalent aggregates, on the ground that there is no reason to think that such a single entity
exists, with which the aggregates have a special relation, but that there may be many such




157, 158]


General discussion of the theory


205


In accordance with the view here advocated of the nature of number, this
definition, or any other one which allows the existence of a cardinal number
to be inferred solely from the existence of a unique aggregate, to which no
other aggregates have been shewn to be equivalent, must be rejected.
158.  The conclusions at which we have arrived in the course of the
above discussion, may now be summarized as follows:(1) The aggregates
1, 2, 3,... n,... co, co+l,... Q,... /3,...
NO, Rl, N2.., **  n+,...  + n  ***, ***. **
of all ordinal numbers, and of all aleph-numbers, in the sense in which
Cantor contemplates them, have no existence. Their existence cannot be
established without the assumption of the principle that every normally
ordered aggregate necessarily has a definite order-type, and a definite
cardinal number, which can themselves be regarded as objects capable of
being elements of an aggregate. This principle leads to contradiction, and
must therefore be rejected as not being a universally valid truth.
(2) Of the aleph-numbers, the postulation of the existence of N0 has
hitherto alone been justified*, by shewing that it is possible to define
aggregates consisting of objects other than the ordinal numbers themselves,
of which it is the characteristic cardinal number. The numbers o), co +- 1...
o. 2,... c)2,... ()w),... of the second class exist, but it has not yet been shewn
that the totality of all such numbers, taken in order, has a definite ordertype or a definite cardinal number; even if it be legitimate to speak of
these numbers as forming a totality. To do this it would be necessary to
shew that a finite set of rules can be set up which will suffice to define a
definite object corresponding to each ordinal number of the second class.
(3) The existence of individual aleph-numbers, other than N0, with the
classes of ordinal numbers belonging to them, may, in the future, be established;
but it is not possible that this should be done beyond some definite stage.
It thus appears that there is at present no sufficient reason for thinking
that any unenumerable aggregate is capable of being normally ordered.
It may be observed that an aggregate which consists wholly of distinct
physical objects which do not penetrate one another must be enumerable:
for each such object occupies some definite volume in space; and it has
been shewn that any set of distinct portions of space is enumerable. It
follows that the objects contained in an unenumerable aggregate, must,
entities, and that there are in fact an infinite number of them. The mind does, however, in
point of fact, in the case of finite aggregates at least, recognize the existence of such a single
entity, viz. the number of the aggregates; and this is a valid creation of our mental activity,
subject to the law of contradiction.
* In Math. Annalen, vol. LX, p. 183 in a paper "Ueber wohlgeordnete Mengen," Schbnflies
has expressed a view which is to a certain extent in agreement with that here stated.




206


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[CH. III


with the possible exception of an enumerable component of the aggregate,
consist of ideal or abstract objects.
159. The regarding of a collection as a " whole " has been- emphatically
declared by Cantor, to be essential to the notion of an aggregate. It is no
doubt true that, in a certain sense, every logical class, or aggregate as defined
in ~ 153, forms a whole, as being dominated by a certain norm; but for
the purposes of Mathematical Science, the fundamental question is, under
what circumstances such an aggregate may be regarded as having a definite
cardinal number, and if ordered, a definite order-type. This question has
been fully discussed in the case of normally ordered aggregates; and the
condition for an affirmative answer in the case of any other aggregate is of
a similar character, viz. that it be possible to define other aggregates which
have either the relation of similarity or that of equivalence with the given one.
Ordered aggregates have been defined, which are not normally ordered;
and of such aggregates, the most important is the arithmetic continuum,
defined in ~ 128, as of order-type 0. The justification for regarding 0 as a
definite object, with a definite cardinal number, must, as has been pointed
out in ~ 158, be regarded as due to a postulation, subject to the law of contradiction. It has been seen that a class of aggregates exists which are similar
to the linear continuum, and thus conform to the type 0, and have c as their
common cardinal number; and this is in accordance with the regulative
principle which we have maintained to be essential to justify our regarding
c as a number.
As has been already remarked, aggregates may be defined, which are
unordered. In such cases no question arises as to the existence of an
order-type; but there is no reason why such aggregates should not have
cardinal numbers, provided that in the case of such an aggregate equivalent
aggregates can be found, the cardinal number in question being then their
common characteristic. The aggregates of which the cardinal number is
f  cC, are an example of this species of aggregate.
Two aggregates which have been independently defined are not necessarily comparable with one another, as regards either order-type or cardinal
number. It cannot be assumed a priori, that the cardinal number of one
of them is necessarily either greater, equal to, or less than that of the other,
in the sense in which these relations have been defined in ~ 112. Further, it
cannot be assumed, that an ordered aggregate, such as, for example, the
continuum, is necessarily capable of being normally ordered. Two aggregates
of abstract objects, which have been independently defined, may belong, no
doubt, to the same universe of thought; but nevertheless, any particular
category of relations may be too narrow to formulate any nexus between the
two systems; so that it is conceivable that, so far as such relations as those




158-160]


General discussion of the theory


207


of order, or cardinal number, are concerned, the two aggregates may be
completely isolated from one another.
160. In some proofs of theorems which have been given by writers on
this subject, which proofs have for their object the establishment of relations
of inequality or equality of cardinal numbers, aggregates are employed, the
elements of which are regarded as being successively defined by an infinite
number of separate acts of choice. When we leave the region of the finite
it would however appear that we have passed beyond the region in which
definitions by arbitrary acts of choice can be regarded as adequate specifications of definite objects; and the existence of a norm would appear to be
essential to our right to regard an aggregate as really defined, and therefore
to justify our making use of the conception of such an aggregate in the
proof of a theorem. The point may be illustrated by a discussion given* by
Du Bois Reymond, in which he contemplates the existence of a number
represented by a non-terminating decimal, in which the figures are determined by no law. He contemplates each figure in the decimal as being fixed
by a throw of dice, and rejects the conception of such a decimal, (ewig
gesetzloses Decimal), as representing a real number. A non-finite, or
endless, process can be conceived of as a completed whole, only when it is
subject to some kind of norm; thus a non-terminating decimal represents a
number, only under the presupposition that a set of rules can be given, which
would suffice to determine the figure that occupies any assigned place in
the decimal. In general, the proof of the possibility of giving a norm is
required before an aggregate of any particular character can be contemplated
as existing, or can be legitimately made use of in a demonstration.
Cantor, in his proof (see ~ 117), that N, is less than any other cardinal
number, has assumed that it is possible to pick out of any given transfinite
aggregate an enumerable component. This proof can only be accepted as
valid in case it is possible to define an enumerable component of the aggregate
in question. In a large class of cases, perhaps in all which are of importance
in Mathematics, this condition can be satisfied; for example in the case of
the continuum. In the aggregate of "all propositions," for example, the
enumerable component might be taken to be that aggregate of propositions
which asserts the existence of the numbers 1, 2, 3,....
G. H. Hardy has extendedt Cantor's method, for the purpose of shewing
that every cardinal number is either an aleph-number, or is greater than
all the aleph-numbers, and in particular that 2:o = c - N1. This proof runs as
follows:-Having given any aggregate whose cardinal number is > No, we can
choose from it successive individuals u, u,,...,,,... u,..., corresponding to
all the numbers of the first and second classes; and if the process came to an
* Allgemeine Functionentheorie, p. 91.
+ Quarterly Journal of Math., vol. xxxv, 1903, p. 88.




208


Transfinite numbers and order-types


[CH. III


end, the cardinal number would be K0. Its cardinal number is therefore
_ KN; and if > N,, - 2, and so on. And if > N^, for all finite values of n,
it must be   N,; for we can choose individuals from the aggregate corresponding to all the numbers of the first, second,... nth,... classes. And by
a repetition of these two arguments, we can shew that, if there is no Ng equal
to the cardinal number of the aggregate, it must be at least equal to the
cardinal number of the aggregate of all N's, and so greater than any Np.
Apart altogether from the question as to what constitutes all the alephnumbers, this argument could only be valid, if it were shewn how the
successive individuals u,, uz,... u,... are to be defined by means of some
norm, and also how the individuals of the aggregate which may correspond
to the numbers of the first, second,... nth,... classes can be assigned by
a norm. The process can neither come to an end, nor be regarded as, in
any sense, a completed one, unless this has been done.
In connection with the definition, given in ~ 116, of the aggregate obtained by covering one aggregate by another one it must be assumed that
each particular element of the aggregate of coverings is defined by a norm.
This point will be exemplified in the discussion, which will be given in
Chapter IV, of the cardinal number of all functions of a real variable.
In the theorem of ~ 120, the aggregate MAB must be regarded as such
that each element of it is defined by a norm. It is further necessary that
the aggregate M'AB be defined by a norm. This point may be illustrated
by referring to the second proof in ~ 56, that c > a, which is a special case
of the theorem of ~ 120. It is there hypothetically assumed that it is
possible to define a number of the continuum corresponding to each integral
number, by means of a norm; and thus the existence is assumed of a finite
set of rules by means of which the nth figure of the number which corresponds to the integer n can be calculated. By introducing an additional
rule, that, when this figure has been calculated, it is to be increased by
unity, unless it be 9, in which case it is to be replaced by zero, the existence
of a norm has been established, by which a number is defined that cannot
correspond to any integer; and thus a contradiction is shewn to arise from
the hypothesis made. It would not be sufficient to say that we may write
down a number which differs, in at least one figure, from any of the numbers
in the correspondence; it is essential to the validity of the proof, that such
a number be shewn to be definable by a finite set of rules.
161. Two proofs have been advanced, that every cardinal number is
necessarily an aleph-number; but this is equivalent to the statement that
every aggregate which has a cardinal number can be normally ordered.
If these proofs could be accepted as valid, the particular theorem would be
established that the arithmetic continuum is capable of being normally




160, 161]        General discussion of the theory                209
ordered; and the only question which would remain open, as regards this
aggregate, would be as to which particular aleph-number is the cardinal
number of the continuum.
The first of these proofs, that of Jourdain*, is founded on the assumption
that, if a cardinal number is greater than every aleph, there must be a part
of the aggregate to which this cardinal number belongs, which can be made
to have a (1, 1) correspondence with the "inconsistent" aggregate of all
the ordinal numbers arranged in normal order. This assumption is regarded
as justified by the process of making the successive elements of the aggregate of ordinal numbers correspond to elements of the given aggregate: it is
then argued, that, if this process comes to an end, the cardinal number of the
aggregate is an aleph; and that, if it does not come to an end, the given aggregate must contain a part that corresponds to the "inconsistent" aggregate
of all the ordinal numbers; and thus that, in the latter case the aggregate
is inconsistent, and has no cardinal number. The objection to this proof is
the fundamental one which has been already stated, viz. that no norm is
forthcoming by which the correspondence in question is defined; and, in
default of such norm, there is no meaning in speaking of an essentially
endless process as a completed one, or as having come to an end.
In the second proof, duet to E. Zermelo, no account is taken of the
possibility that an aggregate may have no cardinal number, nor of the
existence of "inconsistent" aggregates. The proof, which is fundamentally
of a similar character to that of Jourdain, is represented as demonstrating
that every aggregate can be normally ordered, and thus has an aleph as
its cardinal number.
It is assumed that, in each part M' of a given aggregate M, one element
m', called the special (ausgezeichnetes) element of MA', can be chosen.
A part M' must contain one element of M at least, and may contain all the
elements; and the aggregate {M'} of all parts of M is considered. Each
element M' of {M'} corresponds to a special element m' which belongs to M;
and this particular mode of covering the elements of {M'} by elements
of M is called a " covering " 7; the employment of a particular " covering " y,
is essential to the proof. A y-aggregate is then defined as follows:-Let MA
be a normally ordered aggregate consisting of different elements of M, such
that, if a be any arbitrarily chosen element of My, and if A be the segment
of My defined by a, which segment consists of all the ie-enmis of /My that
precede a, then a is always the special element of i -- 4  Every such
aggregate M, is a ry-aggregate. If every element1 of.iMJ anich occurs in a
7-aggregate be called a y-element of M, it is s ie'y  that the aggregate L,
of all ry-elements can be so ordered that it -;il;: i:i:'! I-aggregate, and contains
* Phil. Mag. January 1904, pp. 67, 70.
+ Math. Annalen, vol. LIX, 1904, "Beweiss, dass jede Menge wohlgeordnet werden kann."


H.


14




210


Transfinite numbers and order-types


[CH. III


all the elements of the original aggregate M. It follows then that M can be
normally ordered.
Zermelo himself expressly recognizes the assumption made as to the
existence of a definite "covering" 7. The objection to this assumption is
of the same character as before, viz. that for its validity a norm must be
shewn to be possible; this norm must assign to each part of the given
aggregate a definite " special" element belonging to that part. In the case
of such an aggregate as the continuum it is not clear how such a norm
could be devised; indeed, it seems probable that a proof of the possibility
of establishing such a norm involves difficulties comparable with those which
occur in any attempt to prove the original theorem. The non-recognition
of the existence of "inconsistent" aggregates, which existence, on the assumption of Cantor's theory, cannot be denied, introduces an additional
element of doubt as regards this proof.  The aggregate L, here employed,
is parallel with the normally ordered aggregate which occurs in Jourdain's
earlier proof.
162. As regards the method of G. H. Hardy (see ~ 151) for constructing
a set of points of cardinal number M, it was pointed out by Hardy, that
an infinite freedom of choice arises in the case of each limiting number y,
since there are an indefinite number of sequences of the preceding ordinal
numbers, of each of which sequences y is the limiting number.    Thus, for
example, o2 is not only the limit of o, co. 2,. 3,..., but also of o + 1,
w. 2 + 2,. 3 + 3,.... In the case of the smaller limiting numbers of the
second class Hardy has shewn how to exercise this freedom of choice so as
to obtain distinct sequences; thus o is taken as the limit of 1, 2, 3,...;
w2 is taken as the limit of o,.2,.3,....  In order however that the
method should really suffice to define sequences of integers which shall
correspond uniquely to each prescribed number of the first or of the second
class, it would be necessary to replace this freedom of choice by a definite
norm, or finite set of rules, which would decide, in the case of any particular
limiting number y, of what particular sequence of the preceding ordinal
numbers ry must be regarded as the limit, for the purpose of forming the
sequence of integers which is to correspond to it, in accordance with the
mode of formation employed in the method.
Hardy has given no norm of this character, but has confined himself to
the selection of the sequences which are to correspond to some of the lower
limiting numbers of the second class. When we reach the region of the
* A criticism of Zermelo's pnoof has also been published by Borel, Math. Ann., vol. LX, p. 194,
and is substantially identical with the above, which was published in the Proc. Lond. Math.
Soc., ser. 2, vol. III. Borel however objects to the definition of an aggregate by an infinite number
of acts of choice only when the aggregate is unenumerable; whereas the objection is really valid
in the case of any non-finite aggregate.




161-163]


General discussion of the theory


211


e-numbers of the second class, it is difficult, if not impossible, to imagine
the nature of the norm which would suffice to make the decision referred
to above; and no such norm is in fact forthcoming. On this ground, the
method cannot be regarded as really defining a set of points such that a
determinate point corresponds to each ordinal number of the first, or the
second class.
163. In case the criticisms which have been given above, of the general
theory of classes of order-types and of aleph-numbers, be accepted as wholly,
or in part, valid, nevertheless the debt which Mathematical Science owes to
the genius of G. Cantor will be in no material respect diminished. The
fundamental distinction between enumerable and unenumerable aggregates,
the interpretation of the arithmetic doctrine of limits, the ordinal theory of
the arithmetic continuum, and the conception of the transfinite ordinal
numbers of the second class, with their application to the theory of sets of
points, remain as permanent acquisitions which rest upon a firm logical basis.
This order of ideas has already become indispensable, for purposes of exact
formulation, in Analysis and in Geometry; it is constantly receiving new
applications, owing to its admirable power of providing the language requisite
for expressing results in the theory of functions with the highest degree of
rigour and generality. Cantor's creations have rendered inestimable service
in formulating the limitations to which many results in Analysis, formerly
supposed to be universally valid, are really subject. The outlying parts of
the theory, to which exception has been taken, would not appear to be
comparable in importance, for the general purposes of Analysis, with those
parts to which the criticisms made are not applicable. The latter involve
only a natural extension of the notion of Number, in which account is taken
of all the elements that are essential to the conception of number in its
original form; whereas we have endeavoured to shew that the more speculative general theory of aleph-numbers, and order-types, depends upon an
extension of the notion of number which leaves out of account an essential
element of that conception, viz. the notion of correspondence; and that this
is the origin of the contradiction which arises when an endeavour is made
to contemplate the totality of these new entities. The criticisms contained
in the latter part of the present Chapter are advanced with some diffidence,
on account of the great logical difficulties of the subject, and especially on
account of the philosophical difficulties relating to existential propositions.
It is hoped, however, that they may, in any case, be of utility as a contribution towards the discussion of questions of great interest which, at the
present time, cannot be regarded as having been decisively settled.
The fact that the general theory of the aleph-numbers has received no
applications in the theory of functions, and has indeed remained a purely
14-2




212           Transfinite numbers and order-types           [CH. III
abstract development of the theory of order, differentiates it from the theory
of normally ordered enumerable aggregates, which has now become an essential instrument in the theory of functions of one or more variables. All
aggregates of points in a continuum, which we at present know how to
define, have either the power of the aggregate of rational numbers, or else
that of the arithmetic continuum itself. The theories of these two kinds
of aggregates, including, as they do, a complete arithmetic theory of limits,
would thus appear to afford a sufficient basis for the development of
Analysis.




CHAPTER IV.


FUNCTIONS OF A REAL VARIABLE.
164. IF we suppose that an aggregate of real numbers is defined, the
aggregate being either enumerable or of the power of the continuum, such
an aggregate is said to be the domain of a real variable. It is necessary for
the purposes of Analysis to be able to make statements applicable to each and
every real number of the aggregate, and which shall be valid for any
particular number that may at will be selected. This is done by employing
the real variable, denoted by some symbol other than those used to denote
real numbers; and the essential nature of the variable consisting in its
being identifiable with any particular number of its domain. The symbols
used for denoting variables differ from those employed in the case of numbers
in being non-systematic. Operations involving real variables x, y, z,...,
with or without particular numbers, are carried out in conformity with the
same formal laws as hold in the arithmetic of real numbers. The result
of any such operation is itself a variable with a domain of its own,. which
may or may not be identical with that of any one of the constituent variables.
The numbers being used to designate in the usual manner the points of
a set on a straight line, the variable may then be taken to refer to the points
of the set.
If the given set of points be bounded, in the sense explained in ~ 46,
then the domain of the variable is said to be limited. When the domain of
the variable is not limited, it is said to be unlimited in one or in both
directions.
The variable is said to be continuous in a given interval (a, b) when all
the points of the interval, including a and b, belong to the domain of the
variable. If the points a, b do not belong to the domain, but every internal
point of the interval does so belong, the variable is said to be continuous in
the open interval (a, b), or within the interval (a, b). It is unnecessary to
give in detail the corresponding definitions applying to the case of an
aggregate of any number n of dimensions, which is regarded as the domain
of n independent variables x,, x2,... x,.




214


Functions of a real variable


[CH. IV


The term "variable" has been commonly associated with the conception
of a point moving in a straight line or in a curve. It has however been
pointed out in the course of the discussions of the continuum, contained in
the earlier chapters, that the continuum cannot legitimately be regarded as a
synthetic construction formed by a set of points determined successively.
Successive determination is applicable only in the case of any enumerable
sequence which may be defined within the continuum, and such a sequence
may represent a succession of positions of a point moving in a straight line.
It is however unnecessary to proceed to a detailed analysis of the conception
of motion, because the Theory of Functions has no need of the conception of
temporal succession. The theory makes continual use of simply infinite
sequences determined in the continuum; and any such sequence may be
regarded as a series of distinct determinations of the variable in which the
elements are in logical succession, each element after the first being preceded
and succeeded by definite elements.
THE FUNCTIONAL RELATION.
165. If to each point of the domain of the independent variable x there
be made in any manner to correspond a definite number, so that all such
numbers form a new aggregate which can be regarded as the domain of a
new variable y, this variable y is said to be a (single-valued) function of x.
The variables x, y are called the independent and the dependent variable
respectively; and the functional relation between these variables may be
denoted symbolically by the equation y =f(x). In this definition no restriction is made a priori as regards the mode in which, corresponding to each
value of x, the value of y is assigned; and the conception of the functional
relation contains nothing more than the notion of determinate correspondence
in its abstract form, free from any implication as to the mode of specification
of such correspondence.  In any particular case, however, the special
functional relation must be assigned by means of a set of prescribed rules
or specifications, which may be of any kind that shall suffice for the
determination of the value of y corresponding to each value of x. Such
rules may in any particular case be embodied in a single arithmetic formula
from which the value of y corresponding to each value of x is arithmetically
determinable; or the rules may be expressed by a set of arithmetic formulae
each one of which applies to a part of the domain of the independent variable.
In case these formulae be reducible to a set of mutually independent
formulae, that set must be a finite one. In case the function be defined
by an enumerably infinite set of formulae, each applicable to a part of the
domain, these formulae cannot be mutually independent, but must be subject
to some norm.




164, 165]


The functional relation


215


It must be observed that, when for any particular value of x the
corresponding value of y is given by means of an arithmetic formula, the
numerical value of y is in general only formally determinate; for in practice
only a finite number of elements of a convergent aggregate which defines the
value of y can in general be actually found, and thus the value of y can be
specified only to any required degree of approximation, but it is still regarded
as perfectly determinate.
The domain of x consisting of a set (P) of points, the values of y, in the
case of a given functional relation y =f(x), may be represented by points Q
on another straight line, all such points forming a set (Q). The set (Q) is
said to be the functional image of the set (P), determined by the function
f(x); to each point of (P) there corresponds a single point of (Q), iff(x)
be a single-valued function, but to each point of (Q) there may correspond
a finite or an infinite number of points of (P).
The perfectly general definition of a function which has been given above
is the culmination of a process of evolution which has proceeded largely in
connection with the study of the representation of functions by means of
trigonometrical series. By the older mathematicians a function was understood to mean a single formula, at first usually only a power of the variable;
but afterwards it was regarded as defined by any one analytical expression,
and was extended by Euler to include the case in which the function
is given implicitly by a formal relation between the two variables. In
connection with the problem of the determination of the forms of vibrating
strings, which led to the discussion of functions represented by trigonometrical
series, the conception arose of a single function defined in different intervals
by means of different analytical expressions. The arbitrary nature of a
function given by a graph was distinctly recognised by Fourier; thus the
notion of a function was emancipated from the restriction that an a priori
representation of it by a single formula is necessary.
The idea that a function can be defined completely, in the case when the
domain of the independent variable is a finite continuous interval, by means
of a graph arbitrarily drawn, leaves out of account the essentially unarithmetic
nature of geometrical intuition. A curve that is drawn is indistinguishable
by the perception from a sufficiently great number of discrete points; and
thus all that is really given by an arbitrarily drawn graph consists of more or
less arithmetically inexact values of the ordinates at those points of the
x-axis at which we are able to measure ordinates. In order that a curve
may be really known, sufficiently to serve for the purpose of defining a finction,
a series of rules must be prescribed, by means of which the values of the
ordinates can be formally determined at all points of the x-axis. It is
sometimes said, in order to illustrate the generality of the functional relation,
that a function is definable in the form of a table which specifies values of y




216


Functions of a real variable


[cH. IV


corresponding to values of x, this table being of a perfectly arbitrary character.
The inadequacy of such illustration is manifest, if we consider that even if
the table were an endless one, as has been remarked in ~ 160, no aggregate
of y-values can be defined by an endless set of numbers, apart from the
production of a norm by which those numbers are defined. Moreover, even
if the table were subject to a definite norm, it could only theoretically suffice
to define a function of a variable whose domain consisted of an enumerable
set of points, and would be totally inapplicable to the case in which the
variable has a continuous domain, unless some special restrictive assumptions
as to the nature of the function be introduced, by means of which the values
of the function are made determinate at the remaining points of the continuous domain.
It thus appears that an adequate definition of a function for a continuous
interval (a, b) must take the form first given to it by Dirichlet*, viz. that y is
a single-valued function of the variable x, in the continuous interval (a, b), when
a definite value of y corresponds to each value of x such that a _ x  b, no
matter in what form this correspondence is specified. A particular function
is actually defined when y is arithmetically defined for each value of x.
No elaborate theory is required for functions which retain their complete
generality, in accordance with the abstract definition given above, since no
deductions of importance can be made from that definition which will be valid
for all functions. When, however, the nature of a function is in some way
restricted, either in the whole domain, or in the neighbourhoods of special
points of that domain, there is room for the development of a theory which
shall deal with the peculiarities that follow from such restrictions upon
the complete generality of functions.
166. The functions defined in accordance with the above definition are
known as single-valued functions, since, to each value of x in the domain of x,
there corresponds a single value of y. The definition may be so generalised
as to be applicable to multiple-valued functions. This is done by replacing
the requirement that, to each value of x in the domain of x there shall
correspond a single value of y, by the more general statement that, to each
value of x there shall correspond a definite aggregate of values of y. The
aggregate of values of y may, for any particular value of x, consist of a finite,
or of an infinite, set of numbers. A particular function is then defined when
the aggregate of values of y is arithmetically determinate for each value of x,
in accordance with the criteria for the determinacy of a linear aggregate
which have been developed in the theory of aggregates. Although the
Theory of Functions, as developed in the present work, is mainly concerned
with single-valued functions, it is necessary, or at least convenient, in the


* See Dirichlet's Werke, vol. I, p. 135.




165, 166]


The fjnctional relation


217


course of the examination of particular functions and classes of functions, to
make use of auxiliary functions which are multiple-valued at certain points
of the domain of the independent variable. Moreover, Dirichlet's definition,
in its original form, has the inconvenience that it excludes from the category
of functions such as are represented by analytical expressions which, for
particular values of the independent variable, cease to define a single number.
For example, an infinite series which, for particular values of the variable,
either diverges, or ceases to converge to a single definite limit, does not define
a single-valued function in accordance with Dirichlet's definition, for the
whole domain of the variable, and yet it is convenient to so extend the
meaning of the term function that a function may be nevertheless defined
for the whole domain by such a series.
The distinction has been considered in detail by Broden* between those
functions for which the relation between the dependent variable y and the
independent variable x is formally the same for the whole domain of x, and
those functions for which the domain of x is divisible into a plurality of parts,
for which the forms of the relation between x and y are different. He
remarks that the distinction is one relating to the character of the definitions
rather than to the nature of the functions themselves; in the former case the
function is said to be homonomically defined, and, in the latter case, to be
heteronomically defined. Broden has given a formal proof that, when a
function is heteronomically defined, the number of parts into which the
domain of x is divided, so that the relations of y to x in any one part are
completely independent of the relations in the other parts, must be finite.
The Theory of Functions of a Real Variable is concerned with the
classification of functions, according as they possess various special properties,
e.g. continuity, differentiability, integrability, throughout the domain of the
independent variable, or at, or near, special points which form part of that
domain.   The theory requires the introduction of precise arithmetical
definitions of the scope and meaning of these characteristic properties, and
is largely concerned with the determination of criteria which shall suffice
to decide, in the case of a function defined in some special manner, what can
be inferred as regards the possession by such function of properties other
than those that are immediately apparent from the definition itself. Much
of the theory is concerned with a minute examination of functions, and
of classes of functions, which possess properties that do not occur in the case
of those functions which are employed in ordinary analysis and in its
applications to Geometry and Physics; and the theory has in consequence
frequently been described as the Pathology of Functions. It appears
however from the theory itself that many of those peculiarities, which
from the point of view of traditional Analysis would be described as ex

* Acta Univ. Lund. vol. xxxIII, 1897, " Functionentheoretische Bemerkungen und Satze."




218


8Functions of a real variable


[CH. IV


ceptional, have no claim to be so described; that in fact it is in the functions
of ordinary Analysis that the abnormalities really occur, such functions
occupying an exceptional position in relation to a scientific Analysis of
the properties of functions in general. An important result of the labours
of those who have developed the modern theory of functions of a real
variable has been that restrictive assumptions, which had previously been
unconsciously made in the processes of ordinary Analysis, have been placed
in a clear light; and it has been shewn that modes of reasoning which had
their origin in an uncritical application of ideas obtained from intuition would
fail to yield correct results when applied to cases of sufficient generality,
the unsoundness of the logical basis of such reasoning being thereby
demonstrated.
In ordinary Analysis the domain of the independent variable is taken
to be a limited, or unlimited, continuous interval. In the theory of functions,
on the other hand, it has been found advantageous to consider also the
properties of functions defined for a domain which is not a continuous one.
It appears, in particular, that a non-dense perfect set of points, or more
generally any closed set, is well suited to be the domain of a function, inasmuch
as, for such domains, the principal peculiarities of functions, such as continuity,
differentiability, &c., are capable of precise formulation, and can serve for
purposes of classification, exactly as in the case of functions defined for a
continuous domain. Much of the recent progress in the subject is due to
a recognition of the parity of all perfect sets of points, not only as regards
their internal structure, but also in relation to their fitness for forming the
domains for which functions can be defined, without loss of any of the
characteristic properties that serve for the classification of functions of a
real variable, or of several such variables.
EXAMPLES.
1. A function f(x) may be defined for the interval (0, 1) as follows:-for -1  x > 1 
n     n+l'
f(x)= - 2, and for x=, f(0)= 1, n denoting any positive integer. In this case, the norm
by which the function is defined is expressible by an enumerable set of formulae which are
however not independent of one another.
1       x     1      1
2. A function may be defined as follows:-for 1'x> 2, f(x)=; for ->  >,
2       2     2      3
x     i      1      x                              I        x
f(x);for x3 ', f (x)=5,... and in general, for 1 x  /f(z)=   where
3     3- -4      ~  5                     n     =     AX =   P-II,
P, denotes the nth of the prime numbers 2, 3, 5, 7,.... If the function is to be defined at
the point x=0, this may be done by assigning to f(O) any arbitrarily chosen value we
please. It will be observed that the values p-are in this case not representable by a
single expression which involves n and  only.
single expression which involves n and x only.




166, 167]   Upper and lower limits of functions


219


3. Any number x of the interval (0, 1) except 0, can be uniquely expressed in the form
bl b2c     b
l+2+... +b+...
2  22     21n2
where b, has for every value of n one of the values 0, 1, and it is stipulated that all the b%
are not to be zero from and after any fixed value of n.
1
A multiple-valued function* may be defined by y=xn, where n has all positive integral
values for which b= 1. This is a homonomic definition, although no analytical expression
of a unitary character can be given for the representation of y.
THE UPPER AND LOWER LIMITS OF FUNCTIONS.
167. A function y =f(x), being defined for the domain of x, we have seen
that the values of y form a set of points, determined as usual upon a straight
line, which is called the functional image of that set of points which forms
the domain of x. In case the set of points, which represent the values of y,
is a bounded set, the function f(x) is said to be limited in the domain of x.
When the set of values of y is bounded, either boundary may be a limiting
point, or only an extreme point, of the set. For convenience, and in accordance
with usage, the terms upper limit and lower limit will be applied to denote
the upper and lower boundaries of y, even when the boundary is not a limiting
point of the set, but is only an extreme point, without being a limiting point
in the sense in which this term is used in the theory of sets of points.
Thus we may say that:If the set of points y, which represents the functional image of a function
f (x), defined for a given domain of x, have an upper and a lower boundary,
then the function f(x) is said to be a limited function, and the boundaries are
said to be the upper and lower limits oJ f(x) in the domain of cx.
The upper or the lower limit of a function f (x) in its domain may or may
not be attained, i.e. there may or may not be a value of x, in the domain of x,
for which the functional value is equal to the upper, or to the lower limit,
of the function. An upper or lower limit, which is attained, is an extreme
point of the set of values of y, and may or may not be a limiting point
of such set, in the accurate sense. An upper or a lower limit which is not
attained is certainly a limiting point of the set of values of y.
In case y have no upper limit, or no lower limit, for the domain of x, the
function f(x) is said to be an unlimited function. In this case there exist
values of the finction, of one or of both signs, which are numerically greater
than any arbitrarily assigned number A.
When y has no upper limit in the domain of x, the function is said to
have the improper limit + oo, in the domain of x. Similarly, when y has no


* Brodin, loc. cit. p. 4.




220


Functions of a real variable


[CH. IV


lower limit, it is said to have the improper limit - o. It is frequently said,
for the sake of brevity, that the upper or the lower limit of the function
is infinite.
The excess of the upper limit of a function, in its domain, over its lower
limit, is called the fluctuation (Schwankung) of the function in the domain.
In case the upper or the lower limit is infinite, the function is said to
have an infinite fluctuation in its domain.
Instead of the whole of the domain of x, we may consider that part
which lies in a given interval (a, b), including the end-points a and b, and the
preceding definitions may be applied to this portion of the domain; thus:The upper limit of a function f (x) in an interval (a, b) is the upper limit
of the function when only those points of the domain of x which lie in (a, b) are
taken into account. A similar definition applies to the case of the lower
limit.
The excess of the upper limit of f (x), in the interval (a, b), over its lower
limit in that interval, is called the fluctuation of f(x) in the interval (a, b).
In case one or both of the limits is infinite, the fluctuation of the function
in (a, b) is said to be infinite.
If the upper limit of f(x) in (a, b) is attained, i.e. if there exists a value c
of x such that f(c) is the upper limit, where c is a point of the domain in
(a, b), then this upper limit is said to be the upper extreme of the function in
(a, b); and a similar definition applies to the lower extreme.
If the end-points a, b of the interval be left out of account, in case they
belong to the domain of x, the fluctuation is called the fluctuation in the
open interval (a, b). This is sometimes spoken of as the inner fluctuation
of the function in (a, b), and is determinable as the limit of the fluctuation
in the interval (a + e, b- e), when e is indefinitely diminished.
168. In accordance with the definition which has been given for a
function in any domain, the value of the function at any particular point
of the domain has a definite finite value. It may happen that a point P,
of the domain of x, may be such that in any arbitrarily small neighbourhood
of P either the upper or the lower limit of the function, or both, may not
exist; so that, however small the neighbourhood of P may be chosen, there
exist functional values in that neighbourhood which are numerically greater
than any number that may be assigned. In that case, the point P is said to
be an infinity, or point of infinite discontinuity of the function; although the
function has a definite finite value at the point P itself.
Although f (x) is not properly defined at a point P, (x0), unless a definite
numerical value be assigned tof(x0), nevertheless an improper definition of the
functional value at the point P is sometimes admitted, of the form  (=0;
f (X)




167-169]


Continuity of functions


221


in this case the function is said to possess an infinity at P. This infinity is
said to be removable provided that, when the functional value at P is
altered to some finite value, the function have finite upper and lower limits
in a sufficiently small neighbourhood of P.
There are other cases in which an improper definition of the functional
value at a point x, of the domain of x is admitted. The function may
be defined by means of an infinite series, of which the terms are given
functions of x.  This series may diverge at the particular point x0;
but it is nevertheless frequently convenient to regard the series as defining
the function for all values of x in some interval which includes x0. The
functional value at x0 is then regarded as infinite.
In accordance with strict arithmetic theory, the function is regarded as
undefined at points where no definite finite value of the function is specified.
For the most part, in the theory which will be here developed, this
restriction will be rigidly adhered to. It will be found, however, that in
cases, such as in the theory of infinite series, in which it is convenient
to admit improper definitions of functions at particular points, no essential
change in the main results of the theory will have to be made.
In some cases it will be found convenient to remove the restriction that
at each point of the domain of the independent variable the function shall
be single-valued, and to define the function in such a manner that, at single
points, or at each point of some set belonging to the domain of x, the function
may possess finite or infinite multiplicity. It will be found, in the cases
in which it is convenient to make this extension of the meaning of a function,
that no difficulty arises as regards the use of results primarily applicable to
functions which are single-valued at all points of the domain of the variable,
without exception.
THE CONTINUITY OF FUNCTIONS.
169. Let the domain of the independent variable x be.continuous, and
either bounded or unbounded; and denote the function y at the point x by
J ().
The function f(x) is said to be continuous at the point a of the domain of x,
if, corresponding to any arbitrarily chosen positive number e whatever, a
positive number 8 dependent on e can be found, such that f (a + q) - f(a) I < e,
for all positive or negative values of v which are numerically less than 8, and
which are such that a + ir is in the domain of x. At an end-point of a limited
domain, the values of q will have one sign only.
In accordance with this definition, a neighbourhood (a - 8, a + 3) of the
point a exists, such that the function, at any point in the interior of this
interval, differs numerically from its value at a, by less than e. It follows




222


Functions of a real variable


[CH. IV


that the inner fluctuation of the function in (a - 8, a + 8) is less than 2e, and
it is obvious that the fluctuation in any interval interior to (a -, a + 8)
is less than 2e. The condition of continuity of the function f(x) at the point a
may thus be stated to be that a neighbourhood of the point can be found
in which the fluctuation of the function is as small as we please.
The above definition of continuity at a point is that due to Cauchy, and
is a particular case of the definition of continuity for a function of any
number of variables. If we denote by f(x, y, z,...) a function of the variables
x, y, z,... defined for any continuous domain, the condition of continuity
at the point (a, /, y,...) is that, corresponding to every arbitrarily chosen
positive number e, a number 8 dependent on e, can be found, such that
If(a + h, 3+ k,  +,...)-f(a,/3,y,...)< e, provided h, k,,... have any
values which are numerically less than 8. In this case, a neighbourhood
(a- 8, a + 8) of a point of a linear domain, is replaced by a " rectangular cell,"
which is a square in the case of a two-dimensional domain. The definition
of continuity has been stated by Heine* in a form which depends upon the
notion of a convergent sequence of numbers or of points. Let (P1, P,,... P,,...)
be a convergent sequence of points in the given domain, and of which P is
the limiting point. The condition of continuity of the function at P is that,
for every such convergent aggregate which has P as limiting point, the
numbersf(Pi), f(P),... f(P,,),... form a convergent sequence which represents the number f(P).   That this definition is equivalent to Cauchy's
is seen at once by considering a sequence of values of e which have the limit
zero, and are such that e1 > e, > e3....
A function which is not continuous at a point a may satisfy the condition
that in a neighbourhood of a on the right the fluctuation of the function
may be made as small as we please by taking the neighbourhood small
enough; the function is then said to be continuous on the right at a. A
similar definition applies to continuity on the left.
A function is said to be continuous in the interval (a, b) if it satisfies the
condition of continuity at every point in the interval.
The function is said to be in general continuous in the interval, if, when
arbitrarily small neighbourhoods of a finite number of points are removed,
the functions be continuous in each of the remaining intervals. Either of
the points a, b may be one of this finite number of points.
170. The domain of the independent variable has hitherto been considered to be continuous; it is however clear from a consideration of the
definition of continuity, either in Cauchy's or in Heine's form, that the
* Crelle's Journal, vol. LXXIV, p. 182.
t C. Neumann uses the term abtheilungsweise stetig: see his work "Die nach Kreis, Kugel,
und Cylinder-functionen fortschreitenden Reihen."




169, 170]


Continuity of functions


223


definition is applicable in case the domain of the independent variable is not
continuous, but consists of any set of points which contains limiting points
that belong to the set. It is, of course, only at such a limiting point that the
question of continuity arises; for at an isolated point of the aggregate there
are only a finite number of values of the function in any sufficiently small
neighbourhood of the point. If P be a point of the domain of x which is a
limiting point of the domain, the function is continuous at P when, for every
sub-set (P,, P2,... P,,...), all the points of which belong to the domain, and
which has P as limiting point, the numbers f(P1), f(P,),... f(P),... form
a convergent sequence of which f(P) is the limit. If the function be
continuous at every limiting point of the domain of x it is said to be
continuous relatively to the given domain; and thus the notion of continuity
of a function is applicable whatever be the domain of the independent
variable, except when it consists of an isolated set of points.
Let PI, P2, P3,... be a convergent sequence of points of the domain of x,
of which P, is the limiting point; and let P, also belong to the domain
of x. Supposing the functional image, corresponding to f(x), to contain the
points Q1, Q2, Q3,... which correspond to P1, P2, P3,..., let Q, Q2, Q3,... form
a convergent sequence of which the limiting point Q. corresponds to P..
If this condition be satisfied, however the convergent sequence be chosen in
the domain of x, the aggregate (Q), of values of y, is said to be a continuous
functional image of the domain (P) of x.
It is clear that the continuous functional image of a closed domain is
itself closed. For, corresponding to the points of a convergent sequence
(Q,, Q2, Q3,...), in (Q), there corresponds an aggregate (P,, P,, P,...), in (P),
which must have at least one limiting point, and all such limiting points
belong to the domain (P), and must correspond to the limiting point of
(Q1, Q2, Q3,...), which therefore belongs to the aggregate (Q). Moreover if
(P) be perfect, the continuous functional image (Q) is perfect also; for, corresponding to any particular point Q' of (Q), we may take a point P' of (P), for
which Q' is the image. P' is the limiting point of a convergent sequence
of points of (P), and to this convergent sequence there corresponds a convergent sequence in (Q), of which Q' is the limiting point. It has thus been
shewn that (Q) contains no isolated points, and therefore (Q) is perfect.
If (Q) be a continuous functional image of the closed set (P), and if only
one point of (P) correspond to each one point of (Q), then (P) is a continuous
functional image of (Q).
To the points of any convergent sequence (Q,, Q,,...) in (Q), of which Q.
is the limiting point, there corresponds a convergent sequence (P1, P,, P3,...)
in (P) of which P. is the limiting point, and P, is the functional image
of Q,.




224


Functions of a real variable


[CH. IV


171. The theorem has been given by Weierstrass that, if (a, b) be any
interval containing points of the domain of a function, then one point at
least exists in the interval, which is such that, in any arbitrarily small
neighbourhood of that point, the upper limit of the lunction is the same as
the upper limit of the function in the whole interval (a, b).
This theorem holds for all functions without restriction, and it makes no
difference whether the whole interval (a, b), or only a set of points in that
interval, belongs to the domain of the independent variable.
If M  denotes the upper limit of the function in (a, b), the case of an
indefinitely great upper limit being included, let the interval be divided into
a number n of equal parts. It is then clear that the upper limit of the
function for no one of these parts can be greater than M, and that, in one at
least (al, /3,) of these sub-intervals, the upper limit of the function must
be M. Divide (a,, /3) into n equal parts, then, as before, one of these parts
(a2, /3), at least, is such that M  is the upper limit of the function in it.
Proceeding in this manner, we obtain a sequence (al, 1/3), (a2, /32),... (a,., /,.),...
of intervals whose lengths converge to zero, such that each one is contained
in the preceding one, and such that Mi is the upper limit of the function in
any one of these intervals. In accordance with the theorem of ~ 48, there is
one point x1, which is in all these intervals; and this point x, is such that in
any arbitrarily small neighbourhood the upper limit of the function is M.
A similar result holds for the lower limit of a function.
In the case of a function which is continuous in the interval (a, b),
it follows from the foregoing theorem that the upper and lower limits of the
function in (a, b) are both finite, and thus that a function which is continuous
in an interval is limited in that interval.
For consider that point x, in (a, b), in the arbitrarily small neighbourhood
of which (x, - e, x1 + e) the upper limit has the same value as for the whole
interval (a, b). Since the function is continuous at x,, corresponding to a
given number 8, a number e can be found such that If(x)-f(x) < 8,
provided x lies in (xi - e, x1 + e); consequently the upper limit of f(x) in this
interval must be finite, and hence f(x) has a finite upper limit in (a, b). It
may be shewn in a similar manner that the function has a finite lower
limit.
A function which is continuous in the interval (a, b) is such that its upper
limit and its lower limit are each actually attained at one point at least in
the interval, i.e. the function has an upper extreme and a lower extreme in the
interval.
For suppose, if possible, that f (x,) has a value A different from M; and
consider an arbitrarily small interval (xi - e, xl + e) for which M is the upper
limit of the values of the function; then points can be found in this interval for




171, 172]


Continuous functions


225


which the function differs by less than an arbitrarily small number S from M.
These values of the function would differ from f (xi) by an amount which is
not arbitrarily small, and this would be inconsistent with the condition of
continuity of the function at the point xi. It follows that we must have
f (x) = M. Similarly it may be shewn that the lower limit m is reached at
least once in the interval (a, b).
CONTINUOUS FUNCTIONS DEFINED FOR A CONTINUOUS INTERVAL.
172. It will now be shewn that if f (x) be continuous in the continuous
domain (a, b), and if f (a), f (b) have opposite signs, then there is at least one
value of x in the interval, for which f (x) vanishes.
Suppose f (a) < 0, f (b)> 0; then on account of the continuity of the
function we know that at a point x, for which f (x) is negative, an interval
(x, x + e) can be found for which the fluctuation of the function is as small as
we please; and therefore the interval can be so chosen that for every point of
it the function is negative. Dividing the whole interval into any n equal
parts consider the signs of the function at the points of division. Writing e
for 1/n, let a + (pi + 1) e (b - a) be the first of these for which the function is
positive; thus for a +pe(b- a) the function is negative, or zero. Divide
the interval {a ++pe (b - a), a + (p, + 1)e (b - a)} into n equal parts, and
suppose the point of division, a+(p,e+p2e2) (b-a), is the last of these,
reckoned to the right, for which the function is negative, or zero. Proceeding
in this manner we obtain a series of numbers S, where S, denotes
p1e + p2e2 +... + pnem, which are such that for a + Sm (b- a) the function is
negative, or zero; and for a + (Sm + e'") (b - a) the function is positive. Let
c be the limit of the sequence a + S(b-a); then it can be shewn that
f (c) = 0. For if f(c) were negative, then an interval (c, c + 8) could be found,
for all points of which the function is negative: and by choosing m sufficiently
great, the point a + (S + eem) (b - a) could be made to fall within the interval
(c, c + 8), for which point the function would be positive: hence f (c) cannot
be negative. Again, f (c) cannot be positive; for in that case an interval
(c -, c) can be found for all points of which the function is positive; but
by choosing m large enough the point a +S (b-ca) can be made to fall
within this interval, and then for this point the function is negative, or zero.
Since then the function f(c) cannot be either positive or negative it must
therefore be zero.
From this theorem we can deduce that, whatever values f(a), f(b) may
have, there must be in the interval (a, b) at least one value of x, for which f(x)
has any prescribed value lying between f (a) and f (b).
Let this value be C, and suppose f(a) < C < f(b); then the function
f (x) - C is continuous in the given interval, is negative when x = a, and
positive when x = b; thus it vanishes at least once in the interval (a, b).


H.


15




226


Functions of a real variable


[CH. IV


A continuous function has frequently been defined as a function such that,
iff (a), f(b) be its values at any two points a and b, then the function passes
through every value intermediate between f (a) and f (b), as x changes from
a to b. The property contained in this definition has been shewn above to
hold of every function which is continuous in accordance with Cauchy's
definition; but the converse theorem does not, in general, hold. The definition just referred to is accordingly not equivalent to that of Cauchy,
which is here adopted as the basis of the treatment of continuous functions.
As an example of the non-equivalence of the two definitions, we may consider
1
the function defined by y=sin-, for x  0, and by y =0, for x=0. For
XQ
this function there are values of x between a and b for which f (x) has any
assigned values c lying between f (a) and f (b); but the function is not continuous, in accordance with Cauchy's definition, in any interval (a, b) which
contains the point 0. It is, in fact, easily seen that the point 0 is a point of discontinuity of the function; for an arbitrarily small neighbourhood of the point 0
contains points at which the function has all values in the interval (- 1, 1).
As another example* of a function which satisfies the condition referred
to, but is discontinuous. in accordance with Cauchy's definition, let the number
x in the interval (0, 1) be expressed as a decimal aaaa,... a,...; then consider the decimal 'aca3aca7.... If this last decimal is not periodic, we take
f (x) = 0; if it is periodic, and the first period commences at a2n-,, we take
f (x) = 'a2,a2n+2a2,,,.... The function so defined for the interval (0, 1) of x
has every value between 0 and 1, in every arbitrarily small interval in the
domain of x; thus the function is discontinuous at every point. A value of
x for which f(x) has any prescribed value 'PP2-...P.... is
-1A Aa3B.. Ka2np 1a2,p2a21,+2...,
where 'acca... is any periodic decimal, the first period of which begins at
a2-,_, and A, B,... K are arbitrarily chosen digits. Nevertheless there are
values of x between a and / at which the function takes any assigned value
intermediate between f (a) and f (/).
CONTINUOUS FUNCTIONS DEFINED AT POINTS OF A SET.
173. It will now be shewn that, if a function f (x), having prescribed
values at each point of an infinite set of points in the interval (a, b), be continuous in that interval, then the values of the function are determinate at each
point of the derivative of the set.
Suppose a., a2, a3... a,... to be a convergent sequence of points, for which
xI is the limiting point, and suppose f (a1), f (a2)...f (a,)... to be known; it
will be shewn that these functional values form a convergent sequence whose
limit is f (x). An interval (x - 8, x, + 8) can always be found, corresponding
" See Lebesgue, Legons sur V'integration, p. 90.




172-174]


Continuous functions


227


to any fixed number e, such that the function at any point of this interval
differs from f (x1) by less than the arbitrarily small number e; this follows
from the continuity of the function. A number n can be found such that
all the points An, an+i,,,+2... lie within the interval (x, - 8, x- + 8). It follows
that If(I) -f( (a) | and If (x) -f(an+)..., etc. are all less than e, which is
arbitrarily small; hence f (x,) is the limit of the sequence f (al), f (a,)...
and thus f (x1) is determinate. From this special case it follows that, for all
the limiting points of a given set of points in (a, b), the values of the continuous function are determinate. It further appears that the function is
determinate for all points which belong to any derivative of the given set,
for the points of which set the values of the function are known.
In particular, if a continuous function have prescribed values for points of
a set which is everywhere-dense throughout the interval (a, b), then its values are
determinate for all points of the interval.
A special case of such a set would be all the rational points within the
interval. It follows that a continuous function whose values are known for all
the rational points in an interval is determinate for all the. irrational points.
A continuous function which is known to be constant for all the rational
points has the same constant value for all the irrational points in the interval.
A generalization of the above theorem is, that a function which is
continuous with reference to a domain which consists of a set (P), and
is known for all points of a sub-set which is everywhere-dense in (P), is
determinate for every point of (P). This may be seen by considering that
every point of (P) is a limiting point of the sub-set, and applying the same
reasoning as before.
174. From the theorem established above, that a continuous function is
determinate when its values at an everywhere-dense enumerable set of points
are prescribed, we may deduce that the cardinal number of the aggregate of all
continuous functions of a real variable is the cardinal number c of the continuum.
We may suppose the values of a function to be prescribed at the rational
points. The cardinal number of the aggregate of all functions defined for
the rational points only is the cardinal number of the ways of covering the
aggregate of rational numbers by the aggregate of numbers of the continuum.
This number is ca, which has been shewn in ~ 148 to be equal to c. Only some
of the "coverings" of this kind are such as will give rise to continuous
functions; hence the aggregate of all continuous functions is a part of the
aggregate of all possible coverings of the set of rational numbers by the numbers
of the continuum. It follows that the cardinal number of the aggregate of all
continuous functions is _ c. Again, this cardinal number is > c; for among
the con i-inu.ous functions are those each of which is constant and everywhere
equal ti: anv: assigned number of the continuum; and thus the aggregate of
15-2




228


Functions of a real variable


[OH. IV


all continuous functions contains a part which has the cardinal number c.
Since the cardinal number is _ c, and also - c, it is equal to c.
It has been shewn by Borel* that the aggregate of all continuous
functions and also that of all analytical functions of two or more variables
have the cardinal number c.
The cardinal number of the aggregate of all functions of a real variable
is that of all coverings of the continuum by itself; this is, in accordance
with the definition in ~ 116, denoted by c0, for which we may writef.
Each particular "covering" of the numbers of the continuum by themselves is definable by a definite norm, and corresponding to each such
covering there is a definite function for a continuous domain.  Let the
aggregate of all such functions be denoted by F: it will then be proved that
the cardinal number f, of F, is > c.
First, F has a part which is equivalent to the continuum. This is at
once seen, since the functions f(x)=c, where c is any number of the
continuum, constitute such a part. It follows that f _ c.
Next, let it b6 assumed, if possible, that F is equivalent to a part of the
continuum. As has been just proved, such a part cannot have a cardinal
number < c; we therefore assume that F is equivalent to the set of numbers
of the continuum. This amounts to the assumption that F can be ordered in
the same type as the continuum, so that, to any assigned number | of the
continuum, there corresponds a definite set of rules Re which defines a
function ft (x). The correspondence between | and Re must itself be defined
by a set of rules, so that when | is assigned, Rg, and therefore the function
fI (x) is defined. The aggregate {fi (x)} must contain every definable function
of a real variable. The number | being assigned, f (x) is producible, and its
existence implies that, at, any assigned point |:', the functional value fe(a')
can be determined arithmetically. We may take, for example, '=:; and
thus, if: is assigned, 4/ (|) is known. We may regard fe (:) as a function of
A; for its value at any point: can be arithmetically determined, and it is
therefore an element of the aggregate F of all functions. With this understanding as to f (), choose a fixed number, say unity, then the function
(b (t)- f(A)+ 1 has a definite norm; for we have only to add to the rules
by which fe (:) is defined, the further rule, that, at each point A, unity is to
be added to the value of f (I). We have now a new definable function 4 (x);
but this cannot possibly belong to the aggregate F, for if it do so belong, there
must be some one point |f of the continuum, with which it corresponds; but
b (x) cannot be identical with fel(x), for (b (l) and fel (,)) differ by unity. Since
0 (|) is not contained in F, contrary to the hypothesis, it follows that F cannot
be equivalent to the continuum, and thus the theorem, f> c, is established.
It has therefore been shewn that f > c, and consequently that-the aggregate
of all functions of a real variable has a cardinal number f, greater than c.
* See Lemons sur la theorie des fonctions, p. 127.




174, 175]               Uniform    continuity                     229
UNIFORM CONTINUITY.
175. It will now be shewn that if the domain of x be a continuum, then
a continuous function is uniformly continuous through the domain of x; that
is to say, a number 8 can be found corresponding to any given e, such that, for
all values of x, the fluctuation off (x) within the neighbourhood (x-, x + x ),
or for all of this neighbourhood which lies within the domain, is less than
the number e.
If within (x -, x + 8) the fluctuation of the function be less than e, then
for a given e, the number 8 must have an upper limit (b (x, e), which is in
general a function of x, and is essentially positive. It must be shewn that
j (x, e), for the whole domain of x, has a finite lower limit which is not zero;
this lower limit is then a suitable value for 8. If the lower limit of Q (x, e)
be zero, there must be at least one point of the domain, such that for its
arbitrarily small neighbourhood, the lower limit of b (x, e) is zero. Suppose,
if possible, xl to be such a point; then the values of b (x, e), for a convergent
-equence of points whose limiting point is x1, must form a convergent sequence
with zero for lower limit. Since f (x) is continuous at x,, a neighbourhood
(x - S', + 8') can be found, with 8' finite, such that the fluctuation of the
function within that neighbourhood is less than e; it follows that, for any
point x of the interval (x, - 8', xL + -8'), a neighbourhood (x - 8', x + 8/')
exists within which the fluctuation of the function is less than e. This is
contrary to the hypothesis that b (x, e) becomes arbitrarily small by taking x
near enough to x,; thus the lower limit of (b (x, e) cannot be less than i-', and
is therefore finite.
It has thus been shewn * that it is unnecessary to draw a distinction, as has
sometimes been done, between functions which are uniformly, and those which
are non-uniformly, continuous in the continuous domain of x; for all continuous functions are uniformly continuous.
The theorem may also be stated in the following form:If f (x) be continuous in the interval (a, b), then, corresponding to any
arbitrarily chosen positive number e, a number X can be determined, such that
the condition I f(x1)- f (x2) I < e, is satisfied, where x 2, x2 are any two points in
(a, b), such that I lx- x2 < r.
The following theorem can be immediately deduced:If a function be continuous in a finite interval, then the interval can be
divided into a finite number of sub-intervals in every one of which the
fluctuation of the function is less than a prescribed positive number.
* This theorem was first stated and proved by Heine; see Crelle's Journal, vol. LXXI (1870),
p. 361, and vol. LXXIV (1872), p. 188.




230


Functions of a real variable


[CH. IV


It is in fact clear that, if e be the prescribed number, the condition is
satisfied when the interval is subdivided in any manner such that the length
of the greatest of the sub-intervals is < 7.
Another proof of the above theorem, in an extended form, will be given
in ~ 185, by employing the Heine-Borel theorem.
It is clear that the above proof applies also to the case in which the
domain of x is not a continuum, but is a perfect aggregate, or any closed
aggregate; because the essential point of the proof depends upon the limiting
points all belonging to the aggregate. For aggregates which are not closed
the proof does not apply; thus a function which is continuous, relatively to
an aggregate which is not closed, is not necessarily uniformly continuous.
THE LIMITS OF A FUNCTION AT A POINT.
176. Let a be a limiting point of the set of points which forms the
domain of the independent variable x; the point a may or may not itself
belong to the domain of x. Let (a, a + h) be a neighbourhood of a on the
right, and let U(h), L (h) denote the upper and lower limits of a given
function f (x) for all the points of the domain of x which are interior to the
interval (a, a + h). It will be observed that f (a), if it exists, is not reckoned
amongst the functional values of which U (h), L (h) are the upper and lower
limits.
Let a descending sequence of values be assigned to h, which converges to
zero; denoting this sequence by hA, h2, h3,..., the corresponding numbers
U(hi), U(h2)... U (hn)... form a sequence of which the members do not increase, and therefore they have in general a definite lower limit, which is
called the upper limit of f (x) at a on the right. It may happen that all the
upper limits U (h) are infinite, in which case we say that the upper limit
of f (x) at a on the right is +oo; or it may happen that the sequence
U (h), U (hA)... U (h)... has no lower limit, in which case we say that the
upper limit of f (x) at a on the right is - o. In any case, the finite or
infinite upper limit of f (x) at a on the right is denoted by f (a+ 0).
The numbers L (hz), L (h)... L (h)... form a sequence of which the
elements do not diminish, and they have in general a definite upper limit,
which is called the lower limit of f (x) at a on the right, and may, as in
the former case, have infinite values oo or - o. This limit is denoted by
f (a + 0).
Corresponding definitions apply to the left of the point a; and the limits
of f (x) at a on the left are denoted by f (a - 0), f (a - 0) respectively. In
case the point a is a limiting point of the domain of x on one side only, the
two limits of the function at a on the other side are non-existent.




175-177]


Limits of a function at a point


231


The definitions may be stated shortly as follows:The upper limit f (a + 0) of a function at a on the right is the limit of the
upper limit of f(x) in the open interval (a, a +h), as h is indefinitely
diminished.
The lower limit f (a + 0) of a function at a on the right is the limit of the
lower limit of f (x) in the open interval (a, a+ h) when h is indefinitely
diminished.
The definitions for the left of a may be stated in a precisely similar
manner.
It is to be observed that the four functional limits f (a + 0), f (a + 0),
f (a - 0), f (a - 0) are entirely independent of f (a), in case a belongs to the
domain for which f(x) is defined.  Any arbitrary alteration in the value of
f (a) will not affect these four limits of f (x) at a.
The conditions that the point a may be a point of continuity of the
function f (x) are that f(a + 0), f(a + 0), f(a- O), f(a - O), f (a) must all
have the same finite value.
It may happen that f (a), f(a + 0), f (a +0) have one and the same
finite value, but that either or both of f(a- 0), f (a- 0) may not have this
value; in that case f (x) is said to be continuous at a on the right. Continuity at a on the left is defined in a similar manner.
If the four functional limits at a be all finite and equal, but f (a) have
a different value, then the function is said to have a removable discontinuity
at the point a. In this case the function would be made continuous at a
merely by properly altering the value of f (a).
The four functional limits at the point x = 0 are usually denoted by
f (+ 0), f (+ 0), f (- 0), f ( — 0) respectively.
177. If the upper and lower limits of f (x) at a on the right have the
same value, this common value is called the limit of f (x) at a on the right,
and is denoted* by f (a + 0). If the upper and lower limits of f (x) at a on
the left have the same value, this is called the limit of f (x) at a on the left,
and is denoted by f (a - 0). Both of the limits on the right or left at a
point, when such limit exists, may be either finite or infinite.
The limit at x =0, on the right, is denoted by f(+0); and the corresponding limit on the left is denoted by f(- 0).
The limit at a point P on one side may be also defined as follows:-Let
(P1, P2, P3...) be any convergent sequence of points belonging to the domain
of x; which is such that a, or P, is its limiting point, and such that all the
* This notation was introduced by Dirichlet; see Werke, vol. I, p. 156.




232


Functions of a real variable


[CH. IV


points of the sequence are on the one side of P. The values of f (x) at
P1, P2, P3... form an aggregate which may be a convergent sequence; let us
suppose it to be so, and also that its limit has a value which is independent
of the particular sequence, which is however subject to the conditions above
stated. In that case this limit is denoted by f (a + 0), or by f (a - 0), as the
case may be, and is called the limit of f (x) at a, on the right or left.
It may be observed that the necessary and sufficient condition for the
existence of a definite finite limit on the right at a is that, corresponding to
every arbitrarily small number e, a neighbourhood (a, a + 8) can be found, such
that the difference of the values of the function at every pair of points of the
domain of x, which are in the interior of this interval, is numerically less
than e.
The necessary and sufficient condition thatf(a + 0) should exist and = + oo,
is that, if A be an arbitrarily chosen positive number, then 8 can be so
determined that at every point interior to (a, a + 8), the condition f(x) > A
is satisfied. In order that f(a + 0) may exist and =  - o, the corresponding
condition is that f(x) < - A.
It is possible that one of the limits f (a + 0), f (a - 0) may exist and not
the other. If the domain of x be either a continuum or a perfect set, a may
be taken to be at any point of the domain.
When the condition for the existence of f (a + 0) or of f (a - 0) at a point
a is not satisfied, the convergent sequence (PI, P2,... P,...), of which P (a) is
the limiting point, may be such that f (P), f (P,),...f (P,)... is either not a
convergent sequence, or else that its limit depends upon the particular choice
of the points P, P,... P,.... In this case the fluctuation of f (x) within an
arbitrarily small neighbourhood (a, a+ 8) on the one side of a is either a
finite number which has not zero for its limit when 8 is indefinitely diminished,
or else it is indefinitely great, however small s may be.
178. If x,, ~, x3,... x... be a convergent sequence of points belonging
to the domain of x, with a for its limiting point, then the sequence
f (x1), f (X2),.. f (x,)... may not be convergent; but, if it be convergent, its
limit may have (1) a single value independent of the mode in which the
convergent sequence is chosen, in which case a is either a point of continuity
of f (x), or a point of removable discontinuity of the function; or (2) one
of two values, in which case both the limits f (a + 0), f (a- 0) exist; or
(3) one of a finite, or an indefinitely great, number of values, which all lie
between the greatest and least of the four functional limits at a.
The aggregate of all possible values of the limits of the convergent
sequences f (x,), f (x2),... f (x,)..., corresponding to different convergent
sequences (xI, x2,......), with a as limiting point, is called the aggregate




177-180]


The discontinuities of functions


233


of functional limits (Werthevorrath) at the point a. The aggregate of functional limits will be shewn, in ~ 190, to be necessarily a closed set.
179. If the domain of x be unbounded in one or in both directions, it may
happen that a point xl of the domain can be found, corresponding to every
arbitrarily chosen positive number e, such that the difference between the
values of f(x), for any two values of x which are both greater than xi, is
numerically less than e. In this case the function has a definite limit as x is
increased indefinitely in its domain; and this is called the limit of f(x) for
x= o. Under a corresponding condition f (x) may have a definite limit for
X= - 00.
If, as x increases, a point xl of the domain of x, corresponding to each
assigned positive number A chosen as great as we please, can be found, such
that f (x) > A for all values of x which belong to the domain and are > x1, then
the limit of f(x) is said to be oo, as x is increased indefinitely. If f(x) < - A,
for all such values of xi, then the limit of f (x) is said to be - oo. Similar
definitions apply to the case in which x has indefinitely great values in the
negative direction.
In case the limit f (a + 0), at a point a on the right, do not exist as a
definite number, and be not infinite with a fixed sign, it is frequently
convenient to regard f(a+ O) as still existent, but indeterminate, and
capable of all values belonging to some closed set of which f(a + 0), f (a + 0)
are the extreme values. It is then said that f (a + O) is indefinite in value,
and that f(a + 0), f (a + 0) are its limits of indeterminacy.  A  similar
remark applies to f (a- 0), which may also be either definite, or indefinite,
with f(a - 0),  (a - ) as its limits of indeterminacy. One or both of the
limits of indeterminacy, in either case, may be infinite.
THE DISCONTINUITIES OF FUNCTIONS.
180. Let us suppose the domain of x to include all points in a sufficiently small neighbourhood of a point a; or, in any case, let a be a limiting
point of the domain of x.
The fluctuation of the function f (x) in the neighbourhood (a - 8, a + 8)
of the point a depends in general upon 8, but cannot increase as 8 is
diminished. It therefore has a lower limit for values of 8 which converge to
zero. This limit, which may be zero, finite, or indefinitely great, is called the
saltus (Sprung), or measure of discontinuity, of the function f(x) at a; thus:The saltus, or measure of discontinuity, of a function f (x) at a point a, is
the limit of the fluctuation of the function in a neighbourhood (a - 8, a + 8), as
8 converges to zero.




234


Functions of a real variable


[CH. IV


The upper limit of the function f(x) in the interval (a - 8, a + 8) has a
lower limit, as 8 is indefinitely diminished, which is called the maximum of the
function f(x) at a.
The lower limit of the function in the same interval, has an upper limit,
as 8 is indefinitely diminished, which is called the minimum of f(x) at a.
Either the maximum or the minimum at a point may be indefinitely great.
The saltus of f(x) at a is easily seen to be the excess of the maximum
at a over the minimum.
It is clear that the maximum of f(x) at a is the greatest of the numbers
f(a + 0), f(a - 0), f(a), and that the minimum is the least of the numbers
f(a + 0), f(a - 0), f(a); and thus that the saltus at a is the excess of the greatest
over the least of the numbers f(a + 0), f(a + 0), f(a - 0), f(a - 0), f(a).
At a point of continuity of f(x), the saltus is zero. Any point at which
the saltus has a finite value, or is indefinitely great, is called a point of
discontinuity of f(x), and in the latter case it is said to be a point of infinite
discontinuity.
If the neighbourhood (a, a + 8) on the right of a be taken, the lower limit
of the fluctuation in this neighbourhood when 8 is indefinitely diminished
is called the saltus at a on the right. This is equivalent to the excess of
the greatest over the least of the three numbers f(a + ), f(a + 0), f(a).
A corresponding definition applies to the saltus at a on the left.
181. The points of discontinuity of a function may be classified as follows:(1) If both the limits f(a + 0), f(a- 0) exist and have definite values which
differ from one another, the point a is said to be a point of discontinuity
of the first kind, or. a point of ordinary discontinuity.
The difference between the greatest and least of the three numbers
f(a+O), f(a-0); f(a) is the saltus, or measure of discontinuity, of the
function at a. If a be not a point of the domain of x, I f(a + O)-f(a - 0)
measures the saltus at a; and if a be a point of the domain, and f(a)
lies between f(a + 0) and f(a - 0), then the saltus is also measured by
l f(a + 0) -f(a - 0).
When f(a) does not lie between f(a + 0) and f(a - 0), the function is said
to have an external saltus at a.
In every case, the saltus on the right is measured by I f(a + 0) -f(a),
and that on the left by I f(a - ) -f(a) I.
Whether there be an external saltus at a or not, the number I f(a + 0)
-f(a - 0) is said to measure the oscillation (Schwingung) at a. The oscillation at a point differs from the saltus in that the functional value f(a)
at the point is in the former case disregarded.




180-182]


The discontinuities of functions


235


If f(a)=f(a- 0), whilst f(a) =f(a + 0), the function is said to be
ordinarily discontinuous at a on the right.  If f (a) f (a - 0), whilst
f(a)=f(a+O0), the function is said to have an ordinary discontinuity at a
on the left.
It may happen that f(a + 0),f(a- 0) have equal values which differ from
f(a). In that case the discontinuity at a is said to be removable; since by
merely altering the functional value at the one point a, the function can be
made continuous at the point.
(2) If neither of the limits f(a+ 0), f(a- 0) exists, the discontinuity
at a is said to be of the second kind.
The oscillation* at a is measured by the excess of the greater of the
numbers f(a + 0), f(a-O ) over the lesser of the two numbers f(a + 0),
f (a- 0), the value off(a) being left out of account.
The differences f(a + 0) -f(a + 0), f(a - 0) -f(a - 0) may be spoken of
as the oscillation at a on the right, and on the left, respectively.
By Dinit, a definition of the saltus is adopted which differs from the one
which we have employed; he takes the greatest of the four differences
If(a+ 0) -f(a)  as the measure of the saltus, the greater of the two
differences If(a+O)-f(a) I being taken as the measure of the saltus
on the right.
(3) It may happen that one of the two limits f(a + 0), f(a - O) exists
as a definite number, whilst the other does not. In this case the point a may
be said to be a point of mixed discontinuity.
If f(a) exist and be equal to that one of the two limitsf(a + O),f(a- 0)
which exists, then the function is continuous at a on.one side, and has a
discontinuity of the second kind on the other side.
(4) If one or more of the four limits f(a + 0) be indefinitely great, the
point a is one of infinite discontinuity.
Under infinite discontinuities is sometimes included the case in which
f(a) is defined by l/f(a) =0, or when f(x) is defined as the limiting sum
of a series which, for the value a, becomes divergent.
182. In an arbitrarily small neighbourhood (a, a + h), on the right of a
point a at which the limits f(a + 0), f(a +0) have different values, there
must be an infinite number of points at which f(x) >f(a + 0) - e, where
e is an arbitrarily small fixed number.
* This definition of the " Schwingung" is given by Pasch in his Einleitung in die Differentialund Integralrechnung, p. 139.
$ See Grundlagen, p. 55.




236


Functions of a real variable


[CH, IV


For if there were only a finite number of such points in (a, a + h), h could
be chosen so small that all such points would be excluded from the neighbourhood; thus, in a sufficiently small neighbourhood (a,a + h), we should have
at every internal point f(x) <f(a + 0)- c; and thus the upper limit at a
on the right could not be f(a + 0). In a similar manner it can be shewn
that, in the arbitrarily small neighbourhood (a, a + ), there must be an
infinite number of points at which f(x) <f(a + 0) + e.
In this case we say that, in the arbitrarily small neighbourhood of a on
the right, the function makes an infinite number of finite oscillations. In
case of the infinity of one or of both of the limits f(a + 0) and f(a + 0), and
in the latter case if they be of opposite signs, the function makes an
infinite number of infinite oscillations in the arbitrarily small neighbourhood
of a. A similar remark applies to the case in which f(a - 0), f(a - 0) have
unequal values. It has thus been shewn that:A point of discontinuity of the second kind is one such that, in its arbitrarily small neighbourhood, the function makes an infinite number of finite or
infinite oscillations.
In an arbitrarily small neighbourhood on either side of a point of
discontinuity of the first kind, the function may make an infinite number
of oscillations; but since the neighbourhood can be chosen so small that
the fluctuation of the function in its interior is arbitrarily small, the
oscillations, when they are infinite in number, are arbitrarily small sufficiently
near the point.
EXAMPLES.
1. Let f(x)=sin x/x, when xIO, and f(x)=A, when x=O.
In this case f(+0)=f ( -0)= 1, f(0)= A; thus f(x) has a removable discontinuity at
x=0, unless A = 1, in which case the function is continuous in any interval.
2. Let f(x)= =-; we have thenf(a+0)=oo, f(a- 0)= -oo, andf(a) is undefined.
3. Let f(x)=(x-a)sin ---; then f(a+0)=0, f(a-0)=0. This function is continuous at x=a, and makes an infinite number of oscillations in any neighbourhood of
that point.
1        1
4. If f() = -  cosec —, then
J   x-a  x-a
f(a+0)=oo, f(a+O)=-oo, f(a-O)=oo, f(a-O)=-oo.
This function has an infinite discontinuity of the second kind at the point a.
1                                       1
5. If f(x)=ex, we have f(+0)=oo, f(-0)=0. If f(x)=-,, then f(+0)=0,
1 - ex
f(-)-=l.




182, 183]


Semi-continuous functions


237


6. If f(x) = sin x, lim f(x) is indeterminate, the limits of indeterminacy being +1, - 1.
X=In the case f(x)= x sin x, the corresponding limits of indeterminacy are + oo, - o.
7. Let y=E(x), where E(x) denotes the integral part of x. This function is discontinuous when x has an integral value n; we then have E(n-0) n -1, E(n) =n,
E(n+O)=n.
8. Let (x) denote the positive or negative excess of x over the nearest integer; and
when x exceeds an integer by ~, let (s)= 0. This function is continuous except for values
x=n+, where n is an integer. We have (n+)=0, (n+~-0)=, (n+2I+0)= -.
SEMI-CONTINUOUS FUNCTIONS.
183. If (b (x) be a function defined for a continuous domain, and if,
corresponding to every arbitrarily chosen positive number e, a neighbourhood
(x-h, x+h) of a particular point.(x) can be determined such that for
every point x' in this neighbourhood the condition, 0 (') <  (x)+ e, be
satisfied; then the point x is said to be a point* of opper semi-continuity of
the function q (x).
If a neighbourhood of the point x can be determined, for each e, such
that b (x') > 0b (x) - e, then the point x is said to be a point of lower semicontinuity of the function + (x).
That a point x may be a point of continuity of the function b (x), it is
necessary that both the above conditions be satisfied.
If every point of the domain (a, b), for which the function b (x) is defined,
is a point of upper semi-continuity, then the function b (x) is said to be an
upper semi-continuous function.
A similar definition applies to a lower semi-continuous function.
It is clear that, if f (x) be a lower semi-continuous function, then -  (x)
is an upper semi-continuous function. Thus the properties of the one class
of functions may easily be extended to the other class.
If f(x) be a function defined for the interval (a, b), and if ( (x), r(cx)
denote the maximum and the minimum of f(x) at the point x, then b (x) is an
upper semi-continuous function, and fr (cx) is a lower semi-continuous function.
For a neighbourhood (x - h, x + h) of any point x can be determined, such
that the maximum (see ~ 180) off(x) for every point in this neighbourhood is
less than k (x) + e, where e is a prescribed positive number. At every point
in (x- h, x + hi) where h, is chosen < h, the value of the function b is less
than b (x) +. Since this holds for every value of e, the function  (x) is
upper semi-continuous at x.


* See Baire's memoir "Sur les fonctions des variables reelles," Annali di mat. series III",
vol. in, 1899.




238


Functions of a real variable


[CH. IV


It is clear that the function f (x), where f (x) denotes the minimum of
f(x) at the point x, is a lower semi-continuous function.
The saltus ( (x) - /r (x) of the function f(x), may be taken to be the value
of a function co (x) which is called the saltus-function off(x).
The saltus-function wo (x) of any function f(x) is an upper semi-continuous
function.
For ( (x), -q(x) are both upper semi-continuous functions, and it is
easily seen that the sum of two such functions belongs to the same class.
If ( (x) be any upper semi-continuous function, then the set of points, for
which (p (x) _ a, is a closed set, where a is any fixed number.
For let P1, P2, P3,... P,... be a sequence of points at each of which the
condition ( (x)  a, is satisfied, and let P be the limiting point of the
sequence. Let us suppose that, if possible, ( (P) < a; then a neighbourhood
of P can be determined, such that at every point in it the value of p (x) is
less than a, and hence this neighbourhood cannot contain any of the points of
the sequence; but this is contrary to the hypothesis that P is the limiting point
of the sequence. It follows that (b (P) > a, and thus that the set of points, for
which ( (x) _ a, contains all its limiting points, and is therefore a closed set.
It can be shewn, in a similar manner, or it can be deduced from the above
theorem, that the set of points for which rf (x) < a, is a closed set; where 4 (x) is
any lower semi-continuous function.
If we apply the theorem proved above to the saltus-function o (x), of any
function f(x), we obtain the following theorem:Having given any function f(x) defined for a continuous domain, the
saltus-function o (x) is such that the set of points for which co (x) > a, forms
a closed set.
184. If ( (x) be an upper semi-continuous function, and if at every point
of the domain of x the minimumr of ( (x) be zero, then there exists a set of
points, everywhere-dense in the domain of x, at which ( (x) is itself zero.
For, in any interval (a,,/), the minimum of () (x) is zero, and therefore
a point P in the interior of (a, /3) can be found at which ( (P) < ~ e, where e
is a prescribed positive number. Since ( (x) is an upper semi-continuous
function, an interval (a1, /8i) interior to (a, /3), and containing P in its
interior, can be determined, such that for every point x in it, ( (x) < ( (P) + ~ e < e.
Similarly, it can be shewn that an interval (a2, /3) interior to (al, /8)
can be found, such that at every point in (a2, /2) the condition p (x) <  e,
is satisfied. Proceeding in this manner, we can determine a set of intervals
(a1, /i), (a2, /2)... (an, /n)..., each of which is interior to the preceding one,
and such that at every point x in (an, /n), the condition  (x)<2 -e, is
2nI




183-185]


Semi-continuous functions


239


satisfied. This sequence of intervals, continued indefinitely, determines a
1
point Q, in the interior of all of them, such that ( (Q) < 2-_ e, for every value
of n; and hence  (Q) = 0. It has thus been shewn that, in any interval
whatever contained in the domain of the variable, there exists a point at
which q (x) is zero; and therefore the set of all such points is everywheredense in the domain of the variable.
In particular, we see that if w (x) be the saltus-function of any given
function f (x), and if w (x) has its minimum equal to zero at every point of the
domain of x, then o (x) vanishes at an everywhere-dense set of points. The
points of this set are the points of continuity of f(x).
185. The following theorem is a generalization* of the theorem of ~ 175,
that a continuous function is uniformly continuous in its (closed) domain.
If a function f(x) be defined for the interval (a, b), and if k be a number
greater than the maximum of the saltus-function o (x) in (a, b), then there
exists a number a, such that within every interval in (a, b) of length not
exceeding a, the fluctuation of f (x) is < k.
The theorem of ~ 175 is the particular case which arises when the
maximum of o(x) is zero.
The theorem is most easily established by means of an application of the
Heine-Borel theorem given in ~ 68.
For any point P in (a, b) a neighbourhood can be determined, such that
the fluctuation off(x) within this neighbourhood is < k. If we conceive such a
neighbourhood to be determined for each point in (a, b), then a finite number
of these intervals can be chosen, such that every point in (a, b) is interior to one
at least of the intervals. The end-points of this finite set of intervals form
a finite set of points in (a, b); let then a be the smallest of the distances
between consecutive points of this finite set. Any interval whatever in (a, b) of
length not exceeding a is within one of the intervals of the finite set; hence
within such an interval, the fluctuation off(x) is less than k.
The definitions given above, and the theorems established, are applicable
when the domain of the variable x is not an interval (a, b), but any closed
set of points. In that case, we regard functional values in any interval as
only existing at those points in the interval which belong to the domain of x.
Moreover, the definitions, and theorems are applicable to the case of
functions of a number n of variables. In this case, instead of an interval
(x - h, x + h) used in defining semi-continuous functions and the saltusfunction, the "sphere"


* See Baire, loc. cit. p. 15.




240


Functions of a real variable


[CCH. IV


may be employed. Corresponding to the interval (x -, x + h), we take the
set of points (|f,... * ), for which
(ti- X )2 + (f2 - X2 + *. + (n- Xn)2-< h2.
EXAMPLE.
If*f(x) be any function, / (x) its maximum, and + (x) its minimum at the point x,
and g (x) be any continuous function, then 0 (s) +g (), 4 (x)+g (x) are the maxima and
minima at x, of f(x)+g (x). Also the functions q (x) -f(x), f(x) - + (x) have, each in
any domain, the minimum zero.
THE CLASSIFICATION OF DISCONTINUOUS FUNCTIONS.
186. Let us suppose a function to be defined for all points in a continuous interval (a, b); at each point x the saltus of the function has a finite
value, or is indefinitely great, its value being zero at a point of continuity.
With a view to the classification of functions, in accordance with the distribution of the points of continuity and of discontinuity in the interval (a, b),
the question arises, what is the most general distribution of the points of
continuity?
The answer to this question is contained in the theorem:The points of continuity of a function, defined for a continuous interval,
form an inner limiting set.
To prove this theorem, let e be a fixed positive number, and enclose each
point P of continuity of a function f(x) in an interval so chosen that the
fluctuation of f(x) therein is less than e; all the points of continuity are then
enclosed in a set of intervals which in general overlap. Imagine these sets
of intervals constructed corresponding to a sequence of diminishing values
of e which converges to zero; there exists then a set of points which are interior to intervals of all these sets of intervals, since this set of points includes
all the points of continuity off(x). If Q be any point which belongs to the
inner limiting set so defined, Q must be a point of continuity of f(x); for
corresponding to any arbitrarily small number e,, Q is in the interior of some
interval in which the fluctuation of the function is less than e~, and thus Q
is a point of continuity of the function.
In accordance with the theorems which have been obtained in ~ 97,
relating to inner limiting sets, the points of continuity of a function may form
an enumerable set which contains no component dense-in-itself, or else they
form a set of the cardinal number of the continuum. In the latter case the
set is of the second category, provided it be everywhere-dense.
* Baire, loc. cit. p. 9.




185-187]


Discontinuous functiobns


241


These results lead to the following classification * of functions:(1) A function may have no points of continuity, it is then said to be
totally discontinuous.
(2) The points of continuity may form an enumerable set which has no
component dense-in-itself.
(3) The set of points of continuity may be of the cardinal number of the
continuum, and
(a) non-dense;
(b) everywhere-dense and unclosed, in which case the function is
said to be a point-wise discontinuous function;
(c) everywhere-dense and closed, in which case the function is continuous;
(d) everywhere-dense in each interval of a set, and non-dense in each
interval of another set external to the former one.
This last case (d) is not essentially distinct from the former ones.
By Hankel and others the term "totally discontinuous" has been applied
to all functions which are neither continuous nor point-wise discontinuous.
187. It has been shewn by W. H. Young that a function can be constructed which is continuous at every point of any given inner limiting set
of points, and is discontinuous at every other point of the interval.
Let E denote an inner limiting set, and let the function f(x) be defined
as follows:(1) At every point x of E, let f(x) = x.
(2) It has been shewn that a sequence of sets of non-overlapping
intervals can be constructed such that the only points each of which is in
an interval of every set are the points of E. Let Q be a limiting point of E
which does not belong to E; then a number n exists such that Q is in an
interval of the n - tlh set, but not in one of the intervals of the nth set. Let
this interval of the n -  hll set be of length dQ; and at the point Q let
f(x) = XQ+e eldQ, where e is a fixed positive number less than unity; in
the case n=l, we put dQ=e.
(3) If R be a point which does not belong either to E or to its derivative,
it must lie between two definite points A, B both of which belong to E or to
E', and such that no point of E or of E' lies between A and B. If x, be a
rational number, let f(x)= x  or XB, according as R is nearer to A or
to B; when x_ is irrational, let /(x,)= x,; and if R be the middle point of
AB, let f(x) = x-.
* See a paper by W. H. Young, "Ueber die Eintheilung der unstetigen Funktionen und
die Vertheilung ihrer Stetigkeitspunkte." WIiener Sitzurgsberichte, vol. cxI. Abt. I a, 1903.


II.


16




242


Functions of a real variable


[c-I. IV


It is clear that the function so defined is discontinuous at every internal
point of the interval AB, and at the end-point A it is continuous or discontinuous on the right, according as A does or does not belong to E;
a similar result holds for B. It has thus been shewn that the function is
discontinuous at every point which does not belong to E.
To shew that at any point P of E, the function is continuous, consider
those intervals, one of each set in the sequence, which contain the point P:
the lengths of these intervals will have a lower limit d which may be zero.
In every interior point of d, we have f(x)=x; and thus, if P be interior
to d, P is a point of continuity of the function. If P be an end-point of d, it
is certainly continuous on the side towards the interval; and we have to
shew that it is also continuous on the other side. Choose an arbitrarily small
number ar, and an arbitrarily large integer nm; then a number n1 > ', can
be found such that the nth, and all subsequent intervals of the sequence
which contain P, are of length between d and d + or. The piece of one of these
intervals which is not a portion of d is of length < a; and suppose that Q is a
point in this piece which belongs to E' but not to E: then
I f(xr)-f(ax) I = I xp - Q-e'dQ I < I a + em (d + a) 
since n _??mn > m, and dQ < d + a. From this, there follows
I f(xC)-f(xQ) I < 2a + emd < 3o-,
if in be chosen sufficiently great. If R be an interior point of an interval
AB which contains in its interior no point of E' or of E, and if the points
A, B be both so near to P that their distances from it are less than a, we
have I f(xp) -f(xR) I < a, in virtue of the definition of f(x,); also if only
one of the ends A, B be within the interval of length < d + o-, which has been
chosen, then an interval further on in the sequence can always be found
such that the middle point of AB is exterior to it, and thus the inequality
f(xp) -f(xR) I <a-, holds as in the former case. It has now been shewn
that, for any arbitrarily chosen a, a neighbourhood of P can be found such
that for all points x in it, If(x,) -f(x) I < 3o; therefore P is a point of
continuity of the function. The case in which d = 0, does not require separate
treatment.
EXAMPLES.
1.* Let G denote a non-dense perfect set of points in the segment (0, 1), such that
the end-points of the complementary intervals are rational points. Let f(x) be defined
thus:-at every irrational point inside an interval complementary to G, let f(x)=x; at
every rational point of such interval, let f(x) be equal to the value of x at the middle
point of the interval; and at every point external to a complementary interval, let f(x) = 1.
This function is discontinuous except at the middle points of the intervals complementary


* See W. H. Young, loc. cit.




187, 188]   Point-wise discontinuous functions


243


to G; thus the set of points of continuity is an enumerable set which contains no comlponent dense-in-itself.
2.* With the same non-dense perfect set as in Ex. 1, let AB be a complementary
interval of G, and ilt its middle point. At every rational point of AM except i, let
f(x)-= w,  and at every rational point of iMB except M, let f(x) =xB; also at all points of
(0, 1), except those for which the functional value has been already specified, let f(x)= x.
In this case the points of continuity are non-dense and of the power of the continuum.
POINT-WISE DISCONTINUOUS FUNCTIONS.
188. A function being defined for the continuous domain (a, b), it can
be shewn that, if k be any fixed positive number, those points, at which the
saltus of the function is    s k, jbrm a closed set.
This theorem  follows immediately from the property of semi-continuous
functions established in ~ 183, by considering the saltus-function. It may be
proved directly as follows:If P be a limiting point of the set for which the saltus is  Ic, then in any
arbitrarily small neighbourhood of P there are points of the set; hence the
fluctuation of the function in this neighbourhood is > k, and therefore the
saltus at P is > k.
Moreover, such a limiting point P, of the set of points at which the saltus
is _> k, must be a point of discontinuity of the second kind, at least on one
side of P. If at P the function have a limit on the right, a neighbourhood
PQ can be found such that the inner fluctuation in PQ is <k; hence inside PQ
there can be no point at which the saltus is - k; and therefore P is not
a limiting point on the right, of the set for which the saltus is _ k. A similar
remark applies to the left of P.
We have already defined a point-wise discontinuous function as one of
which the points of continuity are everywhere-dense and unclosed in the
domain of the function; this definition is that given by Dinit, and is
equivalent to the following definition given by Hankel:A point-wise discontinuous function is one for which those points at which
the saltus is > k, ant arbitrarily chosen positive number, form a non-dense
set K, whatever value k may have.
That this set is closed has been shewn above.
To prove the equivalence of the two definitions, let it be assumed that
in any arbitrarily chosen sub-interval (a, /3), a point of continuity x, can be
* See W. H. Young, loc. cit.
t See Grundlagen, p. 81.
+ Math. Annalen, vol. xx (1882), p. 90. This is a reproduction of Hankel's Univ. Programlle,
Tiibingen, 1870, entitled " Untersuchungen fiber die unendlich oft oscillierenden und unstetigen
Functionen."
16-2




244


iFunctions of a real variable


[CH. IV


found. A neighbourhood can be found for x, internal to (a, /), in which the
fluctuation of the function is < k, and this neighbourhood can contain no
point at which the saltus is _ k; hence the points at which the saltus is _ Ic
form a non-dense set K, since, interior to any sub-interval, a sub-interval can
be found which contains no point of the set K.
Conversely, choose a descending sequence of values of k, say k1, I2, k,...
which converges to zero, and let K1, K2, Ks,... be the corresponding nondense closed sets, each of which necessarily contains the preceding one; then
the set M(K1, K2, K3,...) is the set of all the discontinuities of the function.
In accordance with ~ 89, this set is of the first category, and the complementary set, which is the set of points of continuity of the function, is everywhere-dense, and has the cardinal number of the continuum, being a set of
the second category.
It will be observed that the set of all the points of discontinuity may be
either everywhere-dense, or non-dense, in the whole or part of the domain of
the variable. This set may be finite, enumerably infinite, or of the power
of the continuum.
The set K, although non-dense, is not necessarily of content zero. By
Harnack*, the term point-wise discontinuous function was only used for such
functions as possess the property that the set K, for each value of k, has
content zero. It will be seen that this latter case is of special importance in
connection with the theory of integration.
It has been already shewn in ~ 186, that the points of continuity of the
point-wise discontinuous function form an inner limiting set; and if
{1,  t{2}... {8...
be the sets of intervals complementary to the closed sets Ki, K2,... Kn,...,
they form a sequence of sets of non-overlapping intervals which define the
set of points of continuity as their inner limiting set.
The whole theory of point-wise discontinuous functions is applicable to
the case in which the domain of the variable is not a continuum, but is any
perfect set. In this case also, the points of continuity of a point-wise
discontinuous function are everywhere-dense relatively to the perfect domain,
and the points at which the measure of discontinuity is _ I, form a closed
set, non-dense relatively to the domain of the variable. That this is the case
may be shewn by making the points of the perfect set correspond in order
to the points of a continuous interval, as explained in ~ 75. The points
of discontinuity, and those of continuity, relatively to the perfect domain, are
sets of the first and the second category respectively, relative to that domain.


* Math. Annalen, vol. xix, 1882, p. 242, and vol. xxiv, 1884, p. 218.




188, 189]   Point-wise discontinuous functions


189. The domain of the variable being either a continuous interval or
any perfect set, let us suppose that at every point the oscillation (see ~ 181)
of the function both on the right and on the left is < k; there can then
only be a finite number of points at which the saltus is > k; i.e. the set
K  is finite.
For if K  were not finite, it must contain a limiting point P, which
has been shewn in ~ 188 to be a point of discontinuity of the second kind.
Any arbitrarily small neighbourhood of P, on one side at least, must therefore contain points at which the saltus is > k, and hence the oscillation at P
on this side could not be < k.
The domain can therefore be divided into a finite number of parts within
each of which there is no point at which the saltus is _ k; and it follows
that the domain can be divided into a finite number of parts, within each of
which the fluctuation of the function is < k.
If, at each point of a set which is everywhere-dense in the domain of the
variable, there exist a limit of the function on one side at least, then the
finction, is either point-wise discontinuous, or else it is continuous.
In any interval, containing points of the domain, a point can be found
which has a neighbourhood on one side at least in which the inner fluctuation
is < k; within such neighbourhood the saltus is everywhere < k; hence the
points of K are non-dense in the domain, and thus the function is either
point-wise discontinuous, or else it is continuous.
A particular case of this theorem is that a function defined for a continuous
interval, and having ordinary discontinuities only, is point-wise discontinuous.
Among such functions, the monotone functions form an important class.
A monotone function is one such that for every pair of values x,, x, of the
variable, such that x.,> x,, the condition f(x) - f(x,) is satisfied; or else, for
every such pair, the condition f(x,) < f(x,) is satisfied. Since there can be
no oscillations in the neighbourhood of any point, every discontinuity must
be an ordinary one. It follows that every monotone function is either pointwise discontinuous, or else continuous.
If the function be defined for a continuous interval, and all the points of
discontinuity be ordinary ones at least on one side, then the set K of points, at
which the saltus is _ k, is a set of content zero.
The set K can be resolved into a perfect set G and an enumerable set;
the set G contains points which are limiting points on both sides, and at such
a point the oscillation both on the right and on the left must be _ k; it
follows that the set G is non-existent, and that K is therefore an enumerable
closed set, which has necessarily content zero.
The theorem still holds if there be points of discontinuity of the second
kind which form a set of content zero, for these points may be enclosed in a




246


2Functions of a real variable


[CH. IV


finite number of intervals whose sum is arbitrarily small; the theorem can
then be applied to each of the remaining intervals of the domain.
190. At a point P at which a given point-wise discontinuous function is
discontinuous, let xI, x2, X3,... xA,... be any sequence of points converging to P,
and let us suppose that f(xi), f(x),... f(xl,),..., the set of values of the given
function at the points {Ix}, converges to a limit U; the value of U will depend
upon the choice of the particular sequence, subject to the condition of
convergence of {f(xn,)}. If P be a point of discontinuity of the first kind,
U is capable of having two values only, viz. f(x + 0) and f(x - 0); but if the
discontinuity of the function at P be of the second kind, U may have any one
of the values f(x + 0), f(x + 0), f(x - O), f(x - O), or it may possibly have
other values lying between the greatest and least of these four. The possible
nature of the set of all values of U at the point x will now be investigated.
This set is the aggregate of functional limits at x (see ~ 178).
In the first place* the set of all values of U at a point P is a closed set.
For let U1, US, U3,... U,,... be a convergent sequence of values of U, of
which U, is the limit; it will be shewn that U. itself belongs to the set of
values of U, and thus that this set is closed.
Let U,. be the limit of the convergent sequence
f   (x (r)),   f   (   (r) ),...f   ( (r)...;
we may choose r so great that I U, - U,. < fe, for this and all greater values
of r. We can then choose n such that U.-f(xn(') ')  < be; it follows that,
for sufficiently great values of r and n,
I U.s-f /  (.re)) I<,
and as the positive number e is arbitrarily small, the theorem is established;
in fact a sequence {f(x,,('))} can be found which converges to U,, whilst
{x,(T)} converges to x.
A point-wise discontinuous function* can be constructed so that, at a
point xo, the set of values of U may be any prescribed closed set G.
To establish this theorem, we observe that if G be unenumerable it may
be replaced by an enumerable set G, everywhere-dense in G. Let
VI, V,, V3,...
denote the set G,, arranged in the order-type w.
Next, choose an enumerable sequence x,, Cx,... x,,... of values of x having
the single limiting point x0; and arrange the sequence [xn} in the ordertype W2. The sequence {xn} may thus be split up into an enumerable set
of sequences {x,.}, {xr}, *... {xsr},... each of which has the limit x0. The
* These theorems were given by Bettazzi, see Rendiconti di Palermo, vol. vi, p. 173.




189, 190]         Point-wise discontinuous functions                        247
function f(x) may be defined by the specifications, that f(x)=0, for all
values of x which do not belong to the sets {xl.}, {x},.},..; and that
f(x,.)=v,,., where v,1, v2,... Vr,...
is a sequence chosen so as to converge to the limit V,. The function f(x) so
defined is continuous at every point except xo, xi,... x,..., and it has the
required property; since the points of G,, and therefore of G, are all values
of U at the point x0.
EXAMPLES.
1.* If f(x), p (x) be two point-wise discontinuous functions defined for the same
interval, there is an everywhere-dense set of points at each of which both functions are
continuous. This theorem follows at once from the fact that the points common to two
sets of the second category also form a set of the second category, and that this holds for
every sub-interval contained in the given interval.
2. Let (x) denote the positive or negative excess of x above the integer nearest to it,
and if x be half-way between two successive integers, let (w)= 0. Let a function t f(x) be
defined for the interval (0, 1) as the limit of
(x)  (2x)  (3x)     (nx)
1 +  4 + 9            2
when n is indefinitely increased. The function f(x) is a point-wise discontinuous function,
in which the set K of points, at which the saltus is > k, is finite for each positive value of k.
It can be proved that, if x=m/2n, where m and 2n are relative primes, then
IIIn  )    (n)-\ 7r2       f (fm    f7r2
f/\2n o =/          lc~'    / 16 -o I =/I / +   16n —
For values of x not of the above form, f(x) is continuous. The number of points of K is
the number of irreducible proper fractions having even denominators 2n, such that
7r2/8n2 > k. The set of all the points of discontinuity is everywhere-dense in the interval
(0, 1).
3. Let: y-=c, for all rational values of x; and y= d, for all irrational values of x.
This function is totally discontinuous.
4. Let~ f ()= 1, for all values of x in the interval (0, 1), except x=, (n =1, 2, 3,...),
for which f(x) =0. At each of the points (1) there is a saltus equal to unity. This
function is point-wise discontinuous, and the content of K is zero, for every value of k.
5. In~ the interval (-l, 1) of a, let f(x)=l; in the interval (2, ), let f(V)=and in general, in the interval (2^+, -i, let f(x)= n. In this case the point-wise
discontinuous function f(x) is such that the number of points at which the saltus is ` k,
is finite for every value of k > 0.
* See Volterra, Giornale di Mat. vol. xix, 1881.
t See Riemann's Ges. IVerke, p. 242.
+ Dirichlet's WVerke, p. 132.
~ Hankel, Math. Annalen, vol. xx.




248


Functions of a real variable


[CH. IV


6. The* points of a continuous interval (0, 1) may be put into correspondence with
the points of a non-dense set of points, dense-in-itself, contained in an interval (a, b), in
such a manner that the relative order of two points of the interval (0, 1) is the same as
that of the corresponding points in (a, b). Such a correspondence is defined by a pointwise discontinuous monotone function y =f(x.).
7.t  Let the numbers of the interval (0, 1) be expressed as finite or infinite decimals
x= ala2a3... an,... and let f(x)= (-6) + (1- 0) +.... The function f(x) is monotone, and
is discontinuous for every value of x represented by a finite decimal. The set of points K
for a given value of k is finite. The function f(.x) defined by f(x)= 0al Oa2Oa3... has
similar properties.
8.t Let the points of the interval (0, 1) be represented by decimals, and consider the
set Go of those points for which only the digits 0 and 1 occur in the decimal representation,
excluding those points for which all the figures are 0, from and after some fixed place.
The set Go is non-dense in the interval (0, 1), and has the cardinal number c. Any point
x0 of Go is represented by 'a a2a... a,,..., where a, is 0 or 1. Let $ be a fixed point
blb2... b... of Go, and let xt denote xo+2' *Cic2....,...; so that cn=an,+2b,. With
$ fixed, let the set of all points x, be denoted by Ga; the points of Ga are all different from
those of Go, and for two values $, $' of $, the sets Gt, Gt, have no point in common. For
two numbers clc2... c,..., cl'c2'... c,,... are identical only when c,=c,,', which holds only
when an= a,,', and b= bn'. If we read off in the dyad scale the decimal representation of $,
we obtain, by giving $ all the values in Go, every point in (0, 1) except the point 0, and
these once only; let the point which, by thus using the dyad scale, corresponds to $ be
denoted by (t). Now let f(x) be defined by the rules f(x<)=(t), f(xo)=0, and f(x)=0 for
all other values of x. The point-wise discontinuous function f(x) so defined for the interval
(0, 1) is such that at all the points of the unenumerable set G, the saltus is ($); the set of
all the points of discontinuity is non-dense in (0, 1); and f (x) is constant, and =0, in an
everywhere-dense set of linear intervals.
9.4 Let the points x of the interval (0, 1) be expressed as radix-fractions in the scale
of 3. Let Go be the set of points for which all the figures of the radix-fraction are 0 and 1,
excepting those points for which all -the figures are 0 after some fixed place. Let Gn
consist of all the points which contain the digit 2 in at most the first n places, but are also
such that the nth figure is 2; then Ga is non-dense, and of cardinal number c. There are
left only those points for which the radix-fractions contain the digit 2 an infinite number of
times; and these points belong to a set H for which the radix-fractions contain an infinite
number of digits other than 2, or to a set G for each point of which every digit is 2, from
and after some fixed one. Each point of G can be represented by a terminating radixfraction which contains only a finite number of 2's, and can be added to a Gn. Let
Go, G1, G2,..., when so increased, become Go, G1, G2...; and take a sequence of decreasing
numbers go, gl, g2,.... Let the function f(x) be defined by the rules f(x)=n,,, if x is a
point of G,,, and f(x)= 0 for all points of H. The point-wise discontinuous function f (x)
is continuous at all the points of H, and the points of discontinuity are everywhere-dense
in (0, 1), and of cardinal number c.


* Harnack, Math. Annalen, vol. xxIII.              + Peano, Riv. di Mat. vol. I.
+ Schonflies, Gittinger Nachrichten, 1899.




190, 191]


Point-wise discontinuous functions


249


DEFINITION OF POINT-WISE DISCONTINUOUS FUNCTIONS BY EXTENSION.
191. Let us suppose a function f(x) to be defined for a domain which
consists of a set of points which is dense-in-itself but not closed, and further
let us assume that f(x) is continuous in this domain. The new domain
obtained by adding to the original domain those of its limiting points which
do not belong to it may be spoken of as the extended domain. It has been
pointed out in ~ 190 that, at a point a of the extended domain, which does not
belong to the original domain, there is an aggregate of functional limits
which is certainly a closed set, and may consist of a finite, or an infinite, set
of numbers.
Let us now define a function (S(x), for the extended domain, in the
following manner:-At each point of the original domain, which may be
called a primary point, let b (x) =f(x); at each point a, which may be called
a secondary point, and which does not belong to the original domain, attribute
to  (x) the values contained in the aggregate of functional limits off(x)
at a; this function b (x) may then be multiple-valued at any secondary
point.  The new function b (x) defined for the extended domain may be
spoken of as the function obtained by extension of f(x); and those points for
which b (x) is multiple-valued are regarded as points of discontinuity at
which the measure of discontinuity is the excess of the greatest over the
least value of the function at the point.
It will be shewn that the extended function Ob(x) is point-wise discontinuous
in the extended domain, unless it be continuous. This gives rise to a method of
constructing point-wise discontinuous functions which has been employed by
Broden in various special cases. Since we may so choose the original domain
that it shall consist of an enumerable set of points, the method includes
one for the construction of a point-wise discontinuous function from an
enumerable set of specifications.
To prove that the extended function b(x) is at most point-wise discontinuous, it is sufficient to shew that <b (x) is continuous at all points of the
original domain G, which is a set that is everywhere-dense in the extended
domain G'.
Let x be a point of G, and let it be the limiting point of a convergent
sequence (x,', x,', 3',...) of which all the points belong to G'. Consider the
aggregate {1(x '), q(x2 ),...}, where +b(x/), qb(x2'),... have any of the values which
belong to the points x,', x2',,....  Now a point x, of G can be found such
that x' - x, I < ql, and I q (xn) -f(x() | <,e, where r, en are independent
arbitrarily small numbers. If we take a sequence of values of j, such that
i > v2 > 3,..., with zero as its limit, and also a similar sequence of the
e numbers, then the sequence (x,, x2, 3,...) has the same limit x as the
sequence (xi', x',,...), and the aggregate {1 (x1'), b (x2'),...} has the same




250


2.OFunctions of a real variable


[CH. IV


limit as the convergent aggregate {f(xi), f(x),...} viz. f(x) or C (x); and
thus the theorem is established.
It will be observed that the values of b(x) at all the secondary points in
an arbitrarily small neighbourhood of a secondary point a depend only on
the values of f(x) in that same neighbourhood; it follows therefore that a is
a point of continuity or of discontinuity of b (x) according as the aggregate
of functional limits of f(x) at a consists of one number or of more. In
the latter case the measure of discontinuity of b (x) at a is the excess of
the greatest over the least of the numbers belonging to the values of b (x) at
the point.
It can be shewn that a point-wise discontinuous function can be so constructed that, at a given secondary point, the values of the function may be
an arbitrarily assigned closed set.
192. Although a class of point-wise discontinuous functions may be obtained by extension of a continuous function defined for a primary domain,
dense-in-itself but unclosed, yet not every point-wise discontinuous function
can be generated in this manner.
Let f(x) be a point-wise discontinuous function in a domain which is
either a continuum or a perfect set of points.
Consider the function +b(x) obtained by taking the values off(x) as given
only at its points of continuity, and extending this function to the complete
domain, in the manner explained above.
At each point of discontinuity of f(x) there is a saltus Icy, and at that
point the function h (x), obtained by extending the set of values of f(x) at
its point of continuity, has a measure of discontinuity kc,, which will be zero
in case f (x) be continuous at the point; but in any case the condition kc 4  kf
is satisfied, since, within any neighbourhood of the point, the fluctuation of
( (x) cannot be greater than that of f(x).
If k, = 0 at any point of discontinuity of f(x), that point may be said to
be a point of unessential discontinuity of the function f(x); and if k4 > 0,
the point is one of essential discontinuity.
Let now a function X (x) be defined for the whole domain as follows:At every point of continuity off(x), and at every point of discontinuity
at which kf = ki, let x (x) = O; at each point at which yf > kA, let
x () = kf - 1o.
The function X (x) is not necessarily continuous at every point at which it
is zero. At a point xa at which  (x) is continuous, the measure of discontinuity of X (x) is lof, or X (xi); but this is not necessarily the case if b (x)
be not continuous at xi. This function X(x) may be called a point-wise
discontinuous null-function.




191, 192]


Point-wise discontinuous functions


251


By subtracting from f(x) a function -, (x) which never exceeds, at any
point x, in absolute value, the value of    (x), we obtain a function 0 (x) of
which the measure of discontinuity is everywhere = klc (x).
The function * k, (x) may be spoken of as the most nearly continuous
function associated with f (x).
It thus appears that a point-wise discontinuous function can always be
expressed as the sum of a point-wise discontinuous null-function and the
most nearly continuous function associated with the given function.
The latter function 01 (x) has only those discontinuities which necessarily
arise from the values of the given function at its points of continuity,
and is independent of the parts of the discontinuities which arise out
of the functional values off(x) at the points of discontinuity.      The nullfunction depends upon the unessential parts of the discontinuity of f(x).
EXAMPLES.
1 1 1
1.t Let f(x)=0, for x=0, -, -, 3,..., and for all other positive and negative
7rr  27r' 37'  
1                                                    1
values of x, let f(x)=-cos -. The function b (x) associated with f(.x) agrees with cos - at
every point except x=0, where ((x) is represented by (-1, +1). The measure of
I   i   1
discontinuity cf is zero except at -where 1 = 1, and at whe, awhere kf=2;
7rr   ' rr 3-r',,0h
the measure k,, vanishes everywhere except at x=O, where,=2. The function X(x)
1 1
vanishes except at -, 2...' where it is 1; it vanishes at x=0, but is discontinuous at
that point.
111        1
2.t Let f(x) vanish except at the points =2, 3, 4,... -,... where f()-=1. The
function q (x) is everywhere zero, and thus k, is everywhere zero. The function (l (x) is
11       1
everywhere zero, and kg is zero except at 0, p,,...-,..., where klf=1. In this case
f W( - (   -f ( ).
3.t A point-wise discontinuous function f(x) can be constructed S such that the
function 6 (x) may have at a point xvo, the values belonging to a prescribed closed set G,
in accordance with Bettazzi's theorem (~ 190). If G be unenumerable, choose an enumerable set G1 dense in G, and let gl, g2, g3,... be the points of G1. Take a set of intervals
{a,}, where a is ( o+, o0+2-)1, and define f(x) as follows:-in 1, 83, a3,  A7,..
letf (x) =gl; in a2, a6, a10,... let f(x)=g2; in 84, 12, a20,... let f(x)=g3; in general /(x)=g,,
x This definition is not in complete agreement with that of Schinfiies, see Bericht, p. 134, to
whom the term is due. Some erroneous statements of Schinflies, in this connection, were pointed
out and corrected by Hahn; see Monzatshefte f. Math. vol. xvi, 1905.
f See Hahn, loc. cit.
+ This is contrary to a statement of Schonflies, see Bericht, p. 135.




252


Functions of a real variable


[CH. IV


in the first free interval, and in every second of the following free intervals; further let
f(x). )l, for x  xo. The function f(x) is point-wise discontinuous, the points of discontinuity being.x, and the points x0 + 2L. The function q (x) has two values at the points
xn +; and at Xo it has all the values of G1, and therefore all those of G.
FUNCTIONS WITH LIMITED TOTAL FLUCTUATION.
193. Let a function f(x) be defined for the continuous interval (a, b).
Suppose the interval (a, b) to be divided into a number n, of non-overlapping
sub-intervals the greatest of which is d,; let these sub-intervals be divided
into smaller ones, so that the total number of sub-intervals is now n2, and the
greatest of them is d2. Proceed in this manner to continually sub-divide the
sub-intervals according to some prescribed law, so chosen that the numbers
n1, n2, n3,... form a sequence of continually increasing numbers, and that the
numbers di, d2, d3,... form a convergent sequence converging to the limit
zero. Such a system of indefinitely continued sub-divisions of the interval
(a, b) may be spoken of as a convergent system of sub-intervals.
If the function f (x) be such that, any particular convergent system of subintervals of (a, b) being taken, and the sum of the fluctuations of f(x) in the
n,. sub-intervals into which (a, b) is divided at the rth stage of the process of
n,.
szccessive sub-division being denoted by E Al, a, this sum is for every value of
v1z = 1
r less than some fixed finite number, then f(x) is said to be a function with
limited total fluctuation in the interval (a, b).
It will be shewn that, when the condition stated in this definition is
nr
satisfied, then the sums; Ar, n have an upper limit L, which may be called
the total fluctuation of the function in the interval (a, b), for the prescribed
convergent system of sub-intervals.
For, if an interval (a, 3) be divided into two parts (a, y), and (y, 8), by a
point 7, it is clear that the fluctuation in (a, 3) is not greater than the sum
of the fluctuations in (a, y) and (y, /3); and therefore, when (a, /) is divided into
any number of parts, the sum of the fluctuations in those parts is greater
than, or equal to, the fluctuation in (a, /3). It thus appears that, for the
= nr
given succession of sub-intervals, the numbers E A,,,, which are essentially
n= 1
positive, never diminish as n increases.  Therefore, since they are all less
than some fixed number, they have a fixed finite limit L.
If*f (x) have a limited total fluctuation L for a prescribed convergent system
of sub-intervals, then it has also a limited total fluctuation for any other such
convergent system  of sub-intervals.  Further, the total fluctuation L, has an
* See Study, Math. Annalen, vol. XLVII, p. 299.




192-194]   Functions with limited total fluctuation


upper boundary M, and a lower boundary tU _ ~M, when all possible convergent
systems of sub-intervals are considered.
Let Al, A,,... A, denote the fluctuations in those sub-intervals i8, 2),... $,,
which, at any stage, belong to the originally prescribed mode of sub-division,
and let A/, A2',... A'n, be the fluctuations in the sub-intervals 8', 82... 8',,   at
some stage of any other mode of sub-division of the interval. If v be a positive
number smaller than the smallest of the numbers 8', we may take m to be so
large that none of the intervals 8 is greater than y. Now let us suppose
the two sets of sub-intervals to be superimposed; then any interval 8, will be
divided, by means of the points of division in the second set, into not
more than two parts 8.'1", S1.2", with fluctuations A,,", A,", if divided at all;
we have then
A,. Anl + Ar2 - 2A,.
Now every 8' interval is made up of 8 intervals and 3" intervals; therefore LA'&  A"+ S, where S denotes the sum of the fluctuations in the
undivided 8 intervals. It follows that EA' - 2A _- 2L. It thus appears
that the numbers IA', corresponding to any arbitrarily prescribed system
of sub-divisions, are all not greater than 2L; and nA' has therefore, as in
the case of the original system, an upper limit L', and this is < 2L. For
every possible system of sub-divisions, the numbers L' form an aggregate of
positive numbers which do not exceed 2L; and they therefore have an upper
boundary M, and a lower boundary /L. Moreover, a system of successive
sub-divisions can be defined such that the limit of the sum of the fluctuations
is /, in case the lower boundary O/ be attained by the set of numbers L', or is
/L + e, where e is less than an arbitrarily chosen number, in case /L be not
attained. We may therefore take L = /, or, + e, as the case may be; and
then, for any system of sub-divisions 8', 2A' _ 2p/, or 2 (/a + e): it thus follows
that the upper boundary M is not greater than 2/p, or that /u > M. The
numbers M and A/ are the limits of indeterminacy of the total fluctuation
of the function in (a, b), where M cannot fall outside the interval (,/, 2ju).
194. A function with limited total fluctuation can have no points of
discontinuity of the second kind, and thus for such a function the limits
f(x + 0), f(x - ) on the right and on the left must both exist at every point;
except that at the points a, b, only the limits f(a + 0), f(b - 0) can exist.
For, in any arbitrarily small neighbourhood of a point of discontinuity of the
second kind, the function makes an infinite number of oscillations which
are greater than some fixed finite number; and thus, in such a neighbourhood,
the total fluctuation of the function cannot be finite.
It is clear that, in a function with limited total fluctuation, the sums
of the saltuses on the right and on the left
If(x + O) -f(x),    If( (-  ) -O f(x) [,
for all the points of discontinuity, are finite. It follows that the points of




Functions of a real variable


[CH. IV


discontinuity form an enumerable set, since there can be only a finite
number at which either saltus exceeds any arbitrarily chosen positive
number. A function of this kind may however have oscillations in every
sub-interval, but it is necessarily either point-wise discontinuous or else
continuous.
The following theorem will now be established:If a function with limited total fluctuation have at no point an external
saltus, then M= M; and thus the total fluctuation of the function is the same for
every convergent system of sub-divisions.
For let us assume that, if possible, there exists a convergent system of
sub-divisions, such that  A,.,.,, converges to a value G less than M. The
m = 1
interval (a, b) can also, by hypothesis, be divided into a number n of parts,
such that the sum SA' of the fluctuations in those parts is greater than G,
say = G + a. Let this sub-division into n parts be superimposed on the set of
n,.
sub-divisions for which S Al., m is the sum of the fluctuations; we may
assume that r is so great that not more than one of the n - 1 points of the
former sub-division is in any one of the intervals of the latter set, while none
of the end-points of the two sets coincide except at a and b. Let us suppose
that x is one of these n - 1 points, and that it falls in the interval 8 for which
A,., m is the fluctuation, dividing it into two parts, /1' and 82'. Assuming that
there is no external saltus, if 7 be any fixed positive number, we know that if
8/, 82' are sufficiently small, then
A,.,, = I f(x +o) -f( -  ) + a, where oa <;
moreover, under a similar condition, the fluctuations A/', A/' in 8', 23' are
such that,' = i f(x)-f(x - 0) + 0-2, where a-2< 
and             A2 = If(x) -f( + 0') + 03, where a3< <.
Now r may be chosen so great, and consequently d, so small, that these
conditions are satisfied for each of the e - 1 intervals of the originally assumed
set which contain points x. We have then A'/+ A2'- A,,,,= -2 + 3 —1,
provided f(x) lies between f(x+O) and f(x-O); hence the sum of the
differences A,+ A'-A,., taken for all those intervals which contain one
of the n - 1 points x is less than 2 (n- 1)v; and it cannot be negative.
It follows that the sum of the fluctuations in the intervals obtained by
superimposing on the sub-divisions, for which the A,.,, are the fluctuations,
the n - 1 points x, is < G+ 2 (n- 1)v. But the sum of these fluctuations
is certainly > G + a; and since t is arbitrarily small, and independent of n,
these two relations are incompatible with one another, i.e. a must be zero.
It has thus been shewn that, provided there be no external saltus at any
point, G must equal M, and this therefore is the limit of;A for every convergent system of sub-divisions of (a, b).




194]


1Functions with limited total fluctuation


255


In case there be points at which there is an external saltus, the preceding
proof can still be applied to shew that the total fluctuation is the same for
any two convergent systems of sub-divisions of the interval (a, b), provided
no point at which there is an external saltus be an end-point of an interval in
either system. Moreover, in case there be an external saltus at the point x,
the value of Ar, is either I f(x) - f(x + 0) | + a- or I f(x)-f(x- 0) | + -1,;
and we see that A' + A,' - Ar,, = o' + C- - oa- + s (x), where s (x) denotes the
external saltus at x, and is equal to the smaller of the two numbers
f()/f(+ 0) I, |I    (    f(    0). 
If we take the n - 1 points x to consist of all those points at which there is
an external saltus greater than some fixed number /, we see that the sum of
the fluctuations in the set of intervals, obtained by superimposing the n - 1
points x on the system for which SA,.,  is the sum of the fluctuations,
is EA., n + S + /y, where Sp denotes the sum of those external saltuses, all of
which are greater than 83, and y is the sum of the n - 1 numbers -2 + 0-3 -  i,
and is therefore arbitrarily small. It thus appears that the total fluctuation
for a convergent system of sub-intervals, such that no point, at which there is
an external saltus, is ever an end-point, must be a; whereas if those points at
which the external saltus is greater than 3 be end-points of intervals, the
total fluctuation is /, + Sp. If a sequence of descending values be given to 3,
the sum S^ converges to a fixed finite number, which is the sum or limiting
sum of all the external saltuses. Thus we obtain the following theorem:If a function with limited total fluctuation be such that there are points at
which the function has an external saltus, then the difference M -, between the
upper and lower boundaries of the total fluctuation L of the function, for convergent systems of sub-intervals, is equal to the sum of all the external saltuses.
If a convergent system of sub-intervals be such that no point at which there is
an external saltus is an end-point of any sub-interval, then for such a system
L = a. If, however, the system be such that every point at which there is an
external saltus is an end-point of an interval of the system, then L = M.
It thus appears that, provided f(x) be at every point intermediate in value
between f(x + 0) and f(x - 0), the total fluctuation in (a, b), for a function
of the class considered, is a definite number. This number is unaltered by
changing the value of f(x) at a point of discontinuity of the function, provided no external saltus be introduced. For example, we may assign to f(x)
the value I {f(x+0) + f(x - 0) at every point, without altering the total
fluctuation of the function.
If a function f(x), for which an external saltus exists at points of a certain
set, be replaced by a new function ' (x), differing from f (x) only at the points
of the set, and such that ' (x) has nowhere an external saltus, then the new
function ' (x) has its total fluctuation equal to a definite number, which is




256


lFunctions of a real variable


[Cc. IV


independent of the particular system of sub-intervals employed. Consider a
system of sub-intervals for which no point of the set is ever an end-point of
a sub-interval; then the limit of the sum of the fluctuations of f(x) for this
system is A. If at each point of the set we remove the external saltus, by
there substituting b (x) for f(x), we diminish this minimum total fluctuation,/ of f(x) by the sum of the external saltuses, and this is M - /L. It thus
appears that the total fluctuation of the function b (x) is 2u - M, and this is
a definite number independent of any particular system of sub-intervals.
195. A function b (x) defined for the interval (a, b), and such that for
every pair of points xi, x2 in the interval, for which x2 > x1, the condition
b (x2) - b (xi) is satisfied, has been defined, in ~ 189, to be monotone
in (a, b). If for every such pair of points the condition b (x,) (b (x1) is
satisfied, ( (x) is also monotone. In the former case b (x) never diminishes,
and in the latter case it never increases, as x is increased through the interval
from a to b.
A function with limited total fluctuation can always be expressed as the
difference of two functions, each of which is monotone in the interval for which
the function is defined, and neither of which diminishes as the variable increases.
The importance of the class of functions with limited total fluctuation,
in connection with the theory of Fourier's and other series, depends upon
their possession of this property.
To prove the theorem, let the upper boundary of the total fluctuation of
the function, for the interval (a, x), be denoted by Ma-. We then see that
f( + h) -f(x)    Mx+h < MX+h -Ma; it follows that M         -f (x) is a
monotone function which never diminishes as x is increased. The function
Mlx -+f (x) has the same property, since f (x + h) -fJ(x) _  -  MI+t. Hence, if
+ (x) 1 I-     +f ()},  0 (x) =  = {~1 -fx ()},
we can express f(x) in the form 01 (x)- 02 (x), where ~, (x), b2 (x) are both
monotone non-diminishing functions as x increases through the interval (a, b).
The converse property is easily seen to hold, that every function expressible
in this manner has a limited total fluctuation. For the fluctuation of a
monotone function in an interval is the difference of the functional values at
the ends of the interval.
196. The class of functions of which the properties have been investigated above has been defined in a different manner by Jordan*, and it has
been shewn by Studyt that the two definitions are completely equivalent to
one another.
Let us denote by D,.,,, the absolute difference of the functional values at
the end-points of that sub-interval belonging to a convergent system of sub* Cours d'Analyse, vol. i, p. 55.
t Math. Annalen, vol. XLVII, p. 55.




194-196]  Functions with limited total fluctuation


257


intervals, the fluctuation in which has been denoted above by AL,,. Let us
Ur
consider the sum  E D,.,,, which is equivalent to
}?1 = I
I f(a) -f (xr, ) I + I f (X, 1) -f(x., 2) |+.. + If(x/, m-1) -f(X-, m) 
+...+ f(x,,.-) — f(b) f,
where ac, Xr,, xY, 2,..., b are the points of the sub-intervals at the rth stage
nf,
of the process of successive sub-division of (a, b). If the numbers  1 Dr,,,,
nm=1
for every value of r, and for every possible convergent system of sub-divisions
of (a, b) be all less than some fixed finite number, then the function is said
to be a function with limited total variation (a variation bornee) in (a, b); and
Tir
the upper limit of the numbers E D,, m for a particular system  of submn =1
divisions, as r is indefinitely increased, is said to be the total variation off(x)
in (a, b) for that particular convergent system of sub-divisions.
It will be shewn that a function with limited total fluctuation is also
a function with limited total variation; and the converse. The first part
of this theorem  follows at once from the fact that, in any sub-interval,
Dr,,< m, and therefore     D.,,, is certainly less than a fixed finite
n.,.
number, if E A8r, n be so.
2 = 1
To prove the converse theorem that, if E D.,,, be less than some fixed
mw=?ny
finite number, so also is E A, A, let us first suppose that the function
m-=1
f/() has no external saltus at any point in the interval; it will then be
proved that 2 D., r, and; Ar,, for a particular convergent system  of
Wz=1        rn=1
sub-intervals, converge to the same limit, as r is indefinitely increased.
Consider the interval (xi, m-_, x,, m); let U and V be the upper and lower
limits of f(x) in this interval; thus A., n = U - V. We have D,, M  Ar, m;:and it will be shewn that some greater value of r, say r+ s, can be chosen,
*uch that  D~.+s -A, m, where ED2.+s is the sum of the absolute values of
he differences of the functional values at the end-points of the parts into
vhich the interval (x., m_-, xr, an) is divided, when the (r + s)th stage of the
successive sub-divisions of (a, b) is reached. If U, V be the functional values
at the ends of the interval (xr,, -,i,,i) it is clear that D,., = A, in; we
therefore need only consider the case in which one at least of the numbers
/(Xr, _-i), f(xr,,) lies between U and V. If e be an arbitrarily small positive number, there are two points |l, at, in the interval, for one of which the
functional value is greater than U - e, and for the other the functional value
H.                                                          17




258


Functions of a real variable


[CH. IV


is less than V+ e. Now, provided the function have no external saltus at any
point, s may be chosen so large, and consequently d,.+s so small, that there is
an end-point of one of the sub-intervals into which (x,, m-,, xr,,) is divided
at the (r + s)th stage, such that at this end-point the functional value is
> U - 2e, and also such that there is another end-point at which the
functional value is < V+ 2e: and one at least of these end-points does not
coincide either with x,., _ or with X.,. The number s having been so
chosen, we see that for the interval (Xr, m-i, Xr, m), ZD- +s > U - V- 4e + W,
where W   is the absolute difference between the functional value at one
of the end-points of (x,., -,,,,  m) and the functional value at one of
those end-points of a sub-interval at which it is > U-2e, or else < V+ 2e.
Since E is arbitrarily small, it is clear that s can be chosen so great that
D.Dr+s > U- V, for the interval in question. Moreover, s can be chosen so
great that this condition is satisfied for each of the n, intervals (x,, _i-i, xr, in);
and therefore s can be chosen such that D,,+s taken for the whole interval
n.
(a, b) is-   A,., -,. It follows that the limit, as r is indefinitely increased,
in  1
an,                      nr
of 2 D.,,, is._ the limit of X A,, *; hence, since the first of these limits
is < the second, it is seen that the two limits must be identical.
It has now been established that, for a function with limited total fluctuation, and without points at which there is an external saltus, the total fluctuation
and the total variation of the function are identical, being independent of any
particular convergent system of sub-divisions.
Next let f(x) have an external saltus at each point of some set. If we
consider a convergent system of sub-divisions such that no point of this set is
ever an end-point of a sub-interval, it is clear that the total variation of f(x)
for such a system is identical with the total fluctuation of that function 5 (;,),
which differs fromf(x) only in having the functional values at the points of
the set so altered that the external saltus is at every point removed. It ha.s
been shewn that the total fluctuation of > (x) is 2/z - M. If, on the other
hand, a convergent system of sub-divisions be chosen, so that every point;
at which there is an external saltus becomes, at some stage, an end-point of
a sub-interval, the total variation will be identical with the total fluctuation
their common value being M. It thus appears that, for a function of limitec
total fluctuation, which has points with an external saltus, the total variation
is M or 2/, -M, or has some value between these two num'nbers, according to
the particular system of sub-divisions employed.
The necessary and sufficient conditions that a function f(x) defined for
the interval (a, b) may be a function with limited total fluctuation, may be
now stated as follows:



196, 197] Functions with limited total fluctuation


259


(1) The points of discontinuity must all be of the first species, i.e.
f(x +   ), f(x-O) must everywhere exist.
(2) The sum of the absolute values of the external saltuses must be
finite.
(3) A convergent system of sub-intervals must exist such that
tDr, m =   I f(xr, m-l) -f(x, m) I
is, for every value of r, less than some fixed number.
These conditions are clearly equivalent to those which have been given in
the definition of the class of functions with limited total variation.
EXAMPLES.
1.* The function defined by f()=x sin -, f(0) =0, is not of limited total fluctuation
in the interval (0, l/7r), although it is continuous in the interval. For in the interval
(_  —,:-~ ), sin - attains the value (- 1), and thus the fluctuation in this interval is
at least equal to l/(+~)r.- The total fluctuation in the interval (-, -) is at least
r 1      4 +  -+ **    +} or      + 5 +*2s -    ) and it is well known that this
7~ [1+1  2+1~    8i+f +... -3  5  s-1
increases without limit when s is indefinitely increased; therefore the total fluctuation in
(0, l/rr) is not finite.
2.t The function defined by f(x)=x2sin -, f(0)=0, is continuous in any interval
containing x=0, and is everywhere differentiable, but is not of limited total fluctuation.
3.* The function defined by f(x)=x2sin(x-' ), f(0)=0, is of limited total fluctuation in the interval (0, 1/rr4). In the interval (,,the function has a
(r+ 1 r){  (r7r) 
single maximum, or else a single minimum, and the absolute value of the function at this
point is at most 1/(r7r)2. The total fluctuation in (0, 1/rr ) cannot exceed 2 Y, which
1 (r7r)
is finite.
4. Every function defined for a finite interval, which is continuous and of limited total
fluctuation in that interval, is the difference of two continuous functions each of which is
monotone in the interval.
TIHE MAXIMA, MINIMA, AND LINES OF INVARIABILITY OF CONTINUOUS
FUNCTIONS.
197.  Consider a point x, within the interval (a, b), in which a continuous
function is defined; it may happen that a neighbourhood (x, - 8, ax + 8) of
the point x, can be found by taking 8 sufficiently small, which is such that f(x)
* Lebesgue, Legons sur l'integration, p. 56.
t Lebesgue, Annali di Mat., Ser. IIIA, vol. vI, p. 270.
17-2




260


Functions of a real variable


rCH. IV


has the same value at all points in the neighbourhood; then the point x1 is
called a point of linear invariability of the function. If the same holds for
a neighbourhood of xl on the right only, or on the left only, then the point x1
is called a limiting point of linear invariability.
It can be shewn that if a point x1 of linear invariability exist, and the
function be not constant in the whole interval (a, b), then there exist two
limiting points of linear invariability, one of which, however, may be at one
of the ends of the interval (a, b). Suppose the function not to be constant
throughout the interval (x1, b); the points x of this interval may be divided
into two classes, in one of which x is such that in the interval (xi, x) the
function has the constant value f(xi), and in the other class x is such that
(x,, x) contains points at which the function has values differing from f(xi);
a section is thus made of the interval (xi, b), that defines a point which is
the required limiting point of the linear invariability. If the same argument be applied to the interval (a, x1) we see that there is another limiting
point in this interval, unless the function be throughout equal to f(x,).
In the interval (a, b) there may be a finite number, or an indefinitely
great, but enumerable, set of lines of invariability; each point within such
a line is a point of invariability, and the ends of such lines are limiting points
of invariability.
If the point x1 be not a point of invariability, it may happen that a
neighbourhood (xZ - e, x1 + e') exists such that, for every point in the interior
of this neighbourhood not identical with xi, the condition f(x)<f(xl) is
satisfied; in this case xI is said to be a point at which the function has
a proper maximunm.   In case the neighbourhood be such that at every
point x within it, except at x1, the condition f(x) >f(x,) is satisfied, the
point xi is said to be a point at which the function has a proper minimum.
It may happen that when x1 is not a proper maximum, a neighbourhood
(x1 - e, x1 + ') exists which is such that at no point within it the condition
f(x) >f(xl) is satisfied, nor at every point the condition f(x)<f(x1) is
satisfied; in this case x1 is said to be a point at which there is an improper
maximum of the function. If the condition f(x) >f(x,) is satisfied, but
the condition f(x) > f(xl) is not everywhere satisfied, then x1 is said to be a
point at which there is an improper minimum of the function.
A line of invariability of which the end-points are a, 3, and are both
interior to (a, b), is said to be a maximum of the function, if both a, / be
improper maxima, and it is said to be a minimum, if both a, / be improper
minima.
It is clear that, in any arbitrarily small neighbourhood of an improper
maximum or minimum, there are an indefinitely great number of points at
which the functional value is equal to that at the maximum or minimum.




197, 198]  cinMaxima, Minima, and lines of iinvariability  261


At any maximum or minimum      there is a greatest neighbourhood
(X -   x, x- 8') at every interior point of which the condition f(x) <f(xi),
f(x) f (xi), or f(x) >f(x,), f(x) >f(x,) is satisfied. At end-points of such
greatest neighbourhood, it follows from the condition of continuity of the
function, that the functional value is equal to f(x,), unless the end-point
coincides with a or with b.
It has been shewn in ~ 171 that there exists either one point or a set
of points in (a, b) such that the functional value at this point or at all the
points of the set is greater than at all other points in the interval; and it is
to be remarked that this set of points may contain lines of invariability.
Every such point, unless it be an end-point, is said to be a point of absolute
maximum of the function in the interval (a, b), and may be either a proper or
an improper maximum. A similar definition applies to an. absolute minimum.
In case an extreme point of the continuous function (see ~ 167) be at a,
or at b, such point is spoken of as an upper or lower extreme, but not always
as a maximum or minimum of the function. If f(a) and f(b) be equal, and
the function be not constant in (a, b), then there is at least one maximum or
one minimum point, or one line of invariability, in the interior of (a, b). This
is also true when f(a) f (b), unless the function be monotone.
198. If within the interval (a, b) there be two points or two lines of invariability at which the functionl is a maximtmn, proper or improper, then
there is between them at least one point or one line of invariability at which the
function is a proper or improper mininmum; thus maxima and minima occur
alternately.
Suppose that a, /3 are two points at which the function is a maximum,
and that (a, /) is not entirely a line of invariability, also that no maximum
occurs between a and 3. We know that between a and / there is a point or
a set of points at which the function is less than at all other points in the
sub-interval; and since a and / cannot belong to such set, there is therefore
a minimum at a point, or at points on a line of invariability, between a and 3,
and this minimum is less than either of the maxima at a and 3.
Between a maximum and the next miinimum of a fanction the function is
said to make an oscillation, the amplitude of which is the excess of the maximinm over the minimum.
If x, be a point in (a, b), it may be possible to choose e so small that
within the interval (x,, x1 + e) no maxima ori minima occur, so that the
function is monotone in this interval. It may however be the case that,
however small e is taken, there still occur maxima and minima in (x1, xi + e).
In this case the number of oscillations of the function must be indefinitely
great, however small e may be chosen; for if there were a finite number only,
a number e, could be found such that all the maxima and minima were in the




262


22Functions of a real variable


[CH. IV


interval (x, + 61, x1 + c), and thus in (x,, x, + e,) the function would be monotone, which is contrary to the hypothesis made.
It thus appears that, in the neighbourhood of a particular point, a
continuous function may have an indefinitely great number of oscillations.
An improper maximum or minimum, not in a line of invariability, is certainly
such a point.
The proper naximca and minima of a continuous function form   an
enumerable, or a finite, set of points.
Consider (x1 - e, x, + a), the greatest neighbourhood of a point of proper
maximum x,, which is such that for all other points x within the neighbourhood, f/() <f(x). There can in a finite interval be only a finite number
of such points x1 for which e > a, V > a, where a is a fixed positive number;
for if there were an infinite number of such points, they would have a
limiting point:, and we could choose two points xI', x,' of the set, such
that the distance of each from | is less than I a; now each of these points
would lie within the neighbourhood belonging to the other, and thus we
should havef(x1') >f(x,"), and also f(x1,) >f(x,'), which is impossible; thus
the set must be finite. Now choose a sequence of descending values of a
which converges to zero, say a,, a2,... an,...; the number m, of maxima ax
such that for each e > a,, Vr > a,, being finite, we have mi, in2,... m,,... all
finite: and hence the whole set of maxima forms an enumerable set.
If x, be an improper maximum point, and f(x) = A, a neighbourhood
(x1 - e, x1'+ r) can be found which contains an infinite set of points GA such
that f(x) = A, for each point of the set. If x' be an isolated point of the set
GA, then x' is clearly a proper maximum of the function; and if x" be a point
of GA, which is a limiting point of the set, x" is an improper maximum. The
points x, - e, xI + q need not be maxima, even though they be limiting points
of GA. The condition of continuity of the function ensures that the set GA is
a closed one; for, at any limiting point of the set, the functional value is the
limit of a sequence, each member of which is A, and this value is therefore
itself A.
Corresponding to a given A, there may be a finite, or an infinite, set of
detached intervals such as (x1 - c, xI + I), each one of which contains a closed
set such that each isolated point of it is a proper maximum, and each limiting
point (except an end-point) is an improper maximum. The sets GA may
contain perfect components, and thus the improper maxima at which A is the
functional value may form a set of the cardinal number of the continuum. A
similar result holds for minima.
It can further be shewn that the values of a continuous function at all its
maxima and minima form a set which is either finite or enumerably infinite.




198-200] Maxima, Minima, and lines of invariability


263


199. If in the interval (a, b) the function have only a finite number of
maxima and minima, counting any line of invariability which is a maximum
or minimum as one maximum or minimum, the interval can be divided
into a finite number of parts in each of which the function is monotone;
the function is then said to be* in general monotone (abtheilungsweise
monoton ).
If the function have an indefinitely great number of maxima and minima,
which occur either at points or at lines of invariability, the function then
makes an infinite number of oscillations; and these may occur in the neighbourhoods either of a finite number of points, or of an infinite number of points.
It can be shewn that in the case of a continuous function, although there
may be an infinite number of oscillations of the function, there can be only
a finite number of which the amplitude exceeds an arbitrarily small fixed
number a.
For it has been shewn in. 175 that a number e can be determined, such
that, in any sub-interval of length e, the fluctuation of the function does not
exceed a-; therefore in each of the sub-intervals
(a, a + e), (a + e, a + 2e),... (a + ne, b),
the fluctuation of the function is not greater than -. It follows that no
oscillation of the function which is greater than a- can be completed in one
of these sub-intervals, and that such an oscillation must require two at least
of these sub-intervals for its completion; hence the number of such oscillations
in (a, b) cannot exceed the finite number n. As the number a is diminished
indefinitely, it may happen that the number of oscillations of which the
amplitude exceeds a is increased indefinitely.
THE DERIVATIVES OF FUNCTIONS.
200. If a function f(x) be defined for all points in the interval (a, b),
then for a point xi in this interval we may regard the function fx) - (
x - Xas a function F (x) of x, which is defined for all values of x in (a, b), except for
the point xi. This function F(x), although undefined at the point xi, has
finite or infinite functional limits at that point, in accordance with the
definitions in ~ 176.
If the limits F(x,+O), F(x,-O) both exist and have the same finite
value, this value is called the differential coefficient at xi of the function f(x).
At the point a, if F (a + 0) exists, it is frequently said to be the differential
coefficient of f(x) at a; and at the point b, if F (b -0) exists, it is said to be
the differential coefficient of f(x) at b.
v This term is due to C. Neumann; see his work Ueber die nac J Kreis- iKugel- und
Cylinde'fzunctionen fortschreitenden Reihen.




Functions of a real variable


[CH. IV


The condition that f(x) may possess a differential coefficient at x, is that,
corresponding to each arbitrarily chosen positive number e, a neighbourhood
(x, - 8, x + 8) can be found, such that
if() -(X     f(x  ( -f() (X
<Cfbr every pair of points:, 6' which lie within this neighbourhood, or within
such part of it as is interior to (a, b).
In other words, the condition is that a neighbourhood of x1 can be
found such that the fluctuation of the function f (x) - f (  within it, or
X - X
within such part of it as lies in (a, b), may be as small as we please.
When a differential coefficient of f(x) exists at the point x1, then the
function is said to be differentiable at x1, and the differential coefficient at
that point may be denoted by f' (x,).
That a function f(x) may be differentiable at x,, it is necessary, but not
sufficient, that xl should be a point of continuity of the function.
At a point of discontinuity xl of f(x), there always exists a positive
number a, such that in any neighbourhood of x1, however small, points:
exist such that  f(X)-f(x1i) I > a; hence if A be any arbitrarily great
positive number, in the interval (x - 8, x + 8), where 8 < $, there exist
points  such that  f(A  If  )
points: such that     (: / —     > A, and it is thus impossible that
f(x) -f (X) should have a definite finite limit at x1.  On the other
X - XI1
hand, the condition of differentiability, viz. that f() f(x) should have
X - X1
an arbitrarily small fluctuation within a sufficiently small neighbourhood of
x1i, is not necessarily satisfied when the condition of continuity, viz. thatf (x)
should have an arbitrarily small fluctuation within a sufficiently small neighbourhood of,x, is satisfied.
f(gc -f(xio
It may happen that the limit of f (x), on both sides of xa, is
a; - X1
indefinitely great with the same sign on the two sides; in this case it is
usual to say that f(x) has a differential coefficient at xa which is infinite in
value.
A continuous function f(x) defined for the interval (a, b), which has a
differential coefficient at every point of the interval, is said to be differentiable
in its domain. Continuous functions exist, which at no point in their domain
possess a differential coefficient. The first example of such a function was
given by Weierstrass; the construction of such functions will be considered
in Chapter vI.




200, 201]


Derivatives of functions


265


That a continuous function possesses a differential coefficient was formerly
regarded as obvious from geometrical intuition, it being supposed that such
functions were necessarily representable by curves possessing definite tangents
at every point. The first attempt to prove the existence of a differential
coefficient of a continuous function was that of Ampere*; this proof was,
however, insufficient even in the case of those continuous functions which
make only a finite number of oscillations in the intervals for which they are
defined. It is now fully recognized that the class of continuous functions is
much wider than that of functions capable of an approximate graphical
representation; and that the conditions for the existence of definite differential coeffiients are of a much more stringent character than would be the
case if they were included under the bare condition of continuity of the
function.
201. It may happen that at a point x,, the function f()-         (
X - XI5
possesses finite, or even indefinitely great, limits on the right and on the left
at x1 which differ from one another; the function is then said to have derivatives, on the right and on the left at x,. These are frequently spoken of as the
progressive and regressive derivatives respectively. A function may possess a
progressive derivative and no regressive derivative, or the reverse.
When at the point xa    a function is not differentiable, and possesses
neither a derivative on the right nor one on the left, then the function
f (X) - f(XI)
f(x) -f(1) has at x, four functional limits, an upper and a lower on
X - X1
the right, and an upper and a lower on the left; and any one of these may be
either finite or infinite. These four limits are defined to be the upper and lower
derivatives at xl on the right, and the upper and lower derivatives at xa on
the left, and are, in accordance with the notation of Scheeffert, denoted by
Df+(x1), D+f(xl), D-f(xO), D_/(lx) respectively.
It is frequently convenient in this general case to speak of the derivatives
off(x) on the right and on the left as existent but indefinite in value: and in
this case D+f (x), D+f(x,) are regarded as the limits of indeterminacy of the
derivative on the right, and D-f(xA), D_f (x) as those of the derivative on
the left.
The definitions which have been given for the case in which the domain
of the function is continuous are applicable, without essential change, to the
case in which the domain is any perfect set of points.  At a point of the set
which is a limiting point on both sides there exist in general the four derivatives D+f(xl), D+f(ax), D-f(x), D_f(x), two or more of which may have
JourZn,. Ecol. polyt., vol. vi, 1806, p.. 148.
+ Acta Mathematica, vol. v. The same limits were considered by Du Bois Reymond, Progralmm,
Freiburg, 1870, also Miinch. Abh. vol. xII, p. 125, under the name Unbestimmtheitsgrenzen.




266


Functions of a real variable


[CH. IV


equal values; and at a point of the perfect set, which is a limiting point on one
side only, there exist of course only the two derivatives on that side.  If the
domain be any closed set, the derivatives exist only at those points which are
limiting points of the set.
A function defined for a perfect set may, by the method of correspondence,
be correlated with a function defined for a continuous interval, the order of
the points in the continuous interval and in the perfect set being the same;
and thus all properties of derivatives of functions defined for a continuous interval have their analogues in the case in which the domain is any perfect set.
EXAMPLES.
1. If /(x)  sin -, /(0)=0; we have   /-f ()=sing   and for arbitrarily small
values of h, this oscillates between 1 and -1. The function f(x), although continuous
at x=0, possesses no differential coefficient at that point; in fact
D+f(0)=l, D+f(0)=-1,   D-/(0)=1,   Df(0) =-1.
2. If f() -x2 sin-, f(0) =0, the differential coefficient f' (x) exists for every value
of x, and is finite. At the point x=, f' (x) is zero, but has a discontinuity of the second
kind.
3. Let /f (X)= ox' (1 +x sin I), for x>O; f(.v)= - /X-x (1 +X- sin ), for x<O; and
f(0)=0. In this case f'(x) everywhere exists; its value at x=O, is +c, and although
it has a finite value at every point except at x=0, it oscillates in the neighbourhood of
that point between indefinitely great positive and negative values.
4.t The function defined by f (x)=x {l +I sin (log x2)}, and f(0)=0, is everywhere
continuous, and is monotone, but has no differential coefficient at x=0.
1
5.+  Let f(x)=e x sin, f(O)=O0; this function has at every point a differential
coefficient, and this is continuous at x=O. The differential coefficient vanishes at x-O,
and at an infinite number of points in the neighbourhood of x=0. The function f' (x)
like f (x), has an infinite number of oscillations in a neighbourhood of x=O.
THE DIFFERENTIAL COEFFICIENTS OF CONTINUOUS FUNCTIONS.
202.  Let us suppose that a continuous function, defined for a continuous
domain, is such that at every point interior to an interval (a, /) there exists
a differential coefficient; this differential coefficient may at any point have
a finite value which may be zero, or it may have an infinite value of which,
however, the sign is definite.  It will be observed that f(x) is assumed to be
' Dini, Grundlagen, p. 112.
t Pringsheim, Encyklopddie der Math. Tissensch. II A. i, p. 22.
+ Dini, Grundlagen, p. 313.




201-203] Differential coefficients of continuous functions  267


continuous at the points a, /, but it is not assumed that definite derivatives
exist at those points. It will be shewn that, unless the function be constant
throughout (a, IS), there exists at least one point in the interior of (a, /3) at which
the differential coefficient has a definite finite value different from zero.
Suppose f(a), f(/3) to be unequal. If they be not unequal, and the
function be not constant throughout (a, /3), we can replace the interval (a, /),
by another one contained in it, for which the functional values at the ends
are unequal. Let us consider the function
F(x) =f(x) -f(a) _   -af (/3) -/(a)}
F(a) and F (/) vanish, and F (x) is continuous in (a, /3), and has a differential
coefficient in the ordinary sense at each point, with the possible exception of
a and /; therefore it follows by the theorem of ~ 171 that there is at least one
point x1 in the interior of (a, 3), at which F (x) is a maximum or minimum:
this is the case even if F(x) be everywhere zero in the interval. A number e
can therefore be found such that F(x + 8) - F (x1), F(x, - 8) - F(x) have the
same sign, or else vanish, provided 8 < e; and consequently the derivatives at
x, on the right and left must have opposite signs, unless both of them be zero;
therefore the differential coefficient at x1, which must exist, must be zero. It
follows that f' (x,)-f (f) -f (a)
follows that /f (x) - f(  () = 0, and thus the point xa is the point of
which the existence was to be proved. From this theorem we deduce the
following general theorem:If f(x) be continuous in the interval (a, b), and be such that it has a
differential coefficient at every point in the interior of the interval, and if there
be in (a, b) no lines of invariability, then there exists in (a, b) an everywheredense set of points at which the differential coefficient has finite values differing
from zero.
This is proved at once by applying the foregoing theorem to any interval
contained in (a, b). There may be in (a, b) infinite sets of points at which the
differential coefficient is either zero or infinite.
203..If the function f(x) be continuous in the interval (x, x + h), and at
every point in the interior of this interval f' (x) exist, being either finite or
infinite with fixed sign, then a point x + Oh exists, where 0 is some proper
fraction, and is neither 0 nor 1, such that
f (+ h) =f(x) + hff'(x + h).
This is at once seen by taking a = x, /3 = x + h, xl =x + Oh in the proof in
~ 202. This is known as the mean value theorem of the Differential Calculus.
A corollary from the mean value theorem is that, if f(x) =f(x + h), then
f'(x) must be zero at one point at least in the interior of the interval
(x, x+h).




268


Functions of a real variable


[Co. IV


An important extension of the mean value theorem is the following:If f(x) be continuous in the interval (x, x + h), and have a differential
coefficient at every point of the interval, with the possible exception of the
end-points; and if F(x) be another function, which is also continuous in the
same interval, and at every interior point has a finite differential coefficient
different from zero, whilst at the end-points there may be no definite derivatives,
or they may be zero, or infinite, then
f(x+h)-f(x) _ f'(x+h)
F(x +) )- F(x)   F'(x + 0h)
for some value of 0 which is a proper fraction, and is neither 1 nor 0.
To prove the theorem, let
cc)(+ h )   -f(x) fF(x) - F(x) F'
() f(x)-f  h) - F (x)
then, since F'(x) does not vanish in the interior of the interval (x, x + h), it
follows that F(x + h) - F(x) cannot be zero.  Since  ( (x) = q (x + h), and
b (i) satisfies the conditions of the mean value theorem, b' ()) must vanish
for some value x + Oh of:, interior to the interval (x, x + h).  We have
then
f    '   Oh  f( + h) -  (x) F'(x +Oh) _ 0,O
f'(x~Oh)-h2)-J7^ F'(x~Oh)=0,
f (+ o  h) - -F(x)
from which the theorem follows, since F'(x + Oh), and therefore f'(x + Oh)
cannot be infinite. In the case in which f(x + h) =f(x), we have
f'(x + Oh)= 0,
for some suitable value of; and then, since F (x + h) - F(x), F'(x + Oh) are
finite, the theorem  still holds.
204. The last theorem may be applied to obtain a strict proof' of the
legitimacy, under certain conditions, of a well-known method of evaluating
0 o,
limits which appear in the so-called indeterminate forms -, -o 
Let the two functions f(x), F (x) be both continuous at all points interior to
the interval (a, a+~3), and let the limits f(a+0), F(a+0) both exist and be zero;
if finite differential coefficients f'(x), F'(x) exist at every interior point of
(a, a + /), and F'(x) be everywhere within this interval difrerent from zeqro, then
if one of the two limits
i= n(a + h) )   m F' (a + h)
exist as a definite nlumber, or be infinite with a fixed sign, the other limit also
exists, and the two have the same value.




203, 204] Differential coefficients of continuous functions     269


The two functional values f(a), F(a) may both be defined to be zero, and
thus the functions f(x), F(x) are continuous in any interval (a, + h), when
h < 3. We have then, from the extension of the mean value theorem
f(a +h) _f(a+ O0h)
F(ah + h)  F'(a +Oh) '
where 0 is some proper fraction. Since Oh converges to zero when h does so,
the theorem follows at once from this equality.
Let the two functions f(x), F(x) be both continutous at all points interior to
tLe interval (a, a - /3), and let the limits f(a + 0), F(a + 0) both exist and be
inzfnite, each with a fixed sign; if finite differential coefficients f'(x), F'(x)
exist at every interior point of (a, a + 3), and F'(x) be everywhere, within this
interval, different from zero*, then if one of the two limits
lin f(a +  )  im f'(a + h)
h=oF(a+h)' 70-F'(a + h)
exist as a definite tnumber, or be infinite with a fixed sign, the other limit also
exists, and the two have the same value.
Consider the interval (a + 8, a + 82) interior to (a, a +/3); we have then
f(a + 82) -f(a 4 8 )  f'( + 83)
F (a + 82) -F(a +  )   '(a + 83)'
where a3 lies between the numbers 81 and 82: this equation may be written in
the form
f(a+8 )    f(a + )   f'(a + 83) 1      (a + a2)
F(a +  i) F(a+    ) F'(a + 8)      F (a + 8,) '
Taking a fixed value of 82, and an arbitrarily small positive number e, we
can find a positive number 8' (< 82), such that for every value of 81 (> 0) which
is < ', the inequalities
1                         1
|F(a + 81) > - f( + 82),  F(a + 8) > -  (a + 8,) 
are both satisfied: this follows from the fact that F(a + 0) is infinite with
fixed sign.
We have now
f (a + 30)                +
=i f A( +  )f (+
F(a + 81)    = +       (a+83)'
where Ij i < e, and  < e, for all values of 8, which are > 0, and < 8'.
Let us first assume that f'(~ + h)
Let us first assume that F'(a + h) has a definite finite limit k; we may
The unnecessary hypothesis is made by Stolz (see Gzrndziige, vol. I, p. 77), that F' (x) has
everywhere the same sign. For a history of these theorems, see Pringsheim, Encyklopddie d.
Math. Wissensch. II A. I, p. 26.




270


Functions of a real variable


[CH. IV


then choose 83 so small that f'(a +3 - k is numerically less than an
F'(a +63)
arbitrarily chosen positive number }', for every value of 83, which is < 8,; we
have then
f(a + 1)- I =? + /- ~ (I + ), when   ' < l'
F(a + 61)
Since q' and e are both arbitrarily small, the absolute value of
f (a + 8)  k
F(a + 81)
is arbitrarily small, for all sufficiently small values of A8; and it thus follows
that
lim f (a + 81)
Ne x l a t (a + h)
Next let us assume that F(a + h) has an infinite limit, for h = 0; we
f(a o+ h)
may, without loss of generality, take this limit to be of positive sign. We
may choose an arbitrarily large positive number N, and a number N'> N;
and we may then choose 82 so small that f'( +     > N', for all possible
F'(a + 8,3)
values of 83 < 82; then
f(a +       +   -,)_
F(a + 81,)  + (-)( +),
where p is positive.
The number e may now be chosen so small that
X + (1 - ) (N' +p)> N
for all the possible values of X and ~; and therefore an interval on the right
of a can be determined, such that for all interior points the inequality
f(a + h)
F(a + h)
is satisfied. Since N is arbitrarily great, it follows that
lim f(a + h)    oo
a=o F(a+h)
f'(ra  + ~3)  1  f(a + ~,)   v
The relation   F'(a +S)     1 - F(a+ S) 1 -o
may be employed to prove, in a similar manner, that the existence of
lim f(a+ h)
=:0 F(a + h)
involves tha      t of li a  )
involves that of li(m  ( +h)' and that the two limits have the same
=o               (avalue.
value.




204, 205] Differential coefficients of continuous functions  271


205. If the otherwise continuous function f(x) have a discontinuity of
the second kind at the point a, at least on the side which is towards the interval
(a, a + h), but the function have a finite differential coefficient at every point of
the interval (a, a + h), except at the point a, then the absolute values of these
differential coefficients in any arbitrarily small neighbourhood of a have no
upper limit.
By applying the mean value theorem we have
f(a + 82) -f(a + 81)= (2 - S)f'(a + 83),
where 0< 8, < a2 < h, and 83 is some number lying between 81 and 2,.
Now if f(a + 0) and f(a + 0) be unequal, values of 81 and 82, less than
any arbitrarily prescribed positive number e, can be chosen, such that
f (a + 2) -f(a + 8)
is arbitrarily near to f(a + 0) -f(a + 0), whereas 2 - 81 is arbitrarily small;
therefore it follows that f'(a + 83) must have arbitrarily great values, in any
neighbourhood of a.
The mean value theorem f (a + h) -f (a) = hf'(a + Oh), where 0 < 0 < 1,
affords information as to the existence and value of the derivative at a, on
the right, provided f(x) satisfies, in a neighbourhood of a on the right, the
conditions under which the theorem holds. By considering both sides of a,
information may be obtained as to the existence of a differential coefficient
at a.
(1) If the function f' (x) have a functional limit at a on the right, then
f(a + h) -f(a)
h
has a definite limit for h= 0, either finite, or infinite with fixed sign, and
this is equal to that of f'(x). It follows that, in this case, a derivative at a
on the right exists, and is either finite, or infinite with fixed sign.
(2) If the function f'(x) have no limit at a on the right, it may still
happen that f'(a + Oh) has a definite limit at a on the right, because a + Oh
is not necessarily capable of having all values within a neighbourhood of a.
In this case, either (a) the derivative at a on the right may be definite, and
lie between the upper and lower limits of f'(x) at a on the right, or it may
be equal to one or other of those limits; or (b) there may be no definite
derivative at a on the right, but D+f(a), D+f(a) may have different values,
and these are certainly both finite in case the upper and lower functional
limits of f'(x) at a are both finite.
(3) The derivative on the right at a can only exist and be infinite, (a) if
f'(x) have an infinite limit on the right at a, or (b) if it have an infinite upper
limit on the right at a. In either of the cases, (a) and (b), f'(x) may be
everywhere finite within a neighbourhood of a on the right, or it may be
infinite at some points in such a neighbourhood.




272


Functions of a real variable


[ciH. Iv


(4) If the derivative at a on the right exist and be finite, then either
(a) f'(x) has a definite limit at a on the right, equal to the derivative at a, or
(b) f'(x) has no definite limit at a on the right, but a sequence of points
can be determined, of which a is the limiting point, such that the values
of f'(x) for points of that sequence converge to the value of the derivative
at a. At points which do not belong to the sequence, the values off'(x) may
be either finite or infinite.
(5) The non-existence of a definite derivative at a on the right may be
due to the non-existence of f'(x) at all the points of any neighbourhood of a,
or only at an infinite number of points of such a neighbourhood.
206. If f (x) be contin2uous in a given interval and have at every point,
with the exception of an enumerable set G, a differential coefficient of value
zero, the function is constant throughout the whole interval.
At the points of G we may suppose it to be unknown whether a differential coefficient exists, or, if one does exist, what values it has.
A more general form of this theorem is obtained by considering not the
differential coefficient, but any one of the four derivatives, thus:If f (x) be continuous in. a given interval, and one of the four derivatives
D+f (), D+f(x),. D-f(x), D_f(x), be stuch that it is zero at every point of
the interval, with the exception of points belonging to an enumerable set G, at
which nothing is known as to its value, then the function is constant throeughout
the interval.
To prove the generalized theorem for the case of the function D+f(x),
suppose that, if possible, f(x) -f(a) has at some point x, a value different from
zero, say the positive value p; and let b (x, c) denote f(x) -f(a) - k (x - a).
Then b (a, k) = 0, ) (x,, k) =p - k (x, - a). Choose any fixed positive number
q < p, then 0 (x1, i) > q, provided k  <  P —, or say k < K. Since 0 (x, k) is
X - a
continuous in (a, b), and 0 (a, k) is zero, whilst b (x,, k) > q, there exists an
upper limit of those values of x between 0 and xl, for which 0 (x, k) ) q,
and this upper limit is attained for some value | of x, which is such that
< x1, and ( (E, k) = q. Since ( (I + h, k) > q, provided 0 < h: x -:, we see
that, since        h (+h  -'  ) is positive, D+f (, kJ) is positive if it be not
zero. Now if f were a point not belonging to G, the value of D1+(t, h)
would reduce to - k; and therefore ~ must belong to G.
The number q being fixed, | depends only on k; and, corresponding to a
given value of I, there is only one value of k; for
b (), J) - b (, Jo') (D' - J) (d - a),
which cannot vanish unless k = k', since 0 (a, k) is zero and therefore < q. For




205,206] Differential coefficients of continuous functions      273


a given value of k, the corresponding number of values of I, all of which
necessarily belong to G, must be either finite or enumerably infinite, since
every part of an enumerable aggregate is either finite or enumerable. Therefore to each value of k, in the continuous interval (a, K -,), there corresponds
a finite or enumerable set of values of I, and it would hence follow that the
continuum (a, K -/,) is itself enumerable, which we know is not the case.
It has thus been shewn that for no point can f(x) -f(a) have a positive
value; and similarly, by considering f(x) -f(a) + k (x- a), it can be shewn
that f(x) -f(a) can nowhere have a negative value; hence f(x) =f(a)
throughout the whole interval (a, b).. The case in which one of the other
three derivatives vanishes except at points of G can be treated in a similar
manner.
The following theorem* which is of importance in the theory of Integration will now be established:If two functions be each continuous in a given interval, and if of one of
the four derivatives it be known that, for the two functions, this derivative has
equal finite values at each point of the interval, with the exception of an
enumerable set of points at which nothing is known as regards the two
derivatives, then the two functions differ from one another only by a constant,
which must be the same for the whole interval.
It must first be observed that the proof of the preceding theorem suffices
to shew that, if D+f( ()  0, at every point of (a, b) not belonging to the
set G, then f(x) -f(a) a  O, for every point x of the interval. Similarly, if
D+f(x) 0O, everywhere in the interval, except at the points of G, then
f(x) -f(a) - 0 at every point of the interval.
If now f, (x), f, (x) be two continuous functions such that
D+fi (x) = D+f2 (x)
at every point of (a, b) not belonging to G, let f(x) =f(x) - f(x2). If e be
an arbitrarily small positive number, then for any point x not belonging to G,
the condition
f (   h)-fi(x) >
h
is satisfied for a set of positive values of h which are arbitrarily small. Also
we have, for all sufficiently small values of h,
f2(x +  )h- ($) < D+f, () +e;
h
hence, since D+f (x) = D+f, (x), we see that  (x + h) -f() > - 2e, for all
Scheeffer, ta Mat. vol. v, p. 283
* Scheeffer, Acta Mat. vol. v, p. 283.


H.


18




274


Functions of a real variable


[CH. IV


values of h belonging to some set. It follows that D+f(x) >- 2e, and
thence* that D+f(x)    0, since e is arbitrary.  By interchanging f,(x)
and f,(x), we see that D+ {-f(x)} _ O. From these two results we deduce
that f(x)-f (a)   0, and that f(a) -f(x)    O0, throughout the interval
(a, b); therefore f(x) is everywhere equal to f(a), and thus the theorem
is established.
207. At a point x at which the continuous function f(x) is a maximum,
since, for a sufficiently small neighbourhood of such a point x, the differences
f(x + h)-f(x), f(x- h) -f(x)
are both negative or zero for all points x + h in the neighbourhood, it is clear
that each of the derivatives D+f(x), D+f(x) is either negative or zero, and
that each of the derivatives D-f(x), D_f(x) is either positive or zero. In
case the function possess definite derivatives on the right and on the left at
the point x, the first of these is zero or negative, or possibly - o, whilst the
second is zero or positive, or possibly + co.
If at the point x a definite differential coefficient exist, it must consequently be zero. In the case of a minimum the corresponding statements
hold, where the positive sign takes the place of the negative one, and the
reverse. The following theorem has now been established: —
If a continuous function possess a differential coefficient at a point x at
which the function is a maximum or minimum, then the differential coefficient
at x must be zero.
208. A continuous function may be such that in the interval (a, b) there
exists an everywhere-dense set of non-overlapping intervals, each one of which
is a line of invariability of the function. Within each interval of the set,
the function has its differential coefficient equal to zero; it therefore follows
from the theorem in ~ 206, that the closed set of points, of which the
given set of intervals is the complementary set, cannot be an enumerable
set, otherwise the function would be constant in the whole interval (a, b).
It is further clear that no two of the intervals can abut on one another;
for the condition of continuity of the function at their common end-point
would ensure that the values of the function in the two intervals were the
same, and thus the two intervals would really belong to the same line of
invariability. It follows that the end-points and external points of an
everywhere-dense set of lines of invariability of a continuous function must
form a perfect non-dense set of points.
That a continuous function with an everywhere-dense set of lines of
invariability can actually exist can be easily shewn as follows:-Make the
points of a non-dense perfect set correspond in order to the points of
* It is erroneously stated by Dini, that D+f(x) =0. See Grundlagen, p. 275.




206-209]


Successive differential coefficients


275


a continuous interval (a, b), then, as has been shewn in ~ 128, the correspondence may be such that the whole of a complementary interval of the
perfect set corresponds to one point of the continuous interval.   If a continuous function be defined for the continuous interval, we may define a new
function which has at each point of the perfect set the same value as the
original function has at the corresponding point of the continuous interval;
and since all the points of a complementary interval of the perfect set correspond to the same point of the continuous interval, the new function is such
that it has an everywhere-dense set of lines of invariability.
EXAMPLES.
1. Take* the non-dense perfect set defined in Ex. 1, ~ 75, by
C1  C2      Cn
x    +   +...+ 4.+...,
3   P32     3n
where every C, is either 0 or 2. A complementary interval has as its end-points
C. + 0.2  + n,1 Cl          2.    -1  2
3   3... 3n-1 -3n    3 r 32 + *+. 3"-1 i3n _3
which may be denoted by (a,, b,). Let the function f(x) be defined as follows:-For a
point x of the interval (0, 1) belonging to the perfect set, let
2 (    22+*   2n  ***)
when x is in the interval (a,, b,), let f(x)=f(a,)=f(b,). The function f(x) so defined is
continuous, and varies from 0 to 1, and is constant in each of the intervals (a,, b,) complementary to the non-dense perfect set.
2.+ Let the numbers in the interval (0, 1) be expressed in a scale n- 2n - 1, of odd
degree; thus x=- + a2+.., where 0 < a< n, and the number of digits a,. is finite or
n n
infinite. For any number x represented in this manner, for which all the a,. are even
integers, let f(x) equal  (a + a2+...). In case any of the a, are odd, let ak be the
1 al a2       a,__  lak+
first one which is odd, and let f(x) then equal (+ +- +... + k-) +I a  l. This
2 \m  vt2    mnc -    2 mk
function f (x) is continuous and varies from 0 to 1; for an infinite set of points it has no
differential coefficient, and for all other values of a, ' (x) =0.
THE SUCCESSIVE DIFFERENTIAL COEFFICIENTS OF A CONTINUOUS FUNCTION.
209. If a continuous function f(x), defined for the interval (a, b), have
at every point a differential coefficient f'(x), which is itself continuous
throughout the interval, the function f' (x) may itself have a differential
coefficient f"(x), which is called the second differential coefficient or derivative of f(x).
* Cantor, Acta Mat. vol. iv, p. 386. See also Scheeffer, Acta Mat. vol. v, p. 289.
+ Grave, Comptes Rendus, vol. cxxvII, p. 1005.
18-2




276


Functions of a real variable


[CH. IV


The second differential coefficient of f(x) at a point x1, when it exists, is
expressible as a repeated limit
li    lim f(x +  +  ) -f (1 + h)-f ( +  ) +f ()
k=O h=O                   hk -
in which the limit for h = 0 is to be first obtained, and then the limit
for k = 0.
The existence of the repeated limit as a definite number does not necessarily imply the existence of f" (x).
The above definition of f"(x) is applicable at any point xi for which
a neighbourhood (x - e, x + e') exists, such that f'(x) exists everywhere
in that neighbourhood and is continuous. The continuity off' (x) is however
not sufficient to ensure that f" (x) exists.
When f'(x,) has a definite value, butf'(x) fails to possess a definite
value at some or all of the points of the neighbourhood (x1 - e, x, + e), it
may happen that the ratios
D+ f (x + k) -f' (x1)     D+f (x + ) -f' (xi)
k          '              k         '
D-f(x (+ f) -f' (x)       Df(x1 + k) -f' (xi)
all have the same limit for k= 0. In this case we may regard this limit
as defining f"(xi); and thus this extended definition is applicable to cases in
which f'(x) exists at the point x1, and at some only, or at none, of the
points in any neighbourhood of x1, however small that neighbourhood may
be chosen.
210. If in an interval, which contains in its interior the point x,, the
differential coefficient f' (x) off(x) everywhere exist, and be continuous through
the interval, and if further the second differential coefficient f"(x) exist
throughout the interval, being at every point either finite, or infinite with a
definite sign, and be finite at the point xl, then f" (x1) is the limit, when h = 0,
of either of the expressions
f(x1 + h)- 2f (x) + f (x- h)
hA2
f(x   2 + 2h)- 2f (x1 + h) +f(x)
h2
The converse does not hold; for either of these expressions may have a
definite finite limit at h = 0, and yet f" (xi) may not exist, or even f' (x1)




209-211]


Successive differential coefficients


277


may not exist. An illustration of this is the case of the function defined by
f(0)=0, f(x)=x sin'- for 2> 0; at the point 0, f'(0) has no existence,
and yet
lim f(h)- 2f(0) +f(- h)   O.
h=O          h2
To prove the theorem, we may take (x - e, xi + e) as the neighbourhood
of the point xi, through which f'(x) is continuous and f "(x) everywhere
exists. Suppose f"(x) = k; and let us consider the function 0 (x) =f(x) - kx2,
which has similar properties to the function f(x); and thus 4" (1) = 0.
If h < e,         (x + h) -  (x) =  h' (x1 + Oh),
(x1 - h) -  (xi) = - ho' (x1 - 0ih),
where 0, 06 are proper fractions; again
4' (x~ + Oh) - S' (xi) =  OhA" (x1 + 002h),
' (x1 - 0,h) -  ' (xi) = - O1h" (x1 - 0 03h),
where 02, 03 are proper fractions. We find from these results
4) (x + h) - 2) (xi)+ ) (x, - h)0h
h^+ 2al+  l^ = 0i" (x +   62h) + 010" (x - 6,0,h).
Since S" (x1) exists and is zero,
' (, + Oh)-<' () -        (xi- 0h) -  ' ()
Oh        '            -0, h
both have zero as limit, for h= 0; hence the same is true of " (xI + 002h),
4" (x- - 0 0 h), as is seen from the formulae above.
It has thus been shewn that
lim ) (x, + lh)- 20 (x) + Sb (, - h) = 0
h=O             h2
which shews that
limf(Xl + h) - 2f (,) +f(x - h)   =f" (xI).
h=O             h2
A similar proof establishes the theorem which relates to the other limit.
211. The following theorem, due to Schwarz*, is of fundamental importance in the theory of Fourier's series.
If, in an interval (a, /) in which f(x) is continuous, the expression
f(x +h) - 2f (x) tf (x - h)
h2
converge for each value of x in (a, /3) to the limit zero, for h = 0, then the function
f (x) is a linear function in the whole interval, and consequently f' (x), f" (x)
everywhere exist, and the latter is everywhere zero.
* Crelle's Journal, vol. LXXII.




278


Functions of a real variable


[OH. IV


Let us consider the function
( =      ) -f(a) -     [f (/)-f (a)]} + c2 ( - a) (x  3),
where k is a constant. The function  ((x), whichever sign be taken, is
continuous in (a, /3), and vanishes at a and 8. We find at once
lim ( (x + h) - 2( (-) + b (x - h) = 22
h=o            h2
and therefore, for each value of x in (a, 8), a positive number e can be found,
such that ( (x + h) - 20 (x) + ( (x - h) is positive and greater than zero for
all values of h which are numerically less than e.
If ( (x) could be anywhere positive in (a, f/), there must be a point xa
at which it has the greatest positive value, and this point is not a nor /3,
since ( (a), (,/) both vanish. If r be sufficiently small,
j (i, + 97) - ( (XI): 0,  ) (x, - r) -  ( 1) ~ 0,
hence               ( (x1 + V) - 2(t (xi) + < (x - V)
would be, for all sufficiently small values of V7, either negative or zero, which
is contrary to what was shewn above. It follows that ( (x) is everywhere
negative in (a, 3), and cannot be zero except at a and /3.
This holds whichever sign be taken in defining (O(x). Now k2(x - a) (x- /)
is always negative except at a and /, and may be taken to have its
numerically greatest value as small as we please, since k is at our choice.
It follows that
x a
/(^, -  () -   [f(/3)-f(a)]
f () -f (a) /-  a        -f ()]
can nowhere in the interval be different from zero; for, if at any point it
had a value p, by choosing k such that k2 (x - a) (x - 3) is numerically everywhere < p, the function ( (x) could be made positive at the point by proper
choice of the ambiguous sign. It has thus been shewn that f(x) is linear
in (a, /).
212. Schwarzqs theorem can be extended to the case in which there is
an enumerable set of points in the interval (a, /3), at which it is not known
that the limit in question exists, or is zero, provided a certain condition be
satisfied at each point of the enumerable set.  The following theorem will
be established:
If, in an interval (a, 3) in which f(x) is continuous, the expression
f (x + h)- 2f () +(x - h)
h2
converge for each value of x in (a, /3) to the limit zero, for h = 0, except




211, 212]


Successive differential coefficients


279


that for an enumerable set of points G this is not known to be the case, then,
provided that at each point x of G the expression
f ( + h) - 2f (x) +f(x- h)
h
converge to the limit zero, for h = 0, the function f(x) is a linear function in
the whole interval (a, /3).
It should be observed that the condition
lim f (x + h) - 2f/() f (x - h)  0
h=O            h
is certainly satisfied at any point x at which the differential coefficient f'(x)
exists and is finite.
To prove the theorem, let it be assumed that
f() -f (a)-     a {f(/3) -f(a)}
has a positive value p at some point xa interior to (a, /); and let
b (x, k) =f(x) -f(a) -  _  { (/3) -f(a)} + k(x- a)2,
where k is a positive number. We have
cb(a, k)=0, b(), k)=k(/3-a)2, and sb(xl, k)=p+k(xl-a)2;
and hence, providedJ
k <        p        =K(3 - a)2 - (xI - a)2
the number f (x,, k) is greater than b (/, k), and than q (a, k). We shall
suppose k to be so chosen that this condition is satisfied; it then follows
that ( (x, k) has a maximum between a and /.   The absolute maximum
value of b (x, k) may be attained once, or a finite number of times, or an
infinite number of times, in the interval (a, /3).
The points x at which this maximum is attained have an upper extreme
x(<,), which must itself be a point at which the maximum of b (x, k) is
attained, as is seen, in the case in which x is an upper limit, from the
condition of continuity of the function. We have therefore
(x +h, k) — (, k)   0, and    (x-h, k) -    (x, k)  0,
if h be sufficiently small; from which we conclude that, in case
lim  (c + h, k)- 20 ({x, k) + q (x- h, k) exist,
h=O               h
its value is < 0.  It follows that x must belong to G; because the value of
this limit is 2k, and therefore > 0, for any point which does not belong to G.
Since x is a point of G, we have
lim { (x + h, k -  (   ),+  (x - h, k) - p (X, k)} _0;
h=O          h                     h




280


Functions of a real variable


[CH. IV


and since the two fractions have the same sign, it follows that
im     ~ (h + h, k)- b (i, k) _  and li  ( - h k)- d (  = k)
h=O         h                    h=O        h
From this result we deduce that
lim /f(  + h)-.f()  = lim j (- h) -f (~) X2 (-13)  - (a).
h=O       h         h=O     -h            3-a
To each value of k in the interval (0, K), there corresponds one value of a,
and it is impossible that the same value of x can correspond to two different
values k1, k2 of k.  For if this were the case, we should have
1i (X - a)= 12 (c - a),
and therefore k1 =k2, since ~ >a.  Now it is impossible that the set of
points k interior to the interval (0, K) can be such that to each such point
there corresponds a distinct point x belonging to the enumerable set G. We
conclude that it is impossible that
f (x)-f(a) -      {f( () -f(a)}
/ -a
can have a positive value p at any point x, of the interval (a, /); and it can
be shewn in a similar manner that there can be no negative value of the
same function in the interval. It follows that the function must everywhere
be zero, and therefore that f(x) is linear in the interval (a, /).
213. Let us suppose that a continuous function f(x), defined for the
interval (a, 8/), is such that, in every interior point of any sub-interval
belonging to an everywhere-dense set of sub-intervals, the condition
lim f(x + h) - 2f(x) +f ( - h) =  is satisfied.
h=             h2
It follows from the theorem of ~ 211, that in any one of the sub-intervals
f(x) is a linear function of x; and thus the value of f(x) in a sub-interval
(an, bn) is a linear function Anx+Bn. The set of sub-intervals is complementary to a closed set of points G which is non-dense in (a, /3). In case
this closed set G be enumerable, each interval (an, bn) abuts at each end
on another interval of the set; thus we may suppose that (an, bn) abuts on
(an', bn) at the end bn, so that bn = an',
Let us now assume that, at each point of G, the condition
lim f(x + h)-2f(x)f(- h)     0 is satisfied.
h=O            h
We have then, at the point x = b = an,, by applying this condition, and also
the condition of continuity of f(x),
Anbn + Bn = An,bn + Bn,, and An,- An = 0;




212-214]


Oscillating functions


281


from which we deduce that Bn = Bn, and thus that the linear functions
Ax + B,, Ax + Bw
are identical. Therefore it follows that, in case G be a non-dense enumerable set, the function f(x) must be a linear function Ax + B in the whole
interval (a, /3). This is, in fact, a particular case of the theorem of ~ 212.
The condition lira ff(x + h) - 2/(x) +f (x-h)
The condition lim f  +    h)             = 0 being certainly satisfied
h=O            h
at any point x at which f (x) has a finite differential coefficient, we therefore
obtain the following theorem:If f(x) be a continuous function possessing everywhere in the interval (a, /3)
a finite differential coeficient, and the function be linear in each one of an
everywhere-dense set of intervals complementary to an enumerable closed set of
points G, then f(x) is a linear function in (a, 3).
If the closed set of points G were unenumerable, the preceding reasoning
would no longer be applicable, except that, at an isolated point of G, it would
establish that the linear functions in the two intervals which abut on one
another at the isolated point must be identical. Confining therefore our
attention to the case in which G is a perfect set, we see that a continuous function possessing everywhere a finite differential coefficient may
exist, which is linear in each sub-interval complementary to a non-dense
perfect set of points contained in the interval for which the function is
defined, and yet the function need not be linear in the whole interval.
The existence of such functions will be effectively established in
Chapter v, where it will be shewn that they may be obtained by the integration of continuous functions which have an everywhere-dense set of
lines of invariability.
OSCILLATING CONTINUOUS FUNCTIONS.
214. Let us suppose that the continuous function f(x) has no lines of
invariability in the interval (a, /), and that everywhere in this interval it
has a finite differential coefficient. If within (a, 8/) there be a maximum or
minimum of f(x), then at such a point f' (x), which exists and is finite, must
be zero. If the maxima and minima in (a, /3) be everywhere-dense, then
f' (x) vanishes at every point of the everywhere-dense set; and if f' (x) were
continuous throughout (a, 83) it would follow that it was everywhere zero,
which would be contrary to the hypothesis that (a, /3) is not a line of
invariability.
It follows from this that if in an interval (a, /3), which contains no lines
of invariability of the continuous function f(x), the differential coefficient f' (x)




282


Functions of a real variable


[OH. IV


everywhere exists and is continuous, there must be in the interval an everywheredense set of sub-intervals in each of which the function is monotone.
We have further the following theorem:If f(x) be continuous in (a, /3), and have no lines of invariability, but have
an everywhere-dense set of maxima and minima, there must be in the interval
an everywhere-dense set of points at each of which f' (x) either does not exist,
or does exist and is discontinuous.
A continuous function f(x), which in a given interval (a, /3) has no lines
of invariability, but has an everywhere-dense set of maxima and minima,
is said to be a continuous function which is everywhere-oscillating in the
interval (a, /3). Such a function cannot have a differential coefficient which
is continuous throughout the interval.
The continuous functions which are everywhere-oscillating in an interval
may be divided into two classes.
(1) The function may be such that, if the constants 1, mz be properly
chosen, the function f(x) + lx + m is monotone in the interval. In this case
f(x) is expressible as the difference of two monotone functions, and thus
belongs to the class of functions with limited total fluctuation. These
functions may be said to be of the first species, or to be functions with
removable oscillations.
(2) Such functions as do not belong to (1) may be said to be of the
second species, or to be functions with irremovable oscillations.
In order to bring to light the essential distinction between the two classes
of functions, as exhibited by the properties of their derivatives, we first of
all remark that, if D+f(x) have a positive lower limit c for all points x in the
interval (a, /), then at each point f(x + h) -f(x) is essentially positive for all
positive values of h which are less than some number 8 dependent on x;
hence the function is monotone in the interval. The function would also
be monotone in case the specified condition were that D+f(x) has a negative
upper limit for all values of x in (a, /).
Now suppose that D+f(x) has a definite negative lower limit in (a, 3); let
this be - c, and consider the function b (x) =f(x) + lx + m, where 1 > c; we
have then D++ (x) = 1 + D+f (x) >_ I - c; hence the function <b (x) is monotone
in (a, /). Thus f(x) is expressible as the difference of the two monotone
functions  (x) and lx + m. Similarly, if we had taken the condition that
D+f(x) has a definite positive upper limit c, the function f(x) + lx + m, where
I <- c, could be shewn to be monotone.
It is clear that instead of the linear function lx + m we might have used
any continuous differentiable function whose differential coefficient was > c,
or <-c, throughout the interval, in the two cases.




214, 215]


Oscillating functions


283


The argument would have been unaltered if it had been assumed that
there were a finite or infinite set of lines of invariability in (a, 8/).
It has thus been shewn that:If the continuous function f (x) be such that either D+f (x) has a negative
lower limit for all values of x in (a, /3), or that D+f (x) has a positive upper
limit, then all maxima and minima of the oscillating function f(x) are removed
by adding to f(x) a properly chosen linear function, and thus the function is
of the first species, and is of limited total fluctuation.
In particular, the conditions of the theorem are satisfied if the derivative,
on one side, without necessarily having a definite value at any point, be such
that for the whole interval it is numerically less than some fixed positive
number.
A function, such that for a given interval,
D+f(x),    )D+f(x), 1 D-f(x),  D_-f(x) 
are all less than some fixed number, is said to be a function with limited
derivatives. Such a function has a limited total fluctuation in the interval,
and if it be everywhere-oscillating, it is of the first species.
A function with limited derivatives is necessarily a continuous function,
but the converse does not hold.
In the general case, one of the derivatives on the right may at some or
all of the points of the interval have indefinitely great values, this derivative
being the same one for all such points.
If neither D+f(x) have a definite negative lower limit, nor D+f(x) have
a definite positive upper limit in the interval, and the function be an everywhere-oscillating function, then it is of the second species.
215. Let us suppose that, for a set of points G, everywhere-dense in
(a, b), the derivative of the continuous function f(x) is infinite, but not of
fixed sign, i.e. the derivatives at a point of G on the right and on the left exist,
and are infinite, but of opposite signs. At any point x, of G, a neighbourhood
can be found, containing x,, such that for any point x in it f(x) -f(xo) is
of fixed sign for the whole neighbourhood, and is never zero except when
x = x0; it follows that x0 is a proper maximum or minimum of the function.
It will be shewn that, in any interval (a, /3) contained in (a, b), there are
an infinite number of points at which the function has the same value. Let
*  be a maximum point of f(x) within (a, B), and let (I -a, + e) be the
greatest interval enclosing ~, for which f(x) -f (~) is negative; suppose that
the absolute minimum of the function for this interval is in (I -, ~); taking
a maximum point x in the interval (~, 4 + e), then in ( - 7, I) there is
a point If' at which f(') =f(l,), since f(~,) lies between the greatest and
least values of the continuous function in (I -, I).




284


Functions of a real variable


[CH. IV


Now there is a maximum interval (, -W,,+ el) for the point 1:, and
this lies within (, +e); and in this interval we may as before find a
maximum point:, such that a point:,' also exists within the interval, for
which f(:2) =f(.2'). There is also a point 2" in ( -V,:), such that
f(V:") =/f( (') =f (~2).
We may proceed in this manner, until we find n points
~t.to   01 n - 1)
9n-i1,  n-1, 'n- '_
such that          f (.n-) =f('-) =     =f(n - )).
Now let, be a limiting point of t:, 2, ~3...,...; and let,' be a
limiting point of i', L',..., and," be a limiting point of /", 3",...; then
f/() =f/(    w') =f(") =
Thus the points,, A', ",... form an infinite set in (a, f/) at which
the functional values are the same.
The points,,,', d'',... have a limiting point 0o at which the functional
value is the same as for the set itself; therefore
f()o) -f(E ) _f(o) -f(lzL) _ =   o
to0- to       o0-.s'
hence at:o either the derivative is determinate and equal to zero, or else
it is indeterminate with zero. lying between its upper and lower limits.
Thus it has been shewn that*:If the continuous function f(x), have an everywhere-dense set of points at
which the derivative is infinite but not of fixed sign, there is an everywheredense set of points at each of which the derivative is either indeterminate or
else zero. Thus a continuous function cannot at all points have a derivative
which is infinite and not of fixed sign.
If we apply the above theorem  to the function f(x) -cx, where c is
a prescribed constant, then, since f(x) - cx has an infinite derivative at the
same points as those for which f(x) has an infinite derivative, we obtain
the following theorem:If the continuous function f(x) have at an everywhere-dense set of points
a derivative which is infinite but not of fixed sign, there is an everywheredense set of points at each of which the derivative either has the prescribed
value c, or is indeterminate, and such that c lies between its upper and lower
limits.
* Kinig, Monatshefte f. Math. u. Physik, vol. i. The above proof is that given by Schonflies,
Bericht, p. 160.




215, 216]


Properties of derivatives


285


GENERAL PROPERTIES OF DERIVATIVES.
216. A large number of properties of the derivatives of special classes of
functions, chiefly belonging to the oscillating, or to the monotone, continuous
functions, have been given by Dini and other writers; the most important
of these will be given here.
The following general theorem, due* to W. H. Young, includes as a special
case a theorem for continuous functions due to Brodent.
The points at which one at least of the four derivatives of any given
function is infinite, form an inner limiting set. The set of such points is
accordingly of power c, when it contains a component dense-in-itself; and
otherwise it is enumerable, or finite, or zero.
It follows that the set of such points is a set of the second category in
case it be everywhere-dense.
Let x0 be a point at which one of the four derivatives is infinite, it being
immaterial whether the other derivatives are infinite or finite. A sequence
xi, x2,... xn... converging to x0, and on one side of it, can be found, which has
the property that, corresponding to an arbitrarily large positive number a,
an integer m, can be found such that
f (X) -f (x0) > oa, for n > m;
Xn - Xo
further, m' can be chosen so great that
Ixn-xo <,    for n > m' m,.
Let the intervals (xa,, xCn) be prolonged on the side beyond x0, each being
increased by — 1  of its length; and the whole set of intervals so constructed
1r -1
for every point x of the set at which a derivative is infinite, may be called I,.
Let a,, oa, -3... be a set of values of a- which increase without limit; then
the corresponding sets of intervals I.,, I,,,... define an inner limiting set of
points, to which all the points x of the given set belong; and it will be shewn
that no other points belong to this inner limiting set. If possible let I be a
point of the inner limiting set which does not belong to the given set of points
at which a derivative is infinite. There is at least one interval of each of the
sets I,, Ira,... such that M is an interior point of it; let such intervals be
81,,2..., and let b:, _2,...* be points of the given set interior to these
intervals. Let 1,, f2, 3... be the end-points of the intervals on the sides of
those intervals which were not lengthened. We have
(Arkir  atemat, Ast c, - ol., Sto, 
* Arktivfr Matematik, Astronomi och Fysik, vol. I, Stockholm, 1903.
Jr Acta Univ. Lund. vol. xxxIII, p. 31.




286


Functions of a real variable


[CH. IV


thus the points:  2,,... and also the points L, 2,... form a sequence of
which I is the limit.
Since  f(:') -f() > -,, therefore |/f(^ )-f(,) > (a - 1) L.
Also a positive number A can be determined, such that for all values of t,
If  -(f - ) <A,
for otherwise: would be a point with an infinite derivative; and from this
we see that
I f()-f(   1 <A,.
For a sufficiently great value of L,
ar-I>A;
hence for such a value of t,
If(6)-f(f) I > (, - A - 1) 8,
and thus              (>) -f()  >   - A - 1
Now a, - A -1 is arbitrarily large for a sufficiently great L; hence, since 5 is
the limiting point of the sequence {iJ, there is an infinite derivative at:,
which is contrary to the hypothesis made; therefore the points of the
given set constitute the inner limiting set which has been defined.
217. If x1 be a point of the interval (a, b) in which f(x) is defined,
f(X)o -f(xc )
the function( -f(       for points x such that xI < x _ b may be called the
X - X
incrementary ratio at x, on the right; and in case f(x) be a continuous
function, this incrementary ratio is also continuous at every point of its
domain. This incrementary function has an upper and a lower limit for its
whole domain (xI < x < b); and these upper and lower limits may be denoted
by U(x,), L (x), and either of them may be finite or infinite; however U (x,)
can only be infinite with the positive sign, and L (x,) only with the negative
sign. U (x,), L (x,) being regarded as functions of x1, defined for every point
of (a, b) except the point b, the function U(x,) has a finite or infinite upper
limit for its whole domain, which we denote by U; and the function L (x,)
has a finite or infinite lower limit for its whole domain, which we may
denote by L. There exist therefore two finite numbers U, L, which may
have the improper values + oo, - oo respectively, such that
f (x2)-f (,)
X2 - X1
for every pair of values of x1, x2, where x2 >, always lies between them, or
is equal to one of them.




216-218]


Properties of derivatives


287


The incrementary ratio on the left of a point can be defined in a similar
manner; and we thus define two functions U' (x1), L' (x,) at xi, as the upper
and lower limits of these incrementary ratios.
It is easily seen that U', the upper limit of U' (x) in the interval (a, b), is
identical with U, and that L' the lower limit of L'(x) is identical with L.
Thus U, L are the upper and lower limits of
f (x2)- f (xI)
X2 - XI
for every possible pair of points (xi, x2) in the interval (a, b).
218. Let f(x) be continuous in the interval (a, b), and let U and L be
the upper and lower limits of the incrementary ratios above defined. Take
(a, /?) any interval in (a, b), and consider the function
( (x) =f(x) -f(a)      [f () -f(C)].
Since ( (a) = 0, ( (/) = 0, unless (p (x) be constant through (a, /) there must
be within (a, /3) a maximum or minimum of (a, /); and thus at least one
point xa exists within (a, /3) such that
+ (1 + Ah) - ( (xi) C 0,
for all sufficiently small values of h, or else
(xI ~ h)- ( (xi)  0,
for all sufficiently small values of h. At such a point
f (1 + h) -f (XI) f (/) -f (a)
h       =       /3-a
nf (x - h) -f (x) >f(/) -f(a)
and                     -A =h /3-a
or else           f(1 + h) -f() (  f(/3)-f(a)
h        =    3-a
f (x- - h) - (x,) <f(/)-f(a)
and 
-h             - /-a
If ( (x) have an infinite number of maxima and minima in (a, /), there are
in (a, 3) an infinite number of points at which the first of these conditions for
( (x) holds, and also an infinite number at which the second holds. If there
be only a finite number of maxima and minima of < (x) in (a, /), then this
interval can be divided into a number of portions in each of which the
function (p (x) is monotone; and in any one of these portions either
f (x + h) -f(x) f(,) -f(a)
+h      --   /-a




288


Functions of a real variable


[OiH. IV


at all points within the sub-interval, or else
> f(/) -f(a)
=    /-a 
for every x within the portion, and for sufficiently small values of h. Now let
U, L be the upper and lower limits of f(x2) -   (x) in (a, b), then
f() -f(a) and f(x + h) -f(x)
/ -a               +h
lie between U and L. Thus, in every interval (a, /) contained in (a, b), in
which b (x) has an infinite number of maxima and minima, there are (1) an
infinity of points x for which f(x + h) -f(x)  for all sufficiently small
values of h, lies between L and f (3) f(a); and (2) an infinity of points
/3-a
for which the same is true of f(x - h) -f(x); (3) an infinity of points
-h
for which f(x + h) -f(x)
for which      h /( +  -, for all sufficiently small values of h, lies between
U and f() -f(a); and (4) an infinity of points for which the same is true
/3-a
-h
In case f(x) -f(a) -     [f (/) -f(a)] have only a finite number of
maxima and minima in (a, /), there are in (a, /3) finite intervals such that
all the points in one of them belong to both the sets (1) and (2), and also
finite intervals in which all the points belong to both the sets (3) and (4);
each of these sets of intervals is finite, and an interval of one set is followed
by one of the other set.
The number L being the lower limit of the function L (x) in the interval
(a, b), there exists a point xa such that L is the lower limit of the values of
L (x) in any arbitrarily small neighbourhood of x1; and it follows that in such
neighbourhood of x, there are points f such that f( + h) -    for an
infinity of values of h, differs from L by less than a prescribed positive
number e. Therefore there are in (a, b) an infinity of pairs of points (a, /) one
of which is arbitrarily near x, such that  () -f(a) differs from L by less
than e.
Similarly it may be shewn that in (a, b), there are an infinity of pairs of
points (a, /) such that f  -f(a) differs from U by less than the prescribed number e.




218]                 Properties of derivatives                   289
If U or L be infinite, there exists an infinity of pairs of points such that
f () -f (a) is arithmetically greater than a prescribed number c, and has the
/go-a
same sign as the infinite U or L. We can consequently choose the interval
(a, /) so that
f (/) -f (a) = L + (,
f(,)f(a)        +-.
or else so that         f () f (a      U-1,
where  provided U and L -a
where 7 < 6, provided U and L are finite. If one or both of U, L be
infinite, (a, /3) can be so chosen that f(A) (a) has the same sign as U
/3 - a
or as L, and is arithmetically greater than a prescribed positive number c.
We have now obtained the following results:f f (x) be a continuous function, and (a, b) be the whole or a part of its
domain, to which U and L correspond, then (1) if L be finite, there exists in
(a, b) an infinity of points for which both D+(x), D+(x) each lie between L and
L + e, where e is an arbitrarily prescribed positive number; and at these points
D+(x), D+(x) are either equal, in which case a derivative on the right exists, or
else they difer from one another by less than e: (2) if U be finite there exists in
(a, b) an infinity of points for which D+(x), D+(x) each lie between U and
U-; and at these points there exist derivatives on the right, or else D+(x), D+(x)
differ from one another by less than e; (3) if U or L be infinite there exists
an infinity of points at which D+(x), D+(x) are both numerically greater
than an arbitrarily great number c, and have the same sign as the U or L
which is infinite. A similar statement holds as regards the derivatives on
the left.
The above is true irrespectively of the number of the maxima and minima
off (); but iff (x) have in (a, b) only a finite number of maxima and minima
and if the same be true of all the functions f(x) - Ix- m, obtained by the
addition of a linear function, then there exist in (a, b) finite sub-intervals such
that at all points in one of them the above statements hold both as regards
the derivatives on the right and as regards those on the left. The numbers
U and L correspond in each case to the particular sub-interval.
It will be observed that the theorem does not assert the necessity of the
existence of points at which a determinate derivative on the right or on the
left exists, but it states that there are in every sub-interval points at which
the difference between the upper and lower derivatives on one side is less
than a prescribed arbitrarily small number, or else at which both such
derivatives are arithmetically greater than an arbitrarily fixed large number.
There are therefore certainly points in every sub-interval at which there is,
H.                                                         19




290


Functions of a real variable


[CH. IV


so to speak, an arbitrarily near degree of approximation to the existence of a
finite or infinite derivative on the right, and also points at which the same is
true as regards derivatives on the left.
219. It will now be shewn* that, for a continuous function, of which
(a, b) is the whole or a part of its domain, the upper limit of each of the four
derivatives Df (x), D+f(x), D-f(x), D_f(x) for all values of x in (a, b) is U
the upper limit of the incrementary function in (a, b), and that the lower limit
of the four functions is L. If U and L be both finite the function belongs
to the class of functions with limited derivatives.
A function with limited derivatives accordingly satisfies the condition, that
for every pair of points xi, x,, If(x0) - f (%2 ) < kx I - x2 1, where k is a fixed
positive number. It has been pointed out in ~ 214, that such a function
belongs to the class of functions of limited total fluctuation.
It is clear that the upper limit of each of the functions D+f(x), D+f(x),
D-f(x), D_f(x) is a number which cannot be greater than U. Now since it
has been shewn that points exist in (a, b) such that, if e be an arbitrarily prescribed number, both D+f(x), D+f(x) differ from U by less than e, when U is
finite, and are arbitrarily great if U is + oo, it follows that U is in either case
the upper limit of D+f(x), D+f(x). In a similar manner it can be shewn
that U is the upper limit of both D-f(x), D_f(x). The proof that L is the
common lower limit of the four functions is exactly similar.
Each of the four expressions D+f(x), D+f(x), D-f(x), D_f(x) may be
regarded as a function defined for the whole domain of f(x) except at one of
the end-points; but in this case we have to extend the ordinary definition
of a function so far as to admit infinite functional values, instead of only
infinite functional limits as in the case of an ordinary function. It will be
convenient to say that, at a point at which one of the above functions is
infinite, it is also continuous, provided the functional limits of the function in
question, on either side, are definitely infinite and of the same sign as the
functional value at the point.
If, at any point xo, interior to (a, b), one of the above functions, say
D+f(x), be continuous, then at that point the other three functions are also
continuous, and are equal in value to D+f(xo), and thus there exists at Xo, a
differential coefficient.
To prove this, take any interval (x, - e, x, + e); then all four functions
have in this interval the same upper limits, and also the same lower limits.
If D+f (x) be finite, the upper and lower limits of D+f(x) in (x, -, x0 + e)
each differ from D+f(xo) by less than a number q which depends on e in
such a way that, as e is indefinitely diminished to the limit zero, rj also
diminishes to the limit zero. Since all four functions have the same upper
* Du Bois Reymond, Math. Ann. vol. xvi, p. 119, also Scheeffer, Acta Mathematica, vol. v, p. 190.




218-220]


Properties of derivatives


291


limit and the same lower limit in (x - e, xo + e), the upper and the lower
limits of each differ from D+f(xo) by less than 7, and X can be made as small
as we please by taking e small enough. It follows that all four functions
are continuous at x0, and that all four at x0 are equal to D+f(xo); and thus
there exists a differential coefficient at x0.
In case D+f (x0) is + co, e can be so chosen that in (x0 - e, xo + e), D+f(x)
is everywhere greater than an arbitrarily large chosen number c, and
the upper and lower limits of each of the four functions are then greater
than c; by taking a succession of values of c which increases indefinitely, and
considering the corresponding sequence of values of e which converges to
zero, we see that each of the functions D+f(x), D-f(x), Df(x) is infinite at
x0, and is continuous in the extended sense of the term, at that point;
there is then a differential coefficient at x0 which is infinite and of definite
sign.
It follows that, if it be known that any one of the four derivatives is everywhere continuous in an interval, there exists everywhere in the interval a
differential coefficient in the ordinary sense of the term..
220. The derivatives D+f(x), D+f(x) of a continuous function are at
any point Xo, such that a < X0 < b, either both continuous on the right, or both
of them have a discontinuity of the second kind on the right; but they cannot
have ordinary discontinuities on the right.
A similar statement holds as regards the continuity or discontinuity of
D-f(x), Df(x) on the left.
Suppose that D+f(x) has at the point Xo, a limit X at x, on the right;
then if 8 be a prescribed positive number, an interval (x,, x0 + e) can be
found, such that D+f(x), for every point of this interval, except x,, lies
between X + 8, and X - 8. The upper and lower limits of each of the four
derivatives D+f(x), D+f(x), D-f(x), D_f(x) for any interval (o + e, x0 + e),
where e, < e, must all lie between the values X + 3, X - 8; hence the upper
and lower limits of D-f(x) for the interval (x,, x, + e), lie between these
same values, the function D-f (x) being regarded as undefined at the point
X0; and these upper and lower limits of D-f(x) are the same as those of
D+f(x), D+f(x) for (x,, xo + e), the point x0 being included. It follows that
D+f(xo), D+f(xo) both lie between X +  and X -; and as this holds for
every value of 8, we must have
D+f (xo) =    I+f (Xo)= X = X;
where X' denotes the limit of D+f(x) at Xo on the right; and thus D+f(x),
D+f(x) are both continuous at x, on the right. If X=+ oo, then in the
interval (x,, x, + e) at every point except x0, D+f(x) > c, where c is an arbitrarily chosen number on which e depends; the argument then proceeds
as before.
19-2




292


Functions of a real variable


[oH. IV


221. If a continuous function f(x) be such that, of the functions
(b (x) =f(x)- lx, where I has all values, none, or only a finite number, or
only an infinite set which does not fill any interval, however small, have
more than a finite number of maxima and minima in sufficiently small
neighbourhoods on the right of Xo, then the function f(x) has a derivative on
the right at xo, either finite or infinite.
If f(x) - lx have only a finite number of maxima and minima in (x,, X0 + a),
an interval (x,, x,+ e,) can be found, in which the function is monotone.
Since then, at every point x0 + h in such an interval,
f(xo -+ h)-f(x,)- lh  O,
or else at every point
f(xo + h)-f(x0) -lh   0,
it follows that D+f(xo), D+f(xo) are either both _ i, or else both - 1, and
this holds for a set of values of 1, everywhere-dense in any interval whatever.
If D+f(xo), D+f(xo) were unequal, we could find a value of 1 which is
between the two and is not one of the exceptional values of 1, and this
would be contrary to what has been proved; hence D+f(x) = D+f(xo), or
f(x) possesses a derivative d(xo) on the right at x0. This derivative is
finite if some of the functions f(x) - x increase and others decrease on the
right of x0; otherwise it is infinite.
For any point x within (x,, x0 + e1), we have, for a sufficiently small
value of h,
f( + h)-f(x),
+h       -
if I be less than the derivative at x, on the right, and
f(x+h)-f(x) _t
+h
if 1 be greater than the same derivative d (x0). If d (x0) be finite, it follows
that f(x + h) -f(x)
that /(-+)-) lies between d (x,) + o-, d(xo) - a, where       - is an
arbitrarily small number, provided x - x  and h be sufficiently small. If
d(xo) is oo or -o, f( + h) -f () lies between c and +oc, or - c and
t+ h
- o, where c is an arbitrarily great positive number.  In either case
therefore the four functions D+f (x), D+f(x), D-f(x), D_f(x) for points
within (x,, Xo + e1) have the same limit at x0, viz. d (x). It follows, as
a particular case, that if, under the conditions of the above theorem, a
derivative on the right exist at all points in an arbitrarily small neighbourhood on the right of X0, the derivative on the right at x0 is a continuous
function on the right. If derivatives on the left exist at all points in a
neighbourhood of Xo on the right, with the possible exception of X0 itself, these
derivatives have, at the point xo, the derivative d (x0) as their limit..




221-223]


Properties of derivatives


293


222. A continuous function f (x) cannot have, at every point of a whole
interval, a single-valued derivative on the right, which is everywhere infinite and
of the same sign.
For if f(x) had this property in an interval (a, /), so also would
f () -f(a) -       f (3)- f (a)],
and this function necessarily has a maximum or minimum within (a, /3),
which is contrary to the condition that it has a derivative on the right
which is always of the same sign; for this involves the condition that the
function must constantly increase as x increases from a to /3.
Let us now suppose that the continuous function f(x) has at all points
of (a, b) single-valued derivatives on the right (finite or infinite), such that,
in a part (a, /) of (a, b), this derivative is continuous at least on one side; the
function f(x) is then such that, at an infinite number of points, it possesses
an ordinary differential coefficient.
The derivative d (x) on the right cannot at all points of (a, /3) be infinite.
For if we take a point x0 such that it is continuous on one side, in the
extended sense of the term explained in ~ 219, then if it were everywhere
infinite, its sign at all points in an interval on the one side of x0 would be
the same; but it has been shewn to be impossible that, everywhere in any
interval, d (x) should be infinite and of constant sign. It follows that there
are points in the neighbourhood of x, at which d(x) is finite. If x, be
such a point in (a, /), then, since d (x) is continuous on one side at x,, an
interval can be found at all points of which d (x) is finite, and also continuous
on one side. If (a1, /i) be such an interval in (a, /), then since d (x) is
everywhere finite in it, and continuous on one side at least, it is a point-wise
discontinuous function, if it be not continuous in (a,, /3i); and there must
therefore be an infinity of points in (a,, /,) at which d (x) is continuous. At
such points, in accordance with ~ 219, f(x) has a differential coefficient.
223. If we now collect the results obtained in ~ 221 and ~ 222, we can
state the following general theorem, applicable to functions which are in
general monotone, and also to a certain class of everywhere-oscillating
functions.
If a continuous function f(x) in a whole interval (a, b) be such that,
corresponding to each point x, there be of the functions f(x) - lx at most only
a finite number, or an infinite set for which the values of I do not fill any
interval, which, in an arbitrarily small neighbourhood on the right of Xo, contains
an infinite number of maxima and minima, and if the same condition be true
as regards an arbitrary small neighbourhood on the left of Xo, then the function
has at every point of (a, b) a definite derivative on the right, and also a definite
derivative on the left, and there is an everywhere-dense set of points at which
there is a differential coefficient.




294:                unctions of a real variable              [CH. IV
In every part of (a, b) there are finite intervals, in each one of which the
derivatives on the right and those on the left are both definite and finite, and
such that each of them is, in the interval to which it belongs, a point-wise
discontinuous, or else a continuous, function.
As regards everywhere-oscillating functions, the following remarks may
be made.
If a continuous function have in every neighbourhood on the right of
a point x,, an infinite number of maxima and minima, there are in such
neighbourhoods an infinity of points at which the derivatives on the right are
negative or zero, and an infinity of points in which these derivatives are
positive or zero. It follows from this, that none of the derivatives at a
point x on the right of x, can have a definite limit as x approaches the limit
x., unless such limit be zero.
In particular, if at all such points x, definite derivatives on the right and
on the left exist, these derivatives cannot be continuous at x0, unless the
derivatives at x0 are both zero.
If at the point x0, and at every point in a neighbourhood of x, which
contains an infinite number of maxima and minima, a differential coefficient
exist, which is continuous at x, this differential coefficient must be zero at
xo and at an infinity of points in the neighbourhood of x0, and must therefore
itself have an infinite number of maxima and minima in the neighbourhood
of x0.
If a function f(x), which has an infinite number of maxima and minima
in the neighbourhood of x0, have at x0, and in its neighbourhood, differential
coefficients of any number of orders, then they are all functions with an
infinite number of maxima and minima in the neighbourhood of x0, and all
of them vanish at x0, except that the one of highest order may be discontinuous
at x0, not then necessarily vanishing at that point. If differential coefficients
of all orders exist, they must all vanish at x,; and such a function is incapable
of expansion in powers of x- x0 in the neighbourhood of x0. An example*
of a function of this kind is
1        1
x2+ e (x-xo)2 sin
X - Xo
FUNCTIONS WITH ONE DERIVATIVE ASSIGNED.
224. If two functions, defined for a given interval, have each limited
derivatives, and if the two functions have one of their four derivatives, say
the upper derivative on the right, equal to one another at every point which
does not belong to a set of points E of measure zero, then the two functions
differ from one another by a constant, the same for the whole interval.
* Dini, Grundlagen, p. 314.




223, 224]  Functions with one derivative assigned


295


This theorem* differs from that of ~ 206, in the respect that the functions
are restricted to be such continuous functions as have limited derivatives; it
is however more general, in that E is not restricted to be enumerable.
Let the points of E be enclosed in the interiors of intervals of a set, of
which the total length has the arbitrarily small value e. To each point P
of E there corresponds an interval PP', where PP' is that part of the
interval of the set that encloses P which is on the right of P; these intervals
PP' may be denoted by 8'. If the two functions f (x), f2 (x) are such that,
at a point x1, D+f, (x,) = D+f, (x,), it has been seen in ~ 206, that D)+f(x) 0,
D+ f(x) _ 0, where f(x) denotes fi (x) - f2 (x). Since f(x) is continuous at
x1, it follows that there is a set of points xa + h on the right of x1, such that
I f(x1 + h) -f(xi)  eh; if we suppose h to have the greatest value for which
this holds, the interval (x1, x, + h) is an interval on the right of x1, and such
intervals may be denoted by 8.
Let f be any point such that a < =5 b, and consider the interval (a, f).
From the point a lay off an interval 8 or 8', according as a is not, or is, a
point of E; from the end of this interval lay off another interval 8 or 8', as
the case may be. Proceeding in this manner, we may either reach the point
I, after taking a finite number of intervals, or else we obtain an infinite set
of intervals, the end-points of which have a limiting point P,, which may, or
may not, coincide with I. In the latter case, we commence again to lay off
intervals on the right of P,, until we either reach I, or else until another
limiting point P,2 is obtained as the limit of a sequence of end-points.
Proceeding in this manner, the point f is certainly reached as the end of an
ordered sequence of intervals corresponding to a set of ordinal numbers
which comes to an end before some number of the first or of the second
class. The set of points not interior to the intervals is a closed enumerable
set. We can now find f(f) -f(a) as the sum, or limiting sum, of the
differences of the functional values at the end-points of the intervals which
have been defined, and each of which is either a 8, or a 8'. It is clear that
I f() -f(a) [ - eV3 + A  ' < e (- a + A),
where the summations refer to those of the intervals 8, 8' which have been
employed in the construction; and A denotes the finite upper limit of
f(xI) -f(x,) for every pair of points xa, x2 in the interval (a, F), and which
is identical with the upper limit of the absolute value of the derivatives
off(x) in the interval. Since e is arbitrarily small, it follows thatf(t) =f(a),
and therefore fi () -f2 () = i (a)-f2 (a); thus the theorem  has been
established.


* Lebesgue's Lefons sur Vintegration, p. 79.




296


Functions of a real variable


[CH. IV


THE CONSTRUCTION OF CONTINUOUS FUNCTIONS.
225. One of the most fruitful methods of obtaining continuous functions
which exhibit various peculiarities as regards the existence or non-existence
of differential coefficients at all the points, or at sets of points of their domain,
consists of defining the functions by means of series specially constructed
with a view to the purpose on hand; this method will be explained and
illustrated in Chapter vi. Broden, Kopcke and others have however given
direct constructions for continuous functions, which illustrate various possibilities in relation to the existence and properties of derivatives.
The method employed* by Broden is that of defining a continuous function
in the domain (a, b), as the function obtained by extension (see ~ 191) of a
function defined for an enumerable everywhere-dense set of points in (a, b),
the primary points. A continuous function is entirely determinate when the
functional values at such a primary set of points have been assigned. The
necessary and sufficient condition that a function defined for the primary set
should, by extension to the domain (a, b), give a function which is continuous
in that domain, is that the primary function should be uniformly continuous
with respect to the unclosed primary domain. To prove this, let {I} denote
the set of primary points, and {x} the set of secondary points; then the
condition that the function f(t) may be uniformly continuous with respect
to the domain {~}, is that, if t, be any point of {I}, and if X be a prescribed
arbitrarily small number, the condition If() -f(I) I < q be satisfied at all
points f which are such that I t-  < e, where e is a number dependent
on q7, but the same for all points i of [{}. Now assuming that this condition
is satisfied let x, be a secondary point, and let T:,:2,... h...,:', t',....
be any two sequences of primary points each of which has x, as its limit;
we have to shew that each of the sequences
f(f), f(/2.)... f(an)...,
f() f(8,     -. f(*/ ')...
converges to the same number, which will then be the single functional value
f(x'). Enclose x, in the interval (x, - -e, x, + 1e); then, from and after some
particular value of n, all the points of both sequences of values of ~ lie
within this neighbourhood. Let this value of n be m, then
If (W)-f (    )|+r)I < 7,
for all positive integral values of r; hence the first sequence of functional
values is convergent, since 1 is arbitrary; and similarly the second is also
convergent. Also for every VJ there is a definite m such that
I f(es+n.)-i f(' m+.) I  <;
* Crelle's Journal, vol. cxvIII; see also Acta Univ. Lund. vol. xxxIII.




225, 226]   Construction of continuous functions


297


hence the two convergent sequences have the same limit, and this limit
defines f(x.). We have now to shew that the single-valued function so defined
is continuous. We have
If(x1) -f(I:) I < V, provided I xi-  | < Ie,
and            If(x2) -f(2) < q, provided I x2-:  < ~;
also           If(2) -f(~,) | < 7, provided 1 2:-: 1 < e.
Hence it follows that   If(a) -/(xi) I < 37,
and this holds provided     x - x 1 < 2e,
for 1, ~2 can be taken to be between xI and x2; and therefore f(x) is continuous at xa, since 3,7 is at our choice. The extended f(x) is also easily seen
to be continuous at any primary point. It has now been proved that the
condition of uniform continuity is sufficient; that it is necessary follows from
the theorem of ~ 175.
The derivatives at any point depend only on the functional values at the
primary points in the neighbourhood of the point. For let xa be any point,
and consider the limit of f() f(x), when x has any sequence of values
x - XI
which converge to x1. A set of primary values of x can always be found,
such that the ratio converges to the same limit, when x has the values of
this sequence of primary points, as for the prescribed sequence consisting
of secondary points, or of both primary and secondary points. For a primary
point ~ can be found, corresponding to x, such that
f(x)-f(x) 'f()-f(     ) x
x - X-           I
where 8 is an arbitrarily small number. This follows from the fact that
f(x) -f (xi)
x - X1
is a continuous function of x at every point except xz.
226. In order to construct monotone continuous functions, the values of
the function are first assigned at the end-points a, b of the interval, then at
two points xo, x1, where x0 < xw; then at four points x,,, xo, Xio, x11, where
o00 < Xo, Xo < x0o < X1o < xi, and xi < xn;
afterwards at eight points
Xoo000, X001, Xo10 011, Xo10o, X101, X10, X111, &C.
lying in the successive intervals measured from left to right, into which (a, b)
was divided by the four points; and so on. The function may then be
regarded as the limit of a sequence of continuous functions, each of which is
representable as a polygon obtained by joining the end-points of ordinates
which represent the functional values that have been assigned at any stage
of the process.




298


Functions of a real variable


[CH. IV


In this manner Broden has constructed a monotone continuous function
f(x), which is such that it has derivatives on the right, and on the left,
which are everywhere definite, finite and different from zero; and such that
a definite differential coefficient everywhere exists, except at the everywheredense enumerable set of primary points.
He has also constructed a monotone function f(x) which is such that at
the everywhere-dense enumerable set of primary points, the derivative on
the left exists, and is zero, and the derivative on the right exists and is
positive; for an unenumerable everywhere-dense set of points there is a
differential coefficient everywhere zero, and for another such set of points,
there is no definite derivative on the left, but there is a positive one on
the right.
A third case is the following:f(x) is continuous, monotone and increasing; at an everywhere-dense
enumerable set of points the derivative on the left is zero, and that on the
right is + oo; for an everywhere-dense unenumerable set, both derivatives exist
and are positive; for another such set both exist and are zero; for a third such
set, both derivatives exist and are + oo; for a fourth such set, neither derivative
exists; for a fifth such set, the derivative on the left is zero, and that on the
right is indefinite, but has zero for its lower limit; for a sixth such set, the
derivative on the right is + oo, but that on the left is indefinite, with + oo
for its upper limit.
227. For the construction of everywhere-oscillating continuous functions
it is more convenient to successively assign the functional values at sets of points
proceeding by powers of 3 instead of 2 as in the case of monotone functions.
In this manner Broden has constructed such a function f(x), which has the
following properties:At an everywhere-dense enumerable set of points, the derivative on the
left exists, and is positive; that on the right exists, and is negative (or the
reverse), this set corresponding to maxima and minima of the function; for
a certain unenumerable everywhere-dense set, there is a differential coefficient
everywhere of the same sign; and for another such set, there is a differential coefficient which is zero; for a third such set, one or both of the
derivatives are indefinite.
Kopcke* has given the first example of a function which is everywhereoscillating and yet has at every point a definite differential coefficient, thus
confirming the conjecture of Dini that such functions can exist; and Brodent
has also constructed such a function. A general theory of such functions has
* Math. Ann. vol. xxix, p. 123; vol. xxxiv, p. 161; vol. xxxv, p. 104. See also Pereno, Giorn.
di Mat. vol. xxxv, p. 132.
+ Stockholm Vet. Ak. Ofv., 1900, pp. 423 and 743.




226-229]      Functions of two or more variables                299
been given* by Schonflies. The method adopted by Kopcke is to construct
the function as the limit of a succession of polygons of which the sides are
circular arcs. Everywhere-oscillating functions have also been studied by
Steinitzt. A detailed account of all the special cases treated of by these
writers would require a large amount of space; reference can therefore only
be made to the original memoirs. A simplification of Kopcke's construction,
due to Pereno, will be given in Chapter VI.
228. A function f(x) which is of such a character that it can be represented approximately by a graph, which exhibits all the peculiarities of the
function, so that y =f(x) is the equation of a "curve," in the ordinary sense
of the term, must satisfy the following three conditions:(1) The function must be continuous everywhere, with the possible
exception of a finite number of points, at which it may have ordinary discontinuities.
(2) It must be differentiable, except that there may be a finite number
of points at which no differential coefficients exist, but at which definite
derivatives on the right and on the left exist.
(3) It can have only a finite number of maxima and minima; and the
same must hold of every function obtainable by the addition of a linear
function to the one in question. This condition may be expressed in the
form, that the function must be in general monotone with reference to every
possible axis which may be employed for the measurement of abscissae.
A function which satisfies these conditions may be characterisedt as an
ordinary function.  As has been already indicated, there exist functions
which satisfy the conditions (1) and (3), but do not satisfy the condition (2).
Again, there exist functions which satisfy the conditions (1) and (2), but not
the condition (3).
FUNCTIONS OF TWO OR MORE VARIABLES.
229. An association of n numbers (a1, a,... an) being considered to
represent a point in n-dimensional space, any set of such points, whether
continuous or not, may be taken as the domain of a set (x,, x,,... Xn) of n
independent variables. When I xa, I X i,... I n I are, for all points of the
domain, all less than some fixed positive number, the domain is said to be
limited.
A function f(x, x2,... xn), as in the case of a domain of one dimension,
is defined by a set of rules from which a single number, the functional value,
* Math. Ann. vol. LIV; also his Bericht, p. 164.
t Math. Annalen, vol. LII.
+ Du Bois Reymond, Crelle's Journal, vol. LXXIX, p. 32.




300


Functions of a real variable


[CH. IV


can be arithmetically determined for any prescribed point of the given
domain. If, for every point of the domain, If(xi, x,,... xn) I be less than
some fixed positive number, the function is said to be limited in its domain.
Corresponding to a neighbourhood (a - e, a + e) of a point a in a straight
line, the rectangular cell which contains all points (xi, x,,... xn) such that
]       x-ai=< e-61, x 2-a2 - e2,... I x-aj  t  e,, where Ce, e,... en are definite
positive numbers, is taken to be a neighbourhood of the point (al, a2,... an).
A "sphere," which contains all points such that
(x - a)2 + (x2- a2)2 +.. +  (x -an)2  p2,
where p is some assigned number, is also frequently employed for purposes
similar to those for which the interval (a - e, a +e) is used in the case of
a linear domain.
The definitions given in ~167, of the upper and lower limits, and of the
fluctuation of a function in its domain, can be immediately extended to the
case of an n-dimensional domain.
The function f(xI, x2,... Xn) is said to be continuous at the point
(a,, a2,... an), which is a limiting point of the domain of the function,
provided that, corresponding to an arbitrarily chosen positive number e,
a neighbourhood of (a,, a2,... an) can be determined, such that
If/(xI, x2... Xn)-f(aa,... an) < e,
for every point (x,, x2,... xn) in the interior of the neighbourhood, which is
conveniently taken to be a rectangular cell.
A function which is not continuous at a point may satisfy the above
condition for a neighbourhood in which x - a,, x2-a2,... x- an are restricted each to have a definite sign. The 2n different partial neighbourhoods of a point so determined, correspond to neighbourhoods on the right
and on the left, in the case of one-dimensional domains.
Such partial continuity of a function is a generalization of the conception
of continuity on the right, and on the left.
The saltus of a function at a point is defined as in ~ 180, as the limit
of the fluctuation in the neighbourhood when the greatest of the numbers
61, e2,... en converges to zero.
There is a separate saltus for the limit of each of the 2n partial
neighbourhoods.
The domain for which a function is defined will most frequently be taken
to be a continuous limited domain, i.e. one which is limited, perfect and
connex.
The theorem of ~ 171, that for such a domain, there exists one point at




229, 230]


2Functions of two variables


301


least, such that, in any arbitrarily small neighbourhood of the point, the upper
limit of the function is the same as the upper limit of the function in the
whole domain, can be extended to the case of an n-dimensional domain.
The whole domain may be taken to be contained in a single rectangular
cell; this rectangular cell may be sub-divided into nr equal parts each similar
to the whole; these parts may then be similarly sub-divided, and so on,
indefinitely. The proof of the theorem is then precisely similar to that
given in ~ 171.
The theorem that, in the case of a continuous function, the upper limit
of the function is actually attained at some point of the domain, may be
proved as in ~ 171.
That a continuous function is determined by the functional values at the
points of an everywhere-dense enumerable set contained in its domain, may
be proved as in ~ 173.
That a continuous function defined for a closed domain is necessarily
uniformly continuous, may be proved by either of the methods employed in
~ 175 and ~ 185. Thus, if for each point of the domain, a neighbourhood be
determined, within which the fluctuation of the function is less than the
prescribed number e, a finite number of these neighbourhoods can be selected
such that each point of the domain is in the interior of one at least of them.
The finite number of cells which overlap one another determine a finite
number of non-overlapping cells. If X be the shortest of the edges of all
these non-overlapping cells, any rectangular cell such that the lengths of all
its edges are less than q will be contained in the interior of one of the cells
of the finite overlapping set. Thus the theorem is established.
FUNCTIONS OF TWO VARIABLES.
230. Most of the points in which the theory of functions of a number of
variables involves considerations which are not an immediate generalization
of those which occur in the case of functions of a single variable, are
sufficiently illustrated by the case of functions of two variables. Accordingly
the properties of functions of two variables will be considered in some detail.
That a function f(x, y) should be continuous at a point (a, b) which is a
limiting point of its domain, it is necessary, but not sufficient, that the
function f(x, b) of x should be continuous at the point x = a, and that the
function f(a, y) of y should be continuous at the point y=b. Thus a
function may be continuous at a point with respect to x, and also with respect
to y, whilst it is discontinuous with respect to the two-dimensional domain
(, y).




302


Functions of a real variable


[CH. IV


It is not even sufficient to ensure the continuity of f(x, y) at a point,
that it be continuous in every direction from the point. Thus
f(a + r cos 0, b + r sin 0)
may be a continuous function of r, at r= 0, for each value of 0 in the
interval (0, 27r), and yet* the function may be discontinuous at (a, b).
The necessary and sufficient condition that f(x, y) may be continuous at
(a, b) may be expressed in the form, that f(x, y) must be continuous in every
direction at the point, and uniformly so for all directions.
Thus if f(a + r cos 0, b + r sin 0) be continuous at r = 0, for each value
of 0, and uniformly so for all values of 0, then if e be a prescribed positive
number, a number p can be determined, independent of 0, such that
I f(a + r cos 0, b +r sin 0) -f(a, b) I <,.
provided r < p. From   this condition it follows that f(x, y) -f(a, b) < e,
provided  x - a, y - b I are each < p/,2, and thus the condition of continuity of the function is satisfied.
The remarks which have been made as regards the continuity of a
function at a point are applicable without essential change if those functional
values in the neighbourhood of the point are alone taken into account,
which are in one of the four quadrants, the values at points on the axes
bounding the quadrant being either included or excluded from consideration,
as may be agreed upon. Thus the condition of continuity at a point may be
satisfied for one such quadrant and not for another one.
EXAMPLES.
1. Let /(x, y)= _2 x ) and f(O, 0)=0. This function is discontinuous at the point
(0, 0), although it is continuous at that point with respect to x, and also with respect to y,
since f (, 0)=0, f(y, 0)=0. In all other directions the function is discontinuous; for
writing x=rcos 0, y=r sin 0, the function is sin 28 and therefore has a constant value
different from zero on a straight line for which 0 is constant, unless 0 has one of the values
0, I~r,, or r   Ir.
2. Lett f (, y)=      f(0, (0)=0. This function is discontinuous at the point
X2 +Ly4+,
(0, 0), although it is continuous in each particular direction, at that point. We find that
rs2 in2 4 0 <, if r < C cosec2 0 {1 - - 1  42 cos2 0}; and in order that this condition
cos20+r2sin4 0       2E
may be satisfied, the greatest value of r diminishes indefinitely as 0 approaches the value
irr; whereas when 0=rr, the function is, for every value of r, equal to f(O, 0). It is
thus seen that the convergence in different directions is non-uniform.
* See Thomae, Abriss einer Theorie der komplexen Functionen, 2nd ed. p. 15.
t Genocchi-Peano, Calc. Diff., ~ 123.




230, 231]


Functions of two variables


303


231. Let (a, b) be a limiting point of the domain for which a function
f(x, y) is defined, and let a neighbourhood of which the corners are the four
points (a + E, b + e) be taken. Let U, L be the upper and lower limits of the
function for all points of the domain in this neighbourhood, the functional
value at (a, b) being however disregarded in case (a, b) belongs to the domain.
If e, e' be diminished, the number U cannot increase; and when values of e, e'
belonging to sequences e1, 62,... En,..., and e', 62,... n',... each of which
converges to the limit zero are taken, and Un be the value of U corresponding
to the values e,, En' of e, e', the numbers U,, U,,... U,... form a sequence of
numbers which do not increase. This sequence has then a limit U, which
may however have the improper value oo, in case all the numbers Un have
this improper value. It is easily seen that U is independent of the particular
sequences chosen for e, '. This number U is said to be the upper limit of
the function at (a, b), and may be denoted by
lirn f(x, y).
x=a,y=b
The lower limit lim f(x, y) may be defined in a similar manner, as
x=a, y=b
the limit of a sequence of values of L; and it may have the improper value
At a point of continuity of the function, the condition
lim  f(, y)=   lim  f(x, y)
x=a, y=b       x=a, y=b
is satisfied; and further, each of these limits is equal to f(a, b), in case (a, b)
belongs to the domain of the function.
Corresponding pairs of limits may also be defined for the case in which
the functional values in one quadrant only are taken into account, the
functional values on the axes being either included or excluded, in case they
exist, as may be agreed upon.
The saltus or measure of discontinuity at the point (a, b) is measured by
the excess of the greatest over the least of the three numbers
f(a, b), lim  f(x, y), lim  f(x, y).
x=a, y=b      x=a, y=b
The saltus at a point of discontinuity may have a finite value, or it may be
indefinitely great.
In case  lim  f(x, y)=   lim  f(x, y), their common value may be
x=a, y=b       x=a, y=b
denoted by  lim  f(x, y), and the function is then said to have a definite
x=a, y=b
double limit at the point (a, b); this double limit  lim  J (x, y) may be
x=a, y=b
finite, or infinite with a definite sign.




304


Functions of a real variable


CH. IV


When the upper and lower limits have different values, limf(x, y) is
frequently regarded as existent but indeterminate, the upper and lower
limits being regarded as its limits of indeterminacy.
232. In considering the functional values in the neighbourhood of
a point, and the functional limits at the point, it is frequently convenient to
consider one quadrant only; this we may take, without loss of generality, to
be the quadrant in which x- a  O, y- b _ O. The results which will be
established are essentially applicable to any one of the four quadrants, and
can be immediately extended to the case in which account is taken of the
whole neighbourhood of (a, b), by taking the totality of the results for the
four separate quadrants, and for the lines x = a, y = b.
Assuming that   - a > 0, y - b > 0, the function f(x, y) considered as
a function of y only, with x constant, has two functional limits f(x, b + 0),
f(x, b + 0), at the point (x, b); these may be denoted by lim f(x, y),
y=b
lim f(x, y) respectively. In case these two limits are identical, their common
y=b
value may be denoted by lim f(x, y), the functional limit f(x, b + O) having
y=b
in that case a definite value.
If either of the limits lim f(x, y), lim f(x, y) is to be taken indifferently,
2=b        y=b
we may denote them by lim f(x, y). This may be regarded as a function of x,
y=b
such that its value at the point (x, b) is multiple-valued, and has
lim f(x, y), limn f(x, y) for its limits of indeterminacy.
y=b        y=b
It may happen that limf(x, y), considered as a function of x, has a
y=b
definite functional limit at the point x = a; this limit may be either finite,
or infinite with fixed sign. In case such a limit exists, it is denoted by
lim limf(x, y), and it is said to be the repeated limit of f(x, y) at the point
x=a y=b
(a, b), the order of the limits being, that the limit for y= b is taken first,
and then afterwards the limit for x = a.
In case this repeated limit does not exist, either as a definite number, or
as infinite with fixed sign, we may regard lim lim f(x, y) as indeterminate,
x=a y=b
its limits of indeterminacy being lim lim f(x, y), and lim lim f(x, y).
xab                x=a y=b ay
The repeated limit lim limf(x, y), in which the limit with respect to x
y=b x=a
is first taken, and afterwards that with respect to y, may be defined in a
precisely similar manner.




231-233]


Functions of two variables


305


It is clear that the functional values on the straight lines x = a, y = b are
irrelevant as regards the existence, or the values, of the repeated limits.
In case the double limit   lim   f(x, y) for x> a, y>b, exists at the
x==a, y=b
point (a, b), having either a finite value, or being infinite with fixed sign, the
existence of the two repeated limits
lim lim f(x, y), lim lim f(x, y)
x=a y=b          y=b x=a
follows as a consequence, their common value being   lim  f(x, y). In this
x=a, yJ=b
case lim fix, ck(x)} also exists, and is equal to the double limit, where ((x)
x=a
is any function of x, which is > b, and is such that lim      (x)= b. Also
x=a
limf{a)(t), q(t)} exists, and is equal to the double limit; where +(t), +(t)
t=T
are functions of a variable t, such    that 4) (t)> a, 0 (t) > b, and that
lim  )(t) = a, lim +)(t)= b.
t=r          t=r
The converse of these statements does not hold good. In particular, the
existence of lim   f(x, y) is not necessary either for the existence with
x=a, y=b
definite values, or for the equality, of the two repeated limits
lim lim f(x, y); lim lirmf(x, y).
x=a =y=b x=a
EXAMPLES.
1. Let f(x, y) be defined for the positive quadrant by f(x, y)= -. We find
x+y
lim lim f(x, y)=l, lim lim f(x, y)= -1; thus  lim  f(x, y) cannot exist.
x=0 y=O          y=0 x=0                 x=0, y=0
x2y2
2. Let f(x, y)=  2+(        In this case lim lim f (x, y) and lim lim f(x, y)
x2 2+(x - y)             x=O y=O            y=O x=O
are both zero, and yet lim  f(x, y) does not exist; for if yy =, f(x, y)= 1; and therex=0, y=O
fore lima f (, x)=1.
x=O
3. Let* f(x, y) be defined for x >0, y>O, by the expression (x+y) sin sin -. In
x   y
this case lim f (, Sy)=  sin-,lim f (x, y)=- xsin, and lim f(x, y)-lim f(x, y) has
y=o      0         =o -0                  y=O0       y=o
1        1
for x =0 the limit zero. We have then lim lim f(x, y)=0, since xsin-, -xsinx=O y=0                     X        X
have each the limit zero for x=0. It is clear that lim lim f(x, y) is also zero. If
y=O x=0
0<  < E, and 0< y< E, we see that If(x, y) <E, and therefore  lim  f(x, y) exists,
=-0, y/=0
and is equal to zero.
233. An important matter for investigation is the determination of the
necessary and sufficient conditions for the existence and equality of the two
* Pringsheim, Encyklopddie der Math. Wissensch., II A. 1, p. 51.


H.


20




306


Functions of a real variable


[H. IV


repeated limits at a point. A knowledge of such conditions, as also of
sufficient conditions, is required in various fundamental theorems of analysis
which turn upon the legitimacy of inverting the order of a repeated limiting
process.
It will be observed that the existence of lim limf(x, y) does not necesx=a y=b
sarily involve the existence of limf(x, y) as a definite number, since
y=b
lim limf(x, y), lim lim f(x, y) may both exist and have the same value,
x=a y=b       x=a y=b
without it being necessarily the case that limf(x, y), limf(x, y) are
y=b        y=b
identical.  It is however necessary that limfc(x, y)-limf(x, y) should
y=b         y=b
converge to the limit zero, as x converges to the value a.
The necessary and sufficient conditions required are contained in the
following general theorem:In order that the repeated limits limn lim f(x, y), lim lim f (x, y) may
x=a y=b        y=b x=a
both exist and have the same finite value, it is necessary and sufficient, (1) that
limf(x, y) -linmf(x, y) should have the limit zero, for x=a, and that
y=b          y=b
lim f(x, y) - lim f (x, y) should have the limit zero, for y= b; and (2) that,
c=a         x=a
corresponding to any fixed positive number e arbitrarily chosen, a positive
number / can be determined, such that for each value of y interior to
the interval (b, b + -/) a positive number ay in general dependent on y exists,
such that; for this value of y, f(x, y) lies between lim f(x, y) + e and
y=b
lim f(x, y)- e, for all values of x interior to the interval (a, a + ay).
y=b
Let us first assume that the conditions stated in the theorem are satisfied.
A value of y may, in virtue of (1), be so chosen that the difference of the two
limits limf(x, y), limf(x, y) is less than an arbitrarily chosen number r;
x=a        x=a
and this value of y may also be so chosen that it is interior to (b, b + /). For
this fixed value of y, an interval (a, a + a,') for x may be so chosen that
f(x, y) lies between limrf(x, y) + e and limf(x, y)- e, provided y has
x=a                x=a
the fixed value, and a < x < a + ay': this follows from  the definition
of the upper and lower limits. Again, from the condition (1), a number
a" can be determined, such that if x be interior to the interval (a, a + a"), the
difference between the two limits limf (x, y), limf(x, y) is less than a.
y=b         y=b
Now let ay be the smallest of the three numbers ay, as', a"; then, if xi, x,
be any two values of x within the interval (a, a + dy), and y have the fixed
value, by applying the conditions of the theorem, we see that the conditions




233]               Functions of two variables                   307
If(X> ) -f(X2, y) I< q + 2e,
If(x, y)-lim f(x, y) < V + e,
f(2, y)-=limf(b2, y 
are all satisfied. It follows that
ylimf(x, y) - lim f(x,, y) < 3, + 4e
v-b         y=b
for every pair of points X1, x2 within the interval (a, a + ay). Hence, since e, V
are both arbitrarily small, limf(x, y) converges for x = a to a definite value
y=b
which is the limit of both limf(x, y) and of lim f(x, y) when x = a; and
y=b              y=b
thus lim lim f(x, y) exists.
x=a y=b
Again, since lim limf(x, y) has a definite value, an interval (a, a +)
x=a y=b
can be determined, such that for any point x interior to it
lim    f(x, y)lim(, )limf(, y) < e.
x=a y=b        y=b
Now lim limf(x, y)- limf(x, y) is the sum of the three differences
x=a y=b        x=a
lim lim f(x, y) - limf(x, y),
x=a y=b        y=b
limf (x, y) -f(x, y), f(x, y) -lim f (x, y),
y=b                         x=a
and for a fixed y, chosen as before, x may be chosen so that it not only lies
within the interval (a, a + 8), but is also such that
|f(x, y) - limf(x, y)  f f(x, y) - limf(x, y)
- y=b                  x=a
are each less than? + 2e. It follows that
li lim f(x, y)-lim f(x, y) < 5 + 2l,
X=a y=b        x==a
and thus that lim f(x, y) converges, as y converges to b, to the limit
x=a
lim lim f(x, y). It has thus been shewn that the two repeated limits both
x=a y==b
exist and have the same value.
Conversely, let us assume that the repeated limits both exist, and are
finite and equal. We have then lim lim( f(, y)- lim f(x, y) <, provided
x= a y=-b      x=a
20-2




308


Functions of a real variable


[CH. IV


y lies between b and b+~9, where / is some fixed number, ' being an
arbitrarily chosen positive number; from this it follows that
lim f (, y) - lim f (, y)
x=a,        x=a
is < 2', for b<y< b+/. Also
limf(x, y) -lim limf(x, y) <
}==b        x=a y=b
provided x lies within some fixed interval (a, a + 8'); and from this it follows
that lim f(x, y) - limf(x, y) is < 2, for a < x < a + 8'. Since y is arbitrarily
y=b-        y=b
small, we now see that the condition (1) of the theorem is satisfied. Further
we see that
f(, y)-lim/(x, y) l< 2~+ r',
x=a
where r' is any arbitrarily chosen positive number, provided x lies within
some interval (a, a + a'y), where a'y depends upon y, and may diminish indefinitely as y approaches the value b. It follows from the three inequalities,
that
f(, y)- imf(x,y) <4 + r',
E   =b
provided b<y < b +, and provided also x lies within some interval
(a, a + ay) where ay depends in general upon y. Since ' and T' are both
arbitrarily small, it follows that the condition (2) of the theorem is satisfied.
If the condition (2) in the above general theorem be replaced by the
more stringent condition that, corresponding to any fixed positive number e,
arbitrarily chosen, a positive number /3 can be determined, which is such
that for each value of y interior to the interval (b, b + /), a positive number
oy dependent on y exists, such that for this value of y, and for all smaller
values, f(x, y) lies between lim f (x, y) + e, and lim f (x, y) - e, then this
y=b                 y=b
condition and the condition (1) are the necessary and sufficient conditions
that not only lim lim f(x, y), lim lim f(x, y) exist and are equal, but also
x=a y=b       y=b x=a
that the double limit  lim  f(x, y) exists, having a definite value the
x=a, y=b
same as the repeated limits. In case the function be defined for values of
x, y on the lines x = a, y = b, the additional conditions must be added that
the functional values on these lines also converge to the same limit
lim  f(x, y).
x=a, y=b
For, under the conditions stated, we have, provided y lies within the
interval (b, b +,/), where /3 < /,
f(x, y) -lim/f(X, y) < e+,
y=b




233, 234]


Functions of two variables


309


where x has any value in the interval (a, a + '), ' being the lesser of the two
numbers ail and S'; the number $' being so chosen that
lim f(x, y) - lim f(x, y) < rj, for a < x < a + 8'.
|Y=b       y=b
Also  lim(,)l(, y)-lm  lim  (x, y) < e, provided x lies within an interval
Y=b         x=a y=b
chosen sufficiently small. Hence the condition
f(x, y) - lim lim f(x, y) < 2e+ 4
x=a y=b
is satisfied, provided b < y < b + /3, and provided x lies within an interval of
which the length may depend upon e and s/. It follows, since e, X are
arbitrarily small, that f(x, y) has a definite double limit at the point (a, b).
That the conditions stated are necessary, follows at once from the definition
of lim f(x, y).
x=a, y=b
234. The theorem obtained in ~ 233 may be simplified in the case in which
lim f(x, y), lim f (x, y) both have definite values at all points on the straight
y=b        x=a
lines x =a, y = b which are in sufficiently small neighbourhoods of the
point (a, b). We may then state the theorem as follows:If lim f(x, y), limf(x, y) have definite finite values in the neighbourhood
y=b        x=a
of the point (a, b), then the necessary and sufficient condition that the two repeated
limits lim lim f(x, y), lim limr f(x, y) may both exist and have the same
x=a y=b        y=b x=a
finite value is that, corresponding to any fixed positive number e, arbitrarily
chosen, a positive number 3 can be determined, which is such that, for each
value of y interior to the interval (b, b + 3), a positive number ay in general
dependent on y exists, such that for this value of y, f(x, y)- lim f (x, y) < e
y=b
for all values oJ x within the interval (a, a + at).
In case the condition f(x, y) - lim f(x, y) < e for all values of x within
'        y=5b
(a, a + oy) be satisfied not only for the particular value of y but for all smaller
values, and this hold for every e, then the double limit lim  f(x, y) exists,
x=a, y=b
and is equal to each of the repeated limits. In this case the point (a, b) is
said to be a point of uniform convergence of the function f(x, y) to the
limit lim f(x, y) with respect to the parameter x; and thus, for such a point,
y=b
there exists for each value of e, an interval (a, a + a), where a depends in
general upon e, such that for each value of x within this interval, the
condition f, (x -    f(, y ) -,  y) < e, is satisfied, provided y be less than some
y=b
fixed value which is the same for the whole x-interval (a, a + a).




310


Functions of a real variable


[CH. IV


It may happen that, as e is indefinitely diminished, a has a positive
minimum a. In that case the fixed interval (a, a + a) is such that, for each
e, the condition If(, y) - limf(x, y) < e is satisfied for all values of x
y=b 
within the fixed interval (a, a + a), provided y is less than some fixed value
dependent on e, the same for the whole x-interval. In this case f(x, y) is
said to converge to limf (x, y) uniformly within the interval (a, a + a), with
y=b
respect to the parameter x. Not only the point (a, b) but also each interior
point of the interval (a, a + a) is then a point of uniform convergence of
f(x, y) to lim f(x, y) with respect to the parameter x.
y=b
235. The necessary and sufficient conditions for the existence and
equality of the two repeated limits of f(x, y) at (a, b) may be put into the
following form different from that of the theorem of ~ 233.
The necessary and sufficient conditions that lim lirnf (x, y) = lim limmf(x, y),
x=a y=b        y=b x=a
their value being finite, are (1) that lim f(x, y) converge to a definite value
x=a
lim lim f (x, y) when y converges to b, and that lim f(x, y) - lim f(x, y) cony=b x=a                                     y=b         y=b
verge to zero, for x= a; and (2) that, corresponding to any arbitrarily chosen
positive number e, and to an arbitrarily chosen value b + /3 of y, a value
yl < b +,3o of y can be found, and also a positive number a, such that the
condition thatf (x, y,) lies between
lim f( (, y) + e, and lim f(x, y) -e
y=b                 y=b
is satisfied for every value of x within the interval (a, a + a).
In case lim, f(x, y) everywhere exists in the neighbourhood of x = a, the
y=b
condition (2) is that f(x, y) - lim f(x, y) I< e, for every value of x within
y=b        I
the interval (a, a + a).
That the conditions contained in the theorem are necessary, is seen from
the theorem of ~ 233; it will be shewn that they are sufficient. Let us assume
that the conditions are satisfied. We have
lim f (x, y) - lim lim f(x, y) = [lim f (, y)-f(, y)
y=b         y=b x=a         ty=b               J
+ [f(x, y) - lim/f(x y) + [iimf(x, y) - lim limf (, y)1.
L    x=a     =J =a              y=b x=a      J
A positive number /3i can now be chosen, such that if b< y< b + /3, the
condition  lim f(x, y) - lim lim f(x, y) < e is satisfied; moreover we may
x=a oy=b x=a
choose 8/3 so that it is <,o.




234-236]


Partial ditferential coefficients


311


Next, a value yi of y exists, such that f(x, yi) lies between lim f(x, y) + e,
y=b
and lim f(x, y) - e, provided x be within the interval (a, a + a): the value of
y=b,/o may be chosen so small that lim f(, y)- ]imf(x, y) < e, for every value
x=a        x=a
of y which is < b + P/, and therefore for the value y, of y. Again, an interval
for x, possibly less than (a, a + a), can be so chosen that
lim f(, y) - lir f(x, y) < e,
y=b        y=b
provided x lie within the interval. It follows that an interval (a, a + a') for
x can be found, such that limf(x, y) -f(x, yl) < 3e. Further, the interval
y=b
within which x lies may, if necessary, be so restricted that
f(x, ) y- limn f(x, yI) < 2e.
Hence, provided x lies within a definite interval, we see that
linm f(, y) - lilim lmf (, y) < 6e;
y=b &y=b x=a
and since this condition holds for an arbitrary e, it follows that limf(x, y)
y=b
converges for x = a to lim lim f(x, y), and thus the sufficiency of the cony=b x=a
ditions is established.
PARTIAL DIFFERENTIAL COEFFICIENTS.
236. If, at a point (x,, Yo) in the domain for which the function f(x, y) is
defined, the limit limf(x + h, Yo)-f(xo, Yo) exists, having either a definite
h=O 
finite value, or being indefinitely great but of fixed sign, this limit is said to
be the partial differential coefficient of f(x, y) at (x,, y,) with respect to x,
and is usually denoted by af (x0 y
axo
When the limit limf(xo~' o + kf) f(X, Yo) exists, it is said to be the
k=O         ki
partial differential coefficient of f(x, y) at (x,, Yo) with respect to y, and is
denoted by af(xo, o)
DYo
In general, h, k in these definitions are regarded as having either sign.
It is possible that either of the above limits may not exist, but that there
may be two definite limits, one for positive values of the increment h or k,
and the other for negative values. In that case the two limits are said to
be the progressive and regressive partial differential coefficients with respect




312


Functions of a real variable


[CH. IV


to the particular variable.  It is of course possible that, at a particular
point, one of these may exist, and not the other.
af fmes
That the two partial differential coefficients   f may exist, it is
necessary, but not sufficient, that f(x, y) should, at the point (xo, yo), be
continuous with respect to x, and also with respect to y.
To express the increment f(xo + h, yo + k) -f(,o, yo) of the function
f(x, y), when the two numbers x,, y, receive increments h, k respectively, we
have
f (xo + h, yo + I) -f (o, yo) = [f(xo + h, yo + I) -f(Xo, yo + k)]
+ [/(xo, S/o + ) -f(Xo, Yo)].
If we now assume that f exists at the point (xo, yo), and has a finite
value, we have
f(Xo, yo + k) -f (o, yo) of +(xo, yo)
k          =     yo   + 
where a(k) converges to the limit zero, when k is indefinitely diminished.
f(Xo+h, 0~+Ic)-f(xo, 0+ Ic)              ~.., 
Again, f ( + h, y + h - f (,  + k) converges to the limit a-, when k
is first diminished to the limit zero, and afterwards h converges to zero, it
being assumed that f has a definite value, and also that f(x, y) is conb       e      ing assumed thaoo
tinuous with respect to y for the value y=y,, where x has any value in a
neighbourhood of Xo. In order, however, that the double limit
lim  f(xo + h, yo + k)-f(xo0, yo + )
h=O, k=O           h
may exist, in which case its value is, being independent of the mode in
ax)
which h, k approach their limits, it is necessary and sufficient that
f(xo + h, yo + k) -f(xo, yo + k)
h
should be a continuous function of (h, k) at the point h = O, k = 0.
If this condition be satisfied, positive numbers h1, k1 can be determined,
h                X
such that         f     (X        o +  h, yo + k) -  f (xo, Yo + k)  af<
where r is a prescribed positive number, and 0 < h  h, 0  k f IC k.
We have now
f(xo + h, yo + k) -f(x, yo +  )( _  /   ),
h-      o             P (h, ak),




"] 1 C


-  7 *  AS   -_  *,An  *


236]             Partial differential coefficients              And
where p (h, k) converges to zero, independently of the mode in which h, k
converge to zero.
Under the conditions stated, we have
f(+o + A, yo + k) -/(o, yo) =  a + k af + hp + kc
axo   ay0
where p, oa converge to zero, when h and k are indefinitely diminished,
independently of the mode in which they approach their limits. This is
equivalent to the statement that, corresponding to an arbitrarily assigned
positive number V, positive numbers hA, 1k can be determined so that p j and
a I are each < V1, for all values of h and c such that I h I < hi, j  1 < k.
In the notation of differentials, denotingf(xo, yo) by zo, we have
az     az
dz =    dx +-  dy;
-ax,     ay;
the expression on the right-hand side being termed the total differential
of z at the point (x, y), and  z dx,  o dy the partial differentials.  In
axo    ay"
accordance with the arithmetical theory, this equation can only be regarded
as a conveniently abridged form of the result obtained in the present
discussion.
The theorem obtained may be stated as follows:The increment of a function f(xo, Yo) when xo, yo are changed into
Xo + h, Yo + k is h  (  ) + 4  af(0), Yo) + hp + ka, where p, a- converge to
ax0         ay,
zero when h, k are indefinitely diminished, independently of the mode in
af (Xo, yo) af (xoyo), Yh)
which they are diminished, provided that (1) -a(   7~, o  ('    have
axo   '   ayo
definite finite values, and (2) that (x + h, yo + k) - f(xo, yo + k) is a continuous
function of (h, k) at the point h =, k = 0.
af a/ have
It will be observed that no assumption has been made that,  have
definite values except at the point (xo, yo) itself.
If it be assumed that a has a definite value at (xo, y) for all values of
ax
y in some neighbourhood of y, the condition (2) may be expressed in the
form that (a) -f for X=X0 must be a continuous function of y at y= y0, and
that (b) the point h = 0, k = 0 must be a point of uniform convergence of the
function f(xo + h, yo + k) -f(xo, yo + k) considered as a function of h, to its
h




314


Functions of a real variable


[CIH. IV


limit for h =0, with k as a parameter, in accordance with the definition of
such a point of uniform convergence given in ~ 234.
If it be assumed that f exists, not merely at the point (x0, yo), but at all
ax
points in a sufficiently small two-dimensional neighbourhood of the point,
the conditions contained in the theorem may be simplified. For we have, in
that case,f(x0  + h, Yo +   k)-f( o, yo + k) = 3 (I +, 
that case,                                f(xo + oh, _o + k),
af(xo, yo)
where 0 is such that 0    < < 1; and this expression converges tof (x-'-o)
provided af(' Y) be continuous with respect to (x, y), at the point (xo, ye).
ax
It has thus been proved that*, in order that the increment of the function
may be of the form given in the theorem above, it is sufficient that (1) a-  a
have definite values at the point (x0, yo), and (2) that one at least of these
partial differential coefficients have definite values everywhere in a twodimensional neighbourhood of (x,, yo), and be continuous at (xo, yo) with respect
to the domain (x, y).
237. Let it now be assumed that, throughout a perfect and connex
domain D, the two partial differential coefficients f  f everywhere exist,
ax' ay
and that they are continuous functions of (x, y). We have then
f(x+h, y+k)-f(x, y +k)= h      f(x + h, y  k),
f(x, y +c) -f(x, y) = k  f(x, y + 01k),
when 0, 01 are proper fractions, provided (x, y) is a point of D, and h, k are so
chosen that the straight line joining (x, y + k), (x + h, y + k), and the straight
line joining (x, y), (x, y + k) are wholly in the domain D. Since
af(x y), ay f(,     y)
are continuous functions of (x, y), it follows that they are uniformly continuous in the domain D. A positive number 8 can accordingly be determined,
corresponding to a prescribed positive number e, so that
ax f(x+ h, y + )-    f(    ) Y)<
a f( (, Y + ek) - af(x, y) < e,
ay              ay


* Thomae, Einleitung in die Theorie der bestimmten Integrale, p. 37.




236, 237]


Partial differential coefficients


315


provided I h, ikI are each less than 8, whatever be the position of (x, y)
in D.
We thus find that
f(x+h, y+k)-f(x, y)=h       f(x, y) + k  f(x, y) + hR + kR',
where R and R' tend to the limit zero, with h and k, uniformly for all points
(x, y) of the domain D.
This equation holds for every point (x, y) of D, and for all values of h
and k, such that the straight line joining
(x, y), (x, y+k)
and the straight line joining (x, y + k), (x+h, y + k) lie wholly in the
domain D. It is easy to replace the last condition by the less stringent one
that the two points P (x, y), Q (x + h, y + k), of the domain D can be joined
by a number of straight lines PP1, P1P2, Pa2P,... PnQ, each of which is
parallel to one of the axes, all of which belong to D, and are such that all of
them are wholly interior to a rectangle with its corners at P and Q, and
sides parallel to the axes.
We have then
f (P) -f (P)= hA  f (x + 01,h, y),
f(P2) -f(PI) = k  / f(x + h,, y + 02-1),
f(P) -f(P2) = h2  f(x + h, + 03h2, y + k1),
where hA, k,... are the lengths of PP1, PP2,..., and 01, 02,... are proper
fractions.  Now  Ihll, I, 1, I h... all being less than 8, all the partial
differential coefficients on the right-hand side of these equations differ
numerically from the corresponding partial differential coefficient
f (x, y), or  yf (x, y)
by less than e. We thus see that
f(x+h, y+k)-f(x, y)= h             +f(.x y)+k+ hf(. y) R+lkR'
where  RI, I R I tend to the limit zero, with h and k, uniformly for all
points of D.




316


Functions of a real variable


[CH. IV


EXAMPLES.
1. Let* f(x, y)= /lxyl, where the positive value of the square root is to be taken.
In this case f, a both exist at the point (0, 0) and are both=0. We have
f(h, k)- f (O, k)     k
and this has different constant values for different constant values of k/h, and is therefore
discontinuous at the point h=0, k=0. It follows that the equation f(h, k)=hp + kr, when
p and ro converge to zero with h and k, cannot hold.
2. Lett f(x, y)=xsin(4tan-1y/x), for x>0; and f(0, y)=O, for all values of y.
We find    (O' 0)=O  f(0, Y)=0, and thus -f(0, Y) is continuous with respect to y at
oxax            af(0, )                  af(x
(0, 0). Also, we find  (x 0) =4,       = )0, and therefore   0) is discontinuous
ay    =      y                      ay
with regard to x, at (0, 0). The value of f (h k) - f(o0 k) is si  4 tan-l h) and this is
dxf      dy, does not hold at the
discontinuous at l=0, k=O0; hence the relation df=-f d+  fdy, does not hold at the
point (0, 0).
HIGHER PARTIAL DIFFERENTIAL COEFFICIENTS.
238.   If the function f (x, y) have the partial differential coefficient  it
ax'
may happen that, at the point (x0, yo), the function a  possesses a differential
ax
coefficient with respect to x.  This is denoted by yf(x    ), and is spoken of
aX02
as the second partial differential coegicient of f(x, y) with respect to x, at the
point xo. The second partial differential coefficient  (   Y0) with respect to
ayo2
y, is defined in a similar manner.
It may happen that, at the point (xo, Yo), the function  f has a differential
ax
coefficient with respect to y: this may be denoted by   a (fX  ) or ay x0 
ayo    a3o    ay0 axo
Similarly, when   f has, at the point (x0, y,), a partial differential coefficient
ay
a /af\      a2f(xo, Yo)
with respect to x, this is denoted by a of) or                  These partial
dxo \9^o/     8^0 ayo
differential coefficients are said to be the mixed partial differential coefficients
of the second order at (x,, yo) of f(x, y) with respect to x and y, the order of
differentiation being different in the two.
* Stolz, Grundzilge, vol. i. p. 133.
+ Harnack's Introduction to the Differential and Integral Calculus, Cathcart's Translation,
p. 93.




237, 238]     Higher partial differential coefficients           317
Under certain conditions which will be here investigated, the two mixed
partial differential coefficients of the second order with respect to x and y
satisfy the relation
a2f (, y) a2f (, y)
axay  -   ayax 
which is known as the fundamental theorem for partial differential coefficients
of the second order.
The differential coefficient a faf, or a(o' Yis the repeated limit
axo \ayx     axo ayo
lim limf(xo + h, Yo + k) -f(xo + h, Yo) -f(xo, yo + k) +f(xo, Yo)
h=0 k=O                       hk                        '
We may denote this repeated limit by lim lim F(h, k). In order that the
h=O k=O
partial differential coefficient may exist, the value of this limit must be
independent of the signs of h and k.
It should be observed that it is not essential for the convergence of this
repeated limit to a definite finite value, that
lim f/(o + h, yo + k) -f(xo + h, yo)
k=o              k
should have a definite value when h = 0. Thus the repeated limit may have
a definite value when
lim 1 im f(x + h, y + k)-f(x + h, yo) _ lim f(o + h, yo + k)-f(xo+ h, yo)]
7w=o h  =0             k               k=O            k
vanishes. The repeated limit cannot however have a definite finite value
unless limf(x   o ~+ k)-f(x, y~) has a definite finite value, i.e. unless
k=O          k'
af (xmyx Yso               w
(ayY) exists and is finite. It thus appears that  af may exist when
ayo                                          axo oyo
af exists at the point (xo, yo), but is indefinite at points in the neighbouray
hood. The existence of the repeated limit as a definite number implies the
existence of a and of a2f
ayo      axo ayo
If, however, the repeated limit lim lim F(h, k) be infinite, with a definite
h=O k=O
sign, we cannot infer that af   exists, with an infinite value, unless it
ax80 ayo
be postulated that   has a definite value value at (, o); for the existence of
af at the point cannot, in this case, be inferred from that of the repeated
ay




318


8Functions of a real variable


[CH. IV


limit; and unless -exists at the point, D2f  has not been defined. When
ay        eaxo ayo
this condition is satisfied, the value of  f  is infinite with definite sign.
ax0 ay,
239. The differential coefficient of  is the repeated limit
ago ax,
lim li f (xo + h, Yo + k) -f (x + h, y) -f(xo, Yo + k) +f (x, yo) 
k=O h=O                       hk
a3f  _   _f holds are identical
and thus the conditions that the relation   =-     holds are identical
axo ayo 8/o ax,
with the conditions that the two repeated limits may be identical. The
necessary and sufficient conditions may be accordingly obtained by applying
the conditions contained in either of the theorems in ~ 233, and ~ 235, to the
function F(h, k) _ f (x + h, yo + k) -f (xo + h, y) -f(x,, Yo + k) +f(xo, yo)
hk
It is however convenient, for application in particular cases, to have
sufficient conditions relating to the partial differential coefficients in the
neighbourhood of the point (x,, y,).
The following theorem will be established:If (1), af  y) exist and be finite at all points in a two-dimensional
ayax
neighbourhood of the point (x,, y,), except that its existence at (x,, y,) is not
assumed, and (2) the point (x,, yo) be a point of continuity of 'f(, ) with
ay ax
respect to (x, y), the limit of this partial differential coefficient at (x,, yo) being
a definite number A, and (3) f (x, y) be continuous with respect to x at (xo, y,),
then a3f (x' yo) a2f(xo, o) both exist, and have the same value A.
ayo 8x      0 axo ay,
It will be observed that the condition (1) implies the existence of
af(x  ) at all points in a neighbourhood of (xo, y,), except at that point
ax
itself, and that it is continuous with respect to y.
From the condition (2), we have, corresponding to an arbitrarily chosen
positive number e,
p3fi(xo + h, yo + k));
ay ax
where  a | <e, provided h, I k  are each less than some fixed positive
number i7 dependent upon e, and are not both zero.
Let u (k') denote  f(  +      )  A', where   ' lies in the interval
ax




238, 239]     Higher partial differential coefficients


319


du(l')
(0, k); we have then dk7-   = a (h, k'), and this is numerically < e. It
follows that u (k)  ()is numerically < e; for, by the mean value theorem of
~ 203, since u (k') is continuous at I' = 0, and at k' = k, and possesses a definite
differential coefficient at every interior point of the interval (0, I), there
exists a number k in the interval (0, k) such that  (k)- u(0)  du () and
k    -    dk
this is numerically less than e. We have now
af(o + h, yo + k) _ af(xs + h  o)_ A  =  a,  ),
ax             ax
where a" is numerically < e. This holds for each value of h such that
0< Ihl <X.
Let v (h') denote f(xo + h', y + k) -f(xo  h', yo) -Ah', where h' lies in
the interval (0, h); we have then d(h) = a(h',k), and this is numerically
less than e. As before, since v(h') is, in virtue of (3), continuous at h' = 0,
and also at h' = A, and possesses a definite differential coefficient at all
interior points of the interval (0, A), it follows that v (h)  v () is numerically
< e; hence
hkF(h, k) =f(xo + h, yo + k) -f(xo + h, yo) -f(xo, yo + A) +f(xo, yo)
= Ahk + hka"' (h, k),
where a"' is numerically less than e.
We have now, corresponding to the arbitrarily chosen e, |F (h, k)- A  < e,
provided h, k are each numerically less than some fixed number r dependent
on e. It follows that F(h, k) is continuous at the point h = 0, k = 0 in the twodimensional domain (h, k), and has A for its double limit. From this we conclude that the two limits lim lim F(h, k), lim lim F(h, k) exist, and are both
h=O k=O        k=O h=O
identical with A. It follows that, when the conditions stated in the theorem are
satisfied, the two partial differential coefficients a2f(x y, y o) a2f(,,  ) both
ax0 ay,    ay, ax,
7f  af f
exist and are equal to A. The existence of af, a  follows from the existaxo ayo
ence of the above partial differential coefficients at the point (xo, yo).
The sufficient conditions in the foregoing theorem are somewhat simpler
than those stated by Schwarz*, who assumed the additional condition that
af(x, yo) exists and is finite for values of x in the neighbourhood of x = xo, for
ayo
* Gesammelte Abh., vol. II. p. 275; see also Peano, Mathesis, vol. x. p. 153. See further Stolz,
Grundziige d. Diff. Rech. vol. i. p. 147.




320                Functions of a real variable                [CH. IV
the constant value y,. Schwarz's theorem is, however, more general, in that
it is applicable to the case in which the two partial differential coefficients
have an infinite value with definite sign. The method of the above proof
may, however, be extended to this case, as follows:Let us assume that, if M be an arbitrarily chosen positive number, the
condition f(0 + h, Y + k) > M, is satisfied for all values of h and k which
are not both zero, and are both numerically less than some fixed number v,
dependent on M. Defining u (k') as af(x +    +     we see by means of
the mean value theorem, that
U (lk)- (O) > M, or af(xo + h, yo + k)  af(xO + h, yo)M
k (h)   f(x, + h', yo + k) - f(x, + h, y,)
Next, defining v (h) as            I (, we see, as before,
that v (h)  v (0) > M; therefore F(h, k) > M, provided h, k are both numerih
cally less than some fixed number r dependent on M. It follows that F(h, k)
converges to the limit + o, with fixed sign, as h, k converge in any manner,
each to the limit zero; thus both the limits lim lim F(h, k), lim lim F(h, k)
h=O k=O        k=O h=O
are + o. In order that a2(o, yo) a2f y,)   may exist, in which case they
axo ayo   ay" ax"
both have the value + oo, it is necessary to assume that a (x0, o) af(x, Y)
axo       ayo
both have definite values. The case in which the limits are both -oo may
be treated in a precisely similar manner. The following theorem has now
been established:
If (1) a2f ' Y) exist and be finite at all points in a two-dimensional
ay ax
neighbourhood of the point (x0, yo), except that its existence at (x,, yo) is not
assumed, and (2) the function  f (x, y) have the limit +'oo or -oo, with definite
ay ax
sign, at the point (xo, yo), and (3) the differential coefficients  a(Xo, y)
f(lo, ~oax,)  f( o
f(xo, Y) both exist and have definite values, then a2f(x  yo) af(y, o)
ay8 x0 ay,                                                  ayo ax,
both exist, having the value + so, or - o, with definite sign.
240. The partial differential coefficients of higher order n of a function
fnf x, y)
f(x, y) are of the form  y ar           where p, q, r,... are positive
integers, including zero, such that p  q...    n. Here,   is first
integers, including zero, such that p + q + r +... + l= n. Here, f is first




239, 240]     Higher partial differential coefficients


321


differentiated I times with respect to y, then k times with respect to x, and
so on.  The total number of possible partial differential coefficients of order
n is 2'1; the number of those in which r differentiations with respect to x,
and n - r with respect to y are involved is 
r! (n - r) 
Sufficient conditions for the existence of all the partial coefficients of
order n may be obtained by extending the theorem of ~ 239, which refers to
the case n = 2. The following criteria* which can be proved by induction,
will be sufficient for the purpose:If the n -1 differential coefficients  a         a nf          a     have
ax'n-i ay I  ' x-2 ay2  a x xayndefinite finite values for all points in a two-dimensional neighbourhood of the
point (x,, Yo), and are continuous at the point (x,, yo) with respect to (x, y), then
all the other mixed partial differential coefficients of order n exist at the point
(X0, yo); and each one of them has the same value at the point as that one
of those given above in which the same nutmber of differentiations with respect
to x, and with respect to y, occurs, as in the one considered.
EXAMPLE.
x2 _ y2
Lett the function f(x, y) be defined by f (, y)-=xy 2  2' for all values of x and y
except when x=0, y=0; for which f(O, 0) =0. At the point (0, 0), the partial differential
coefficients a ", af both exist, and have different finite values.
axay' ayx
The function f(x, y) is continuous at the point (0, 0); for, writing x= r cos 0, y r sin 0,
the function becomes jr2 sin 40; and this is numerically less than e, provided r<2 "/E.
We find     )= x{    +2 + ( x2y2 )}, at any point except (0, 0); at which point
We find af(,          2    (cc + y, ).f    i, f(x, O -f(  O) which is =O.
is lim     -which is =0.
OX   X=o       X
The value of af(a  ) is -y, and that of f (X 0) is x.
We then. find      (      li 1af (0   ) af (0      - 1,
aya-x    y=o y  ax       ax
and                       )     1f(, 0) af(0, 0)} -
and                =2y(0,_~x  lim 1   - ~       '   =1.
aca3y   X0x      ay       ay
T ve af(X, )2 a2fl(x,, Y).         2-y2 fl   8X22,
The value of   a.aq), as also that of  ay   is --        (        I2+y)2j, at every
ax?y                  ayax      y2    \ yY  (X2 + 2)2
point except (0, 0). This value is cos 20 (1+2 sin2 20), which is constant for a constant
value of 0, but has different values for different values of 0; and thus the partial differential
coefficients are discontinuous at the point (0, 0). The conditions of the theorem giving
sufficient conditions for the equality of  f  and  are therefore not satisfied for the
ax-ay   ayax
point (0, 0).
* See Stolz, Grundzuge, vol. i., p. 153.        t Peano, Calc. Diff., p. 174.
H.                                                                21




322


Functions of a real variable


[OH. IV


MAXIMA AND MINIMA OF A FUNCTION OF TWO VARIABLES.
241. Let us suppose that a function f(x, y) is defined at all points in
a two-dimensional neighbourhood of the point (xo, Yo).
If the function be such that f(x0 + h, yo + k) -f(xo, yo) < 0, for all values
of h, k which are not both zero, and are such that lh, Ik are both less
than some fixed positive number 8, then the function f(x, y) is said to have
a proper maximum at the point (;x, yo).
In case the fixed number 8 can only be so determined that the condition
f(xo + h, yo + k) -f(xo, yo) _ O, is satisfied, the function is said to have an
improper maximum at the point (x,, yo).
If the conditions contained in these definitions be replaced by
f(xo + h, yo + k) -f(xo, yo) > 0, and f(xo + h, yo + k)-f(xo, yo) > 0
respectively, the function f(x, y) is said to have, in the first case, a proper
minimum, and in the second case, an improper minimum, at the point
(xo, o).
A proper or improper maximum or minimum may be spoken of as an
extreme of the function.
At an extreme (x,, yo), f (xo + A, yo) -f(xo, Yo), f (o - h, Yo) - f (, yo)
both have the same sign, or are zero, for all sufficiently small values of h; it
follows that, if af(xo~ Yo) exist, it must be zero. A similar remark applies
aXo
af/(x, yo)
to Yo
These conditions are necessary, under the hypothesis of the existence of
the two partial differential coefficients, but not sufficient, for the existence
of an extreme at the point (xo, yo).
If we write x = x0 + r cos 0, y = yo + r sin 0, f(x, y) =  (r, 0), it is clearly
necessary for the existence of an extreme of f(x, y) at (x0, yo), that b (r, 0) for
each constant value of 0, should have an extreme at r = 0. Thus, for an assigned
value of 0, a positive number ao can be determined such that one of the four
conditions q (r, ) -f(xo, yo) < O, 4(r, 0) -f(xo, yo) - O, b(r, ) -f(xo, Yo) > 0,
(r, 0) -f(x0, y) yo), according as the point is a proper maximum, an
improper maximum, a proper minimum, or an improper minimum, shall be
satisfied for all values of r different from zero, and such that j r I < a. Thus
an extreme of a function is necessarily an extreme for values of the function
on each straight line drawn through the point.
This condition, though necessary, is however not sufficient; for ao may
have a definite value for each value of 0, and yet the lower limit of as for
all values of 0 may be zero. In this case, no value of 8 can be determined,
as required in the definition of the extreme in the two-dimensional domain.
It has thus been shewn that, in order that (xo, Yo) may be an extreme point




241, 242]


Maxima and Minima


323


for the function f (x, y), it is necessary and sujicient that (1) r = 0 should be
an extreme point of b (r, 0) for each value of 0, and (2) that* the number ao
which is so determined for each value of 0 that for I r I < a the condition as to
q (r, 0) -f(xo, yo) may be satisfied, should have a finite lower limit when all
values of 0, (0 - 0 <  7r) are considered.  If the lower limit of ao be zero, the
point is not an extreme point of the function.
When the lower limit of as is d (> 0), the neighbourhood of (x0, ye), which
must exist in accordance with the definition, is the square of which the
corners are the four points x0 + -2, yo + d)
EXAMPLE.
As an example of a function which possesses no minimum at a point, although the
point is a minimum for each straight line
through the point; we may take the function                                Q
(y- ax2) (y- bx2) =y2-y(a2+ b.V2)+ abx4,
where a and b have positive values.                             +
The function is positive outside the two
parabolas                                                \               +
y-ax2=0, y-bx2=0,                                    A
and in the space interior to the inner parabola;      +             +
in the space between the parabolas, the function
is negative.  Along any straight line QAR
through A (0, 0), the function exceeds f(0, 0)        R
at all points interior to AP, and everywhere
in PA produced; thus for the line QAR the
function has a minimum at A. The point (0, 0) is not a minimum of the function, since
the lower limit of AP for all positions of QAR is zero; and thus there exists no twodimensional neighbourhood of A, in which the function is never less than at A.
242. We may without loss of generality take the point at which the
conditions for the existence of an extreme of the function f(x, y) are to be
investigated as the point (0, 0). It will be assumed that, at all points in the
neighbourhood of (0, 0), f(x, y) is continuous with respect to x, and also with
respect to y. The following theorem contains a criterion for the existence of
a proper maximum (minimum) at the point (0, 0).
The necessary and sufficient conditions that the point (0, 0) may be a
point at which f(x, y) has a proper maximum        (minimum) are the followingt:-(1) A positive number 8 must exist which is such that, if x be any
* The necessity for this condition has been disregarded in many text-books. The insufficiency
of (1) was first pointed out by Peano, Calcolo diff., Turin 1884, p. 29, in connection with the
example given in the text. See also Dantscher, Math. Annalen, vol. XLII., p. 89, and Scheeffer,
Math. Annalen, vol. xxxv., p. 541.
t See Stolz, Wiener Berichte (Nachtrag), vol. 100, also Grundziige, vol. I., p. 213.
21-2




324


Functions of a real variable


[CH. IV


number different from zero, and numerically less than 8, the upper (lower)
limit of f (x, y), for such constant value of x, and for all values of y for which
- xy    x, being f(x, + (x)), this upper (lower) limit is for every value of
x (- ( < x $ 0 < 3) less (greater) than f(0, 0).
(2) A positive number 8' must exist which is such that, if y be any number
different from zero, and numerically less than 8', the upper (lower) limit of
f(x, y), for such constant value of y, and for all values of x for which
- y   x _- y, being f (r (y), y), this upper (lower) limit is for every value of
y (- Y' < y 4 0< 8) less (greater) thanf (0, 0).
It will be observed that, since f (x, y) is assumed to be continuous with
respect to x, and also with respect to y, the limit f(x, ( (x)) is actually
attained for some value + (x) of y in the interval (-x, x), and the limit
f(4 (y), y) is actually attained for some value  (y) of x in the interval
(-y, y). It is clear that, unless both the conditions stated in the theorem
be satisfied, f (0, 0) cannot be a proper maximum (minimum) of the function.
If, for example, no such number as 8 in (1) can be determined, there are
points in every neighbourhood of (0, 0) at which f(x, y) is _ (-) f(0, 0).
The conditions are sufficient. For, if 3, 8' exist, the value of f (x, y) at every
point, except (0, 0) within the neighbourhood the corners of which are the
four points (+ 8", + 8") is less (greater) than f(O, 0), where 8" is the lesser of
the two numbers 8, 8'.
The necessary and sufficient conditions that the function f(x, y) may
have an improper maximum (minimum) at (0, 0) are similar to the above.
In this case f(x, b (x)) must be less than, or equal to (greater than, or equal
to) f(0, 0) for all the values of x in the interval, and f ( (y), y) must be less
than, or equal to (greater than, or equal to) f(O, 0) for all values of y in the
interval. Further, corresponding to every positive number 8 < 8, there must
be a value of x(< 8), for which f(x, qb (x))=f(0, 0); or else a similar
condition must hold for f (f (y), y); or in both cases, the condition may be
satisfied.
Other methods of determining whether (0, 0) be a point at which there is
a maximum or minimum of f(x, y) will be dealt with in Chap. vI.
PROPERTIES OF A FUNCTION CONTINUOUS WITH RESPECT TO
EACH VARIABLE.
243. Let a function f(x, y), defined for all values of x and y in a continuous domain, be everywhere continuous with respect to y, and be also
continuous with respect to x along each straight line parallel to the x-axis,
and belonging to a set cutting the y-axis in an everywhere-dense set of points.




242, 243]  Functions continuous in each variable


325


Let A be the point (x, y), and let BC be drawn with A as its middle
point, parallel to the y-axis, and of length 2p. If w (p) be the fluctuation of
f(x, y) in the interval BC, then o (p) is a continuous function of p; and
lim w (p) = 0, since f(x, y) is everywhere continuous with respect to y. Let cr
p=o
be a fixed positive number, and let /3 (x, y) denote the upper limit of those
values of p for which w (p) _ a-: thus co (p) _ a, if p _ /, (x, y); and w (p) > a,
if p >  (x, y).
The function, (x, y), thus defined for every point (x, y), is everywhere
positive; and it will be shewn to be an upper semi-continuous function with
respect to the two-dimensional continuum (x, y), in accordance with the definition in ~ 183.
Take BoAo = A    = /A, (x0, yo); and also BBo = CC, = le, where e
is a fixed positive number.  The fluctuation of
f(x, y) in B1C7 is greater than a; let it be o- + k.
If ki be a fixed positive number < k, two points
M, N can be found in BC,, such that                   N    Q
If(M)-f(N) [ >   + k                       C.
Moreover, these points M, N can be so chosen as to
lie on two straight lines parallel to the x-axis, which 
belong to the set along each of which f(x, y) is           A
continuous with respect to x; this follows from
the fact that this set of straight lines cuts BiC1 in
an everywhere-dense set of points. Since f (x, y) is
continuous with respect to x, at each of the points
M, N, two segments M'M", N'N", with M and N
as their middle points, can be determined, so as to    Bo
have equal lengths 28, and to be such that
If(P)-f (M) I < k,,                M'      M        M"
If(Q)/-f(N) ] < k-,provided P be any point in M'M", and Q be any          B
point in N'N".
From these inequalities and the former one, we deduce that
If(P)-f(Q) > c.
Take the square of which Ao is the centre, and of which the sides are
parallel to the axes, and are at a distance from Ao less than the smaller of
the two numbers le and S. If A be any'point in this square, the distance
of A from each of the straight lines M'M", N'N" is less than /3 (0x, yo) + e.
Through A let a straight line be drawn parallel to the y-axis, and mark off
on it the segment of which A is the centre and of which the half-length is




_ _ __




326               Functions of a real variable             [CH. IV
/3( (x,, yo) + e; this segment will cut M'M" and N'N", and therefore contains
two points P, Q which are such that
If (P) -f (Q) I > 
Therefore the fluctuation of f(x, y) in this segment is > a, and hence at
the point A (x, y) we have /3 (x, y) < /3 (x,, yo) + e. A square having been
determined with its centre at Ao, such that for every point in this square
1a (x, y) </ r (S0, yo) + C,
it follows that /3 (x, y) is an upper semi-continuous function at Ao with
respect to the two-dimensional domain (x, y).
244. Let us now consider the linear set C of points (x, y) defined by
y=   (x), where ~ (x) is a continuous function of x.  At each point of
C, the function B,(x, y) is defined, and has at every point a minimum
relatively to the set C, the term being used in accordance with the definition
given in ~ 180.
If, at a point Ao (x0, y0) of C, the function /, (x, y) have its minimum with
respect to C positive, then we shall prove that the saltus of f(x, y) at A.,
with respect to the two-dimensional domain (x, y), is - 2-. Let y denote
this minimum, and let y7 be a positive number < y.
Let an interval (0 -, xo + 8) on the line y = y, be so determined that
in this interval
Q[                   I/!      \I


M
A
-  N
Bl  M  A1  N  C1
_.. 1


This interval may, if necessary, be so
reduced, that for all values of x in it,
/3a X,  (W) > 7l.
Describe the rectangle R, with A0 as
centre, the sides parallel to the axes of
x and y being 28 and 7y respectively. On
every segment PQ of R, parallel to Oy,
the fluctuation of the function is _ -:


P                                       -— _
for there is on PQ a point A of the set
C, and the segment with centre A, and
length 2/3,(A)> 27y, contains the whole
segment PQ.
Taking a fixed positive number e, an area surrounding Ao can be determined, in which the fluctuation of f(x, y) is _ 2or + e. To effect this, take a
point Al on the ordinate through A, and in the rectangle R, such that A, is
a point of continuity of f (x, y) with respect to x; then on the straight line
y = yi, take a segment BCG with centre Al, and of length 28' 28, such that
the fluctuation of f in B(JC is < e. Consider the rectangle R' contained in






243, 244]    Functions continuous in each variable                 327
R, such that the sides of R' are of lengths 28' and ry parallel to the axes, its
centre being at A0. The fluctuation of f(x, y) in this rectangle is < 2o + e.
For if M, N be any two points in it, let MI, 1V, be their*projections on
BC(7; then
I / (M)-/(MI) |-,    f (N)- f(N,) I a,.f (M) -f (N) < e;
and from these inequalities we deduce that
f/(M)-f(N) i < 2 + e.
Since this holds for every e, the saltus off(x, y) at A0 is = 2o-. If, at a point
A0, the saltus off (x, y) be > 2r, then at Ao the minimum of /3 with respect to
C must be zero.
Since a, is positive at every point of C, and is an upper semi-continuous
function of (x, y), it follows from the theorem of ~ 184, that in every arc D of
the curve C, there exists an arc D1 in which the minimum of 3, is positive.
Let us take a sequence t1, 2,... -n... of positive decreasing numbers of
which the limit is zero.  It is then clear that in every arc D there exists a
point where /3~ has its minimum with respect to C positive, for every on.
At this point the fluctuation of f(x, y) with respect to the two-dimensional
continuum (x, y) is < 2o-n, for all values of n, and is therefore zero. This
point must be a point of continuity off(x, y) with respect to (x, y).
The following general theorem * has now been established:If f (x, y) be a function of the two variables x, y which is everywhere
continuous with respect to y, and is continuous with respect to x along straight
lines parallel to the x-axis, which cut the y-axis in an everywhere-dense set of
points, then in every portion of a curve y = q (x), where q (x) is a continuous
function, there exist points at which f (x, y) is continuous with respect to the
two-dimensional domain (x, y).
It follows from this theorem that points of continuity exist in every area,
that is f(x, y) is at most a point-wise discontinuous function.
The whole of the reasoning above is applicable, if only those points of
(x, y) are taken account of, which belong to a perfect set G. It thus appears
that, under the conditions stated in the above theorem, f(x, y) is a point-wise
discontinuous function relatively to every perfect set G of points in (x, y).
The points of continuity of f(x, y) on the curve y= k (x), are everywheredense with respect to every perfect set of points on the curve.
EXAMPLES.
1. If + f(x, y) be a function which is everywhere continuous with respect to each of
the variables x, y, then the points at which the saltus of f (x, y) with respect to the twodimensional continuum (x, y) is _ o- form a set of points such that the projection of the
set on either axis, by lines parallel to the other axis, is a non-dense set.


* Baire, Annali di Mat. Ser. Ia, vol. III., 1899, p. 27.


t Baire, loc. cit., p. 94.




328


Functions of a real variable


[CH. IV


2. If* a function f(x, y, z) of three variables x, y, z be everywhere continuous with
respect to each variable, then f(x, y, z) is at most a point-wise discontinuous function
relatively to the? three-dimensional continuum  (x, y, z). Further, on every surface
x =( (y, z), where q is continuous with respect to (y, z), the function f (x, y, z) is at most
a point-wise discontinuous function with respect to (y, z). The set of points at which
the saltus of f(x, y, z)  a- may contain all the points of a continuous curve.
3. Let* ) (x, y) be a function which is continuous with respect to each of the
variables x and y, and let (0, 0) be a point of discontinuity of 4) (x, y) with respect
to (x, y). Define f (s, y, z) by the condition f (, y, z) =) (S, /); then the function f(x, y, z)
is continuous with respect to each of the three variables, but every point on the z-axis is
a point of discontinuity with respect to (x, y, z).
4. Let* f(x, y, z) be a function which is constant along any straight line parallel to
the straight line x=y=z, and is such that f(x, y, 0)-=  x 2)f, /(O, 0, 0)=0. This
(X2 +Y 2) '
function is discontinuous at every point on the straight line x=y=z.
245. The methods developed by Baire of dealing with functions of two
or more variables, in relation to the distribution of the points of discontinuity, have been applied by him to the consideration of the following
three problems:(1) VWhat must be the nature of a function ( (x), defined for a < x _ /,
in order that a finction f (x, y) can exist which is defined for all points in the
square a _ x - /, a - y,/3, and is continuous at every point with respect to
x and with respect to y, and moreover is equal to  ( (x) on the straight line
x=y?
(2)  What must be the nature of a function q (x) defined for a _ x c /3,
in order that a function f(x, y) can be defined for all points in the square
a   x, /3 a  y  /3, and which shall satisfy the conditions that it is continuous with respect to (x, y) at every point for which y > 0, is continuous
with respect to y at the points of y = 0, and is equal to b (x) when y= 0?
(3)  A function f(x, y) is defined in the rectangle a < x   /3g, 7y  y < 8,
and is everywhere continuous with respect to y. Further, there is a set of
parallels to the x-axis, along each of which f(x, y) is continuous with respect
to x; these parallels intersecting the straight line x= a in a set of points
which is everywhere-dense in the interval (y, 8).  What is the nature of the
function f(x, y) on a continuous curve drawn in the rectangle?
The problems (1), (2) are particular cases of (3). It has been shewn
above that a necessary condition satisfied by f(x, y) in (3) is that it should
be a point-wise discontinuous function relatively to every perfect set of
points.  That this condition is also sufficient, has been demonstrated by
Baire in his memoir quoted above. A proof of this will be given for the case
of problem (2), in Chapter VI, in connection with the theory of functions
representable as the limits of sequences of functions.
* Baire, loc. cit., p. 99.




244-246]


Curves filling a space


329


THE REPRESENTATION OF A SQUARE ON A LINEAR INTERVAL.
246. Let a point of a square whose side is unity be denoted by (x, y),
where 0 - x   1, 1 0  /1; and let t denote a point of a linear interval
(0, 1). An account has been given in ~ 58 of Cantor's method of establishing a (1, 1) correspondence between the points of the square and those of
the linear interval.  Such a correspondence denotes functional relations
x =f(t), y =  (t) between x, y as dependent variables, and t as an independent variable. It will be shewn however that no (1, 1) relationship
between the two sets of points can be a continuous representation*; i.e. it
is impossible that the functions f (t), c (t) can be both continuous.
Let us assume that such a continuous representation can be defined.
To any closed set of points It} in (0, 1), there will correspond a closed
set in the plane area. For if t,, t2,... tn,... be a convergent sequence of
points t, of which t, is the limiting point, then the point f(tw), > (t.) is
the limiting point of the set of points (x,, y,) (x,, y)... (, yn)... which
correspond to t1, t2,... t',... respectively; therefore to a closed set {t} there
corresponds a closed set {(x, y)}. Again, to a convergent sequence (x1, y1),
(x2, y2)... of points in the plane area, there corresponds a set of points
t1, t2,... in the linear interval, the latter of which has a limiting point t, which
must correspond to (x,, y,); and since only one value of t corresponds to one
set of values of (x, y), there can be only one such limiting point t,.  Thus, to
a closed set in the plane, there corresponds a closed set in the linear interval.
Take two points tj, t, in the interval (0, 1); these points correspond to two
points P1, P2 in the square area. To the closed linear interval (t4, t2) there
corresponds a closed set S which contains the points P1, P2. It can be
shewn that there are points other than P1, P2 on the frontier of S. Denote
by C(S) the set of those points of the square area which do not belong to S.
Two points Q, R in the square can be determined, such that Q lies on the
straight line PP2,, and R does not lie on this straight line; such that neither
Q nor R coincides with P1 or P2, and such that one of the two belongs to S
and the other to C (S). The closed set consisting of the straight line QR
contains points both of S and of C(S); those points of S which lie on it
form a closed set, and there must be one such point of S at least which is
on the frontier of S; such a point may or may not coincide with Q or R.
Since then S contains points on its frontier besides PI and P2, we can take a
point t within the linear interval (t4, t2) such that the point T in the square
which corresponds to it is on the frontier of S. Since T is the limiting point
of a sequence of points of C (S), it follows that t must be the limiting point
* See Netto, "Beitrag zur Mannigfaltigkeitslehre," Crelle's Jl., vol. LXXXVI.; also Loria,
Giorn. di Mat., vol. xxv., p. 97. In the proof given by these writers it is assumed that a closed
curve corresponds to a linear sub-interval of (0, 1); this is not necessarily the case, for a nondense closed set may correspond to the closed curve.




330


Functions of a real variable


[CH. IV


of a sequence of points all of which are external to the interval (t1, t,); and
this is impossible. It has thus been established that:No continuous (1, 1) correspondence can exist between all the points in a
square and all the points in a linear interval.
In particular, the correspondence shewn by Cantor to exist, must be
discontinuous.
247.  The reasoning of ~ 246 would be inapplicable if the correspondence x =f(t), y = b (t) were such that, to a given point (x, y) more
than one point t may correspond, the functions f(t), + (t) being still onevalued continuous functions, so that if t be assigned, (x, y) is uniquely determined. In this case, the limiting point of the set of points external to the
interval (t,, t2) would be not i, but another value of t which also corresponds
to the point T.
Peano* gave the first continuous correspondence of the kind just indicated,
thus defining a continuous curve which passes through every point of the
square at least once.
Let the points in the interval (0, 1) be expressed in the form
t= ala2a3... n.,
in radix fractions in the ternary scale, so that each a is either 0, 1, or 2.
Let k(a) denote the number 2 - a, so that k(2)=0, k(1)= 1, k(0)= 2;
and let kP (a) denote the result of performing this operation n times, so that
kcn (a) is a or 2 - a, according as n is even or odd.
Let x, y be defined for a prescribed t by
x =blbab3..., y = 'C1C3...,
the ternary scale being again employed; the numbers b, c being defined by
the relations
bl = a,, b, = ka2 (a3),... bn= ka + a4+... + a2-2 (a2-_),
l = Ical (a,), C2 = kal + as (a4),... Cn = kal a3+.. + a2n-l1 (an);
thus bn is equal to a,,_ or to 2- a2,n_, according as a, + a4 +... + a2n-2 is
even or odd.
The numbers t may be divided into two classes:(1) Those, other than 0 or 1, which are capable of a double representation
t = 'acxaa,... an2 2 2... 'aa2... an +  0 0 0....
(2) Those which have a single representation only.
If t be a number of the second class, x and y are uniquely defined. If t be
a number of the first class
t = aa q... an2 2 2..  'aila... la n +   0a  0... 
* " Sur une courbe, qui remplit toute une aire plane," Math. Ann., vol. xxxvi., 1890.




246, 247]             Curves filling a space                     331
let bbb,..., 'b2/bb'... denote the numbers obtained by applying the
definition of x to the two modes of representation of t. If n is even, say
2m, it is clear that
bl = b', b, = b2',... bm = bm;
also      bm+ - ka2 + a4 +... + a2m 2  bmi+l = ka, + a4 +. + a.+  + 1 0,
b+2 = ka2 + a4 +.. + a2 + 2,  b2 =ka2+ a4 +...+ a2m +;........................
hence               bn+b        62= b',m+, b+ = b *+2,..;
and thus x has the same value whichever of the two forms for t is employed;
the case in which n is odd may be similarly treated.
The same result can readily be shewn to hold for y. Therefore, corresponding to any assigned t, x and y are uniquely determined.
Next, let us suppose x and y to be assigned. We have
a, = bl, a, = kbl (c,), a, = C (b2), a = kb + b (),...
an-1 = kC1+ C2+ --- +C (b),   a, = kb, +tb+...+ b (cn);
for, ifp = cr (q), then p + q is an even number.
In case x, y are both of the second class, t is uniquely determined.
If x is of the first class, and y of the second; let
x = -blb2... b, 2 2 2... = -blb2... b, + 1 0 0...,
=  'i C... CnCn+l -...,
and let the two values of t be denoted by 'aaaS..., a,'a' a....
It is clear that
a, = a,', a, = a2... a2n-i = at21-1;
also a2,= k (a',2), an+ =kl+ 2 ++ n (bn), a'2n+l = kc+c+++C(bn+ 1);
thus a2n+l, a' 2+, are not identical, although an, an' will be so if each is unity.
It is thus seen that t has two distinct values corresponding to one point (x, y)
when x is a number of the first class, and y is of the second class. It can be
shewn in a similar manner that there are four points t corresponding to a
single point (x, y) such that x, y are both numbers of the first class.
The correspondence is continuous. For if t, t' are identical as regards the
first 2n figures, x and x' are identical as regards their first n figures, and the
same is true of y and y'.
The curve which has thus been defined is a continuous curve which passes
through each point in the square at least once; there is an everywheredense enumerable set of points through each of which the curve passes twice,
and another everywhere-dense enumerable set of points through each of which
it passes four times; through each of the remaining unenumerable set of
points, the curve passes once only.




332


Functions of a real variable


[CH. IV


The plane measure of an arc of Peano's curve which corresponds to
an interval (t0, tl) is not zero, i.e. the area which a number of rectangles
enclosing all the points of the arc have in common has a lower limit greater
than zero.
The two continuous functions f (t), 4 (t), which define x, y as functions of
t, possess for no value of t definite differential coefficients, and are perhaps
the simplest examples of continuous non-differentiable functions.
248. It might at first sight appear that a curve having the same
properties as that of Peano might have been defined by restricting t = 'aa...
to be such that an infinite number of digits other than 0 are present, and
then defining x, y by
x = 'aa3a5..., y= 'ca4a.....
If however the double representation of x, y were not restricted, as in the
case of t, there would be no value of t corresponding to, say,
x=    1000..., y=-2000....
If (x, y) were on the other hand so restricted, there would be no values of
(x, y) corresponding, for example, to
t = *111010101....
It thus appears that some such rule as that given by Peano is necessary to
obviate the difficulty caused by the double representation of a certain class of
rational numbers, in a given scale.
The method may easily be extended to obtain a continuous correspondence between the points in a cube and those in a linear interval.
A somewhat different method of establishing correspondence between
the points of the square, and those of the linear interval, is the following*:Let t, denote one of the perfect set of points defined by
a, aa   a3
t:      + - -  +...,
3 +32   33+*
when every a is either 0 or 2.  For such a point tj, x and y may be
defined by
=~      +   + +....
1 al  + as + a +
Y2 e2 222     23 s 'g 
A point t which does not belong to the perfect set is interior to one of the
* See Lebesgue, Legons sur N'integration, p. 44.




247-249]


Curves filling a space


333


complementary intervals (t,', tl") of the set; in such an interval we may
define x, y as linear functions of t, thus
XI - XH 
x = X' + t  t7, (t-t ),
Y =y' + t-, (t- t),


where (x', y'), (x", y") correspond to tj', ti" respectively.
249. A method of constructing a continuous curve which
has been given in a geometrical form by Hilbert*.
2                3
1                4


fills a square


1        2       3      4
FIG. 1.
Divide the interval (0, 1) into four equal parts, and number them in order
as 1, 2, 3, 4. Then divide the square into four equal parts, as in Fig. 1, and
number them 1, 2, 3, 4, to correspond with the segments of the linear
interval.  Next divide each segment of the straight line into four equal
parts, and each of the four squares into four equal parts as in Fig. 2.  The
sixteen squares so formed are then numbered in order so that each square has
one side in common with the one next in order; the squares then correspond
with the segments numbered in the same way. At the next stage there are
(Fig. 3) 64 squares corresponding to 64 segments of the interval (0, 1).
Proceeding in this manner indefinitely, any point of (0, 1) is determined by
the intervals of the successive set of sub-divisions in which it lies. The
corresponding point in the square area is determined by the succession of
squares each containing the next in which it lies.  The curve is thus determined as the limit of a sequence of polygons denoted by the thickened lines
in the figures.  The curve thus obtained is continuous, but has no tangent.
* See Math. Annalen, vol. xxxvIII., p. 459.




334               Functions of a real variable              [CH. IV
Hilbert remarks that if the interval (0, 1) be taken as a time interval,
a kinematical interpretation of the functional relation between the curve
and the segment is that a point may move so that in a finite time it passes
through every point of the square area.
6            7      10           11
5          8   -   -    9         -12
4            3       14           13
1 -2                15           16


1  2-3


16


FIG. 2.


FIG. 3.




249]


Curves filling a space


335


Continuous curves of this kind can be constructed by any method by
which an everywhere-dense enumerable set of points in the square can be
made to correspond with a similar set of points in the linear interval;
provided the functional relation x =f(t), y = 4 (t), in such correspondence, is
uniformly continuous. For, when this condition is satisfied, the functions
obtained by the method of extension of f(t), q (t) to the remaining points
of (0, 1) as secondary points (see ~ 225) will yield a correspondence of all
the points of the square with those of the linear interval, of the required
character.
Another method differing from that of Hilbert has been given by Moore*
and by Schonfliest.
Let m be an uneven number (in the figure, m =3); divide the linear interval
(0, 1) into m2 equal parts, and also the square into m2 equal parts. Let these


1 
0
i  ' 1  -+ r  1 " I ~" 1  i-  i




0   i  2   3   4   5  6   7   8   9
squares be passed through by a polygonal line, of which the sides are
diagonals of the squares, as in the figure; in this manner the squares are
arranged in order 1, 2, 3,... m2, and are placed into correspondence with the
segments bearing the same numbers. At the same time the end-points of a
diagonal so traversed are made to correspond with the end-points of a
segment of the linear interval. Thus m2 + 1 points in the linear interval are
placed in correspondence with points in the square, so that to each of the
m2 + 1 points of the linear interval there is one point in the square; but the
converse is not the case. Next, divide each of the m2 linear intervals into m2


* Trans. Amer. Math. Soc., vol. I., p. 77.
t Bericht iiber die Mentgenlehre, p. 121.




336               Functions of a real variable               [CH. IV
equal parts, and the corresponding squares into m2 equal parts; then construct as before a polygon traversing diagonals of all the m4 squares, making
their end-points correspond to the end-points of the corresponding m4 parts
of the linear interval. Proceeding in this manner, we gradually place points
in the square, consisting of an everywhere-dense enumerable set, into correspondence with a set in the linear interval which possesses the same
property; and the functional relation so set up is uniformly continuous. The
definition of the functions for the whole linear interval is then obtained, as
explained above, by the method of extension. The case m = 3, corresponds
to Peano's analytical method. In the method of Moore and Schonflies, the
curve is determined as the limit of a sequence of polygons inscribed in
the curve. In Hilbert's method the polygons which approximate to the
form of the curve are not inscribed in the curve, but are otherwise determined.




CHAPTER V.


INTEGRATION.
250. THE fundamental operation of the calculus, known as integration,
regarded from one point of view consists essentially in the determination
of the limit of the sum of a finite series of numbers, as the number
of terms of the series is indefinitely increased, whilst the numerically
greatest of the individual terms of the series approaches the limit zero.
The laws which regulate the specification of the terms of the series
must be supposed, in any given instance, to be assigned, and to be of
such a character that the limit in question exists. It is in this form that
the problem of integration naturally presents itself in ordinary problems
of a geometrical character, such as the determination of lengths, areas,
volumes, &c. The method of integration, so regarded, has its origin in the
method of exhaustions employed by the Greek geometers, and was developed
later in forms of which the exactitude depended at various epochs upon the
stage which the development of Analysis in general had reached. In the
hands of Cauchy, Dirichlet, and Riemann the definition of the definite integral
attained to the exact arithmetic form in which it is employed in modern
analysis; and in fact the definition given by Riemann, which is now held to
be fundamental in the calculus, leaves nothing to be desired as regards precision. Riemann gave not only a precise definition, but also a necessary and
sufficient condition, for the existence of the definite integral. Although a
more general definition of integration has recently been developed by
Lebesgue, in accordance with which classes of functions are integrable,
which are not so in accordance with the definition of Riemann, the latter
is the definition which lies at the base of almost all the developments of the
theory of integration that have been made during the last half century, and
will therefore be adopted for full tieatment in the present Chapter. An
account will however be given of the recent more general theory due to
Lebesgue.
Integration has also usually been regarded as the operation inverse to that
of differentiation; and the fundamental theorem of the Integral Calculus
formulates the relation of this mode of regarding integration with the one
referred to above. Many important investigations are concerned with the
relation between these two modes of regarding integration, with the establishment of the fundamental theorem, and with an examination of the limitations
to which it is subject.


H.


22




338


Integration


[CHo. V


THE DEFINITE INTEGRALS OF LIMITED FUNCTIONS.
251. Let f (x) be a limited function, defined for the continuous domain
(a, b), where b ' a; so that there exists an upper limit U and a lower limit L
of the functional values in the whole interval. Let the interval (a, b) be
divided into any 7? sub-intervals 810), 82(1): 8(, ( -) 821,(, so that
81(S) + 8^(1) +... +  '8,(1) = b -a,
and let A, be the greatest of these sub-intervals. Let these sub-intervals be
further sub-divided in any manner so that the whole interval (a, b) then consists of n2 sub-intervals 8(2),, 82(... 8,12(2 whose sum is b - a, and the greatest
of which is A; let further sub-divisions of these sub-intervals be made, and
so on continually, so that at any stage of the process the interval (a, b) is
divided into n,,, sub-intervals, 8(), 82(?),.. m ^,,( (), the greatest of which is A,,,.
If this system of continual sub-division be made in any manner whatever,
which is such that the sequence A,, 2,... A,,,... has the limit zero, we shall,
as in ~ 193, speak of it as a convergent system of sub-divisions of the interval
(a, b). Let M (8("')) denote any number whatever which is so chosen as to be
not greater than the upper limit of the function f (x) in the closed interval
8(ro), and so as to be not less than the lower limit off(x) in the same interval;
and consider the sums
Si = 81(1) I(81()) + 82()lM(8() ) +... +,,(M (8I(s,()),
AS2 81(2'1J (81(2) ) +  (2M.82 (2))+  + &2 (2)IV(48 12(2)
8, =     8 (f(8)+      ( ) + 8... +  8,nm (M) M (82,n () )
If the sequence S1, S,... Sn,,... be convergent and have the same number S
for limit whatever convergent system of sub-divisions of (a, b) be eimployed, and
however the numbers M(8&s(')) be chosen, subject only to their limitation in
relation to the upper and lower limits of f (x) in the intervals &S(?m), then the
function f (x) is said to be integrable in the interval (a, b), and the number S
defines the value of its integral.  This integral, when the limit S exists, is
rb
denoted by   f(x) dx.
It will be observed that Mf(8) is not necessarily the value of f (x) at any
point in the interval 8; for all that is necessary is that it should not be
greater than the upper limit, nor less than the lower limit, of f (x) in the
interval 8. In this respect the definition is a slight generalization of that
given by Riemann, who restricted M () to have the value of f(x) at some
point in the interval 8.
The definition of a definite integral, of which Riemann's definition is a
* We7rke, 2nd ed. p. 239.




251, 252]


Definite Integrals


339


development, was given by Cauchy, for the case of a continuous function.
Cauchy's definition is in fact that which arises when M(8) is in every case
restricted to be the functional value at one end of the interval 8; thus it may
be expressed by
rb
f(x) dx = lim [(xi - a)f(a) + (x2 - x)f(xi) +... + (b - x) (x)],
where a, X,, x2,... x,, b are the end-points of the sub-divisions, and the limit
is determined under the same conditions as have been stated above.
252. The investigation of the necessary and sufficient conditions that the
integral of f (x) in (a, b), as above defined, may exist, is considerably simplified
by the introduction of the notions of the uppert and lower integrals of the
function f (x) in the interval (a, b).
If, in the successive sums which are formed corresponding to a convergent
system  of sub-divisions of (a, b), we identify every number M (3) with the
upper limit U(8) of the function in the interval 8, it can be shewn that for
any limited function whatever, the sequence of numbers
81 ) U(81(1)) +  (1 U(82(')) +.. *. + 811( U(38()) = 
1(2) U(81(2)) +2(2) U(2(2)) +... + &22(2) U(82 (2)) =2
(mn) U(81 (n)) + 82(') U(82(')) +... +  nzm(") U(&Smn() = Em
has a definite limit when m is indefinitely increased, which is independent of
the particular convergent system  of sub-intervals.  This limit is called the
upper integral of f (x) in the interval (a, b), and may be denoted by
Jf (x) dx
A similar theorem holds if M(S) be in every case identified with the lower
limit of f (x) in the interval, the corresponding sum converging to a number
which is also independent of the particular convergent system of sub-intervals
chosen. This limit is then termed the lower integral of f(x) in (a, b), and is
rb
denoted by     / (x) dx.
To prove that the upper integral of a limited function always exists, we
observe that when any sub-interval is subdivided the upper limit in no one of
the sub-divisions can be greater than in the original sub-interval, and consequently Em+i cannot be greater than fm. It thus appears that El, 2,...,,...
* Journal de 'Ecole Polytechnique, cab. 19 (1823), pp. 571 and 590.
t The upper integral (oberes Integral) and the lower integral (unteres Integral) are named by
Jordan, "l'integrale par exces," and "l'integrale par defaut" respectively; see Cours d'Analyse,
vol. I, p. 34. They were introduced by Darboux, Annales de 'ecole normale, ser. 2, vol. iv, and
also by Thomae, Einleitung, p. 12, and by Ascoli, Atti di Linedi, ser. 2, vol. In, 1875, p. 863.
22-2




340


Integration


[CH. v


form a sequence of numbers which do not increase, and moreover none of
them is less than L (b - a); consequently they form a convergent sequence
of which we may denote the limit by N. It must now be shewn that N is
independent of the particular convergent system of sub-divisions. Suppose,
if possible, that another system of sub-divisions leads to another limit N';
we may without loss of generality suppose that N' < N. Take a system of
intervals El, e2,... e, belonging to the second system of sub-divisions, where
we may suppose s to be so great that the sum for this system is < N'+,
where ~ is an arbitrarily small number, and we choose it so that N' + ' < N.
Let n,, > s, and suppose the two sets of sub-divisions
81 () 82 (M) I * nn (M)  e1,  e2,  
to be superimposed, so that (a, b) is divided up by all the points which
are end-points of sub-intervals of either set; the new division of (a, b) may
be regarded as a continuation of either set of sub-intervals into further
sub-division. Since s < nt, at most s -  of the nm intervals are divided
by introducing the points belonging to the e, and the diminution thus produced in E,, is less than or equal to (s - 1) A, (U- L), and thus the new
sum for the combined sub-divisions is I, - (s - 1)A,, (U - L). Now ml
can be chosen so great that A,, <  -1), where r is an arbitrarily
(s -    1) (U- L)'
chosen positive number as small as we please; and if this be done the sum
for the combined sub-divisions is > 2,, - r > N -. Again, since the same
sum may be regarded as belonging to a further sub-division of the intervals
e6, 2,... C,, it is < N'+ N. It is now clear that, since q can be chosen so
that N-   > NV' + ', the sum for the combined system of sub-divisions cannot
be both >N- / and less than N' + C; and it is thus impossible that N and
N' should be unequal: therefore the limiting sum which has been shewn to
exist for any prescribed system of sub-divisions has the same value for all
such systems. The existence of the lower integral may be proved by similar
reasoning, or is immediately deducible from the existence theorem for the
upper integral by considering the function -  (x).
253. It has now been shewn that a limited function f(x), defined for the
continuous interval (a, b), always possesses an upper and a lower integral in
the interval. The necessary and sufficient condition that f (x) should possess an integral as defined in ~ 251 is that the upper and lower integrals in the
interval be equal. That this condition is necessary follows at once from the
fact that all the numbers M (8) may be made identical with U (8), or all may
be made identical with the lower limits L (8) of the functions in the intervals
8; and that the condition is sufficient follows from the fact that S, lies beS —'m           S=?nm
tween E 8s(m) U (8((m)) and E 8S(m)L (8s(m)), and thus that when the two latter
s=l              s=l
sums have the same limit, that limit is also the limit of Sn,.




252-254]


Definite Integrals


341


The necessary and sfficient condition*z that f (x) may be integrable in the
interval (a, b) may now be expressed as follows. —Let D (8(n)) denote the fluctuation U(8(n) ) - L (6(in)) of the function in the interval 6 8(); then it must be possible
to define a convergent system of sub-divisions of the interval (a, b) such that, if
at any stage these sub-divisions are denoted by  (), 82(),,..., l('l), the sum
81 () D (81 (in)) + 82(n) D (8 (n)) ++  +,8n(?")D (8,,, (in)
should have the limit zero, as m is increased indefinitely.
That this zero limit exists is equivalent to saying that, corresponding to
any arbitrarily small positive number e, a number m can be found such that
S= 9~ml
the absolute value of v (8('W)) D (8)("')), or of (b - a) M, where M is a certain
s=l
mean of the numbers D (8sm)), for this value of m and for all greater values
of m, shall be less than e.
254. The necessary and     sufficient condition  for the existence  of
b
ff (x) dx, may be stated    in  a somewhat more convenient form, as
follows:If any convergent system of sub-divisions of the interval (a, b) be taken,
then, corresponding to any arbitrarily chosen positive number kA, the sum of
those sub-intervals of (a, b) in which the flactuation off (x) is greater than or
equal to k,, must, as the successive sub-division advances, become arbitrarily
small, and must then have the limit zero.
To see that the condition so stated is sufficient, we observe that, if s(n) be
the sum of those sub-intervals of $,(n), 82(, 7),...   ' 1( in which the fluctuation
is _ kl, then
('    D (8t(m)) ~ s(m) (U- L) + k (b - a - s().
t=1
Since s(m) has the limit zero as q is increased indefinitely, the limit of
'8t sD(8t(in)) is  k(b - a); and as k is arbitrary, the limit must be zero.
To shew that the condition is necessary, we observe that
I ft (n) D (8t(')) ks(in) + (b - a - s(n,) )D _ ks(),
t=l
where D is the least of the fluctuations in all the sub-intervals. Unless
therefore s(n) has the limit zero it is impossible that YS('n)D(80(')) can have
the limit zero.
Another form of the condition for the existence of  f (x) dx which is for
many purposes more convenient than the above forms of statement, involves
the saltus or measure of discontinuity at points of the interval instead of
the fluctuations in sub-intervals; it may be stated as follows: —
* Riemann's Werke, 2nd ed. p. 240.




342


Integration


[CH. V


The necessary and sufficient condition that the limited function f (x) may
be integrable in the interval (a, b) is that, for any value whatever of the
positive number k, those points of the interval at which the saltus a- is - k form
a set of points of zero content.
To see that the condition is necessary, let any system of sub-divisions of
(a, b) be taken; then the sum of the products of the sub-intervals multiplied
by the corresponding fluctuations is greater than k times the sum of those
sub-intervals which contain points of the set for which o-? k; unless therefore
the sum of these sub-intervals have the limit zero as the sub-division advances,
it is impossible that the sum of the products of sub-intervals and fluctuations
should have the limit zero. To shew that the condition is sufficient, we observe that if the content of the set of points a- > k be zero, all these points can
be included in a definite number of sub-intervals whose sum is less than the
arbitrarily small number e, so that all the points of the set are interior points
of these sub-intervals; and the rest of the interval (a, b) consists of a definite
number of sub-intervals whose sum is greater than b-a-e, and at every
point of which ar < k. Consider one of these latter sub-intervals 8. In
accordance with the theorem established in ~ 185, 8 can be divided into a
definite number of parts in each of which the fluctuation is less than k.
Since the same reasoning applies to every sub-interval 8, therefore the whole
interval (a, b) can be divided into a definite number of sub-intervals such
that the sum of those in which the fluctuation is > k is less than e, and this
however small e may be; and this is the condition of integrability established
above.
The most succinct form in which the condition of integrability of a limited
function may be stated is the following*:The necessary and sufficient condition that a limited function defined for a
given interval may be integrable is that the points of discontinuity of the
function form a set of measure zero.
For if kD, k,,... k,,... be a sequence of diminishing positive numbers which
converges to the limit zero, and GU, G,... G,,... be the closed sets of points
at which the saltus of the function is _ i,, - c,...  > A>,..., then the set of
all the discontinuities of the function is the limit of the set G,,, when n is
indefinitely increased, and this set must, in accordance with the theorem of
~ 88, have the measure zero, since n,, has the measure zero, for every value
of n.
This condition is equivalent to the condition that every closed set,
contained in the set of points of discontinuity of the function, may have
content zerot.
' Lebesgue, Annali di Mat. ser. 3, vol. vII, p. 254.
+ See W. H. Young, Quarterly Journal of Math. vol. xxxv, p. 190. See also Hobson,
Quarterly Journal, vol. xxxv, p. 208.




:254, 255]


InJtegrable functions


343


PARTICULAR CASES OF INTEGRABLE FUNCTIONS.
255. The following classes of limited functions satisfy the condition of
integrability which has been expressed in various forms above.
(1) All functions which are continuous in the intervals for which they
are defined.
(2) All functions with only a finite number of discontinuities, or with
any enumerable set of discontinuities
(3) Monotone functions, and all functions with limited total fluctuation.
For, as has been shewn in ~ 194, the points of discontinuity of a function
with limited total fluctuation form an enumerable set.
(4) Generally, every point-wise discontinuous function which is such that
the closed set of points for which the saltus is -i k has content zero, whatever
positive value k may have.
Dini* has given the theorem that a function is integrable, if at all points
where the discontinuity is of the second iind, it is so for all such points only
on one and the same side of the point; and at these points the finction may
be continuous on the other side, or may have ordinary 'disconttitnities on that
side. In particular, any ftnction which has only ordinary discontinuities is
integrable.
To prove this we observe that it has been proved in ~ 189, that, for such
a function, the set of points for which the saltus is _ k has content zero,
whatever positive value k may have. Therefore the condition of integrability
is satisfied.
Riemann's definition of an integral, and the condition for the existence of
the integral, are applicable, without essential change, to the case of a function
which, for particular values of the variable, has indeterminate functional
values lying, in the case of each such point, between finite limits of indeterminacy. At each point of indeterminacy of the function, it is immaterial
whether the function be capable of having all, or only some, values between
the limits of indeterminacy; thus there is no loss of generality, if the function
be regarded as having two values only at each such point, viz. the two
limits of indeterminacy at the point. In estimating the fluctuation of the
function in a prescribed interval, the upper limit is found by taking the upper
limits of indeterminacy of the function at the special points as functional
values at those points, whilst the lower limit is found by taking the lower
limits of indeterminacy at the special points as the functional values at
those points. As in the case of a function which is everywhere singlevalued, the saltus at any point is defined as the limit of the fluctuation


* See Grundlagen, p. 335.




344


Integration


[Cc1. v


in a neighbourhood of the point, when that neighbourhood is diminished
indefinitely. The conditions of integrability are exactly the same as for a
function which is everywhere single-valued, viz. that the function be limited
in its domain and that the set of points of discontinuity of the function must
have zero measure.
PROPERTIES OF THE DEFINITE INTEGRAL.
256. We proceed to consider the properties of the integral f (x) dx, of
a limited function f (x), defined for the interval (a, b), and such that the
condition for the existence of the integral is satisfied.
ra         ^                     rb
(1) The integral   f (x) dx exists and has the value -  f (x) dx.
For the former integral when it exists is defined by means of the limit of
S&M (8), where 8 is one of a set of finite intervals into which (b, a) is divided;
any such interval differs from a corresponding interval in (a, b) only in sign,
and the numbers M(8) may be taken to be the same for corresponding
intervals in the two cases. It is thus clear that the existence of the one
limit follows from that of the other, and that they differ only in sign.
(2) Iff (x) be integrable int (a, b), so also is f (x) 1, and
f:(x) dx:_-f(x) dx.
For the fluctuation of f (x) I in any interval S cannot exceed that of f(x)
in the same interval; hence, if EDS for a convergent sequence of sub-intervals
have the limit zero when D is the fluctuation off(x) in 8, it has also the limit
zero when D denotes the fluctuation of f(x) '; and thus the latter function is
integrable. Again, U the upper limit of f(x) in 8 cannot numerically exceed
U', the upper limit of If(x) I in the same interval; thus I E  U8 i  c  U'8, and
hence the absolute value of the limit of S US is < that of E U'8.
(3) If the values of the integrable function f (x) be arbitrarily altered at
each point of a measurable set of points G, the new function / (x) so obtained
is integrable, provided it be limited, and the measure of the derivative G' of the
set be zero.
For the only points of discontinuity of b (x) which are not points of
discontinuity of f (x) are points of G or of G', and therefore form a set of
measure zero; hence all the discontinuities of b (x) form a set of points
of zero measure, and b (x) is therefore integrable provided it be limited. In
particular, the theorem holds for any reducible set G.
Also, if p (x)=f (x), at all points belonging to a set which is everywheredense in (a, b), then, provided q (x) be integrable, its integral is identical with
that of f (x).




255, 256]


Properties of the definite integral


345


For, in the finite sum  SfM(8), we may take the value of M1(S) in any
interval 8 to be one of the values which the two functions f (x), b (x) have
in common in that interval; hence the sums may all be chosen so as to be the
same for the two functions. Thus, if the functions be both integrable, their
integrals are identical.
(4) A function f (x) which is integrable in (a, b) is also integrable in any
interval (a, /9) contained in (a, b).
For the measure of the set of points of discontinuity of f (x) in (a, b)
being zero, the measure of the set of those points of discontinuity which
are in (a, 3) is also zero, and thus the function is integrable in (a, /3).
If c is any point in (a, b), we have
f/ (x) dx =f f (x) dx +    f (x) dx.
J a       J acJ c
For the two integrals on the right-hand side both exist; also a convergent
sequence of sets of sub-divisions of (a, b) can be so chosen that the point c is
always an end-point of two of the sub-divisions. If this be done, the sum
SfM(t) for (a, b) may be divided into two parts, one of which contains all the
intervals on the left of the point c, and the other all those on the right of that
point; thus  ES8M(8)= E28M(8)+E4281(8). The limits of the three sums
are the three integrals of f (x) in (a, b), (a, c), and (c, b) respectively; thus
the theorem is established.
(5) If' f,, f,, f2,... fn be a finite enumber of limited functions, each of
which is integrable in (a, b), and if F (ft, f,... f,,) be a continuous function
of the n variables f, f2,... f, then the function F is integrable in (a, b).
For the only points of discontinuity of the function F (x) are those of the
functions f (x), f2 (x),... f,n (x); hence the set of points of discontinuity of
F (x) has measure zero; and thus F (x) is integrable, since it is also a limited
function.
Important particular cases of the general theorem are the following:(a) If f (x) = J (x)  ( + 2  (.) +... +f (x), where all the functions J;. (x)
rb         n rb
are integrable, then   (x) dx =      (x) dx.
a           1 a
(b) If f (x) = f, (x).f2(x)... f,, (x), where all the functions f. (x) are
integrable in (a, b), then f (x) is also integrable in (a, b).
(c) If f (x), ( (x) be integrable in (a, b), and! b (x) I always exceed
some fixed number A, so that  (x) is a continuous function of f and q, then
b (x)) is integrable in (a, b). (
is integrable in (a, b).
( )


- Du Bois Reymond, Math. Annalen, vol. xx, p. 123. See also W. H. Young, Quarterly
Journal of Math. vol. xxxv, p. 190.




346


Integration


[CH. V


(6)  If two futnctions f + (x), f -(x) be defined as follows:-Let f +(x) == (x)
for all values of x such that f (x) > 0, and let f+(x) = 0, when f (x) < 0; let
f-(x) = -f (x) for all values of x such that f (x) < 0, and f -(x) = 0, when
f (x) 2 0; then if f (x) be integrable in (a, b), the functions f +(x), f-(x) are
integrable in (a, b), and  f (x) dx  f + ()dx-    f -   (x) dx.
For the fluctuation of f+(x) in any interval 8 cannot exceed that off (x)
in the same interval; hence, since S8D(8) for f (x) has the limit zero, the
corresponding sum for f +(x) has the limit zero, and thus f+(x) is integrable.
In a similar manner it can be shewn that f-(x) is integrable.
Since f (x)=f+(x)-f-(Ix), we see from (5) (a) that
I,         rb)           b
f/ (x) dx =f +(x) d -       f-(x) dx.
It should be observed that it is not in general true that, if f (x) be
integrable in (a, b), and be expressed as the sum fj (x) + Jf (x) of two limited
functions, then f; (x), f2 (x) are also integrable in (a, b). For it is clear that,
f (x) being given, we may take for J1 (x) any arbitrarily defined non-integrable
function, then Jf (x) is also determinate and non-integrable.
(7) If f (x), q (x) be both integrable, and be such that f (x)  < j (x) 
for every value of x, then  J (x) dx      (x) l dx.
Jfa          fa
In particular, if 0 (x) is constant and equal to P, the uzpper limit of f (x) |
in (a, b), thenz  (x) dx < P (b- a).
I J a
rb
FoIr     (x) - f (x)  I dx is?0, since in every interval 8 no value of
I () I- f(x) is negative, and thus the sums of which the integral is the
limit are all g 0. Also from (2), we have  (x)     x <  jj(x) dx, and this
J a.     a
[b
is   j! f) (x) dx. The particular case follows by assuming p (x) = P.
If U, L denote the upper acd lower limits of f (x) in (a, b), then
L (b - a) - J f (x) dx '- U (b - a).
For I87U(S), ZSL (8) each lie between UZ$ and LZ8, or between U(b- a)
and L (b - a); the same must hold of the common limit, which is the inrb
tegral   f (x) dx.
(8) If ql 9, W,...... be an enumerable set of non-overlapping intervals
contained in (a, b) in descending order of length, then the sum of the integrals
of f (x) taken through rTh,,,... Vn, converges to a definite finite limit, as n is
increased indefinitely; f (x) being a function which is integrable in (a, b).




. 256 257]9


Integrable nUill-functions


347


Let us denote by S,, the sum of the integrals of' (x) taken through the
intervals 1, v2,...  1. Since m, + A, +... + 2, increases with n, and is always
less than b - a, it has a definite limit as n is increased indefinitely; we can
therefore choose n so great that 77,+1 + Vq+2 +... + q1,r+m < e, for every value of
m, where e is an arbitrarily chosen positive number. With this value of n,
we see that S,,n+, - S, i < e. P, where P is the upper limit of f (x) I in (a, b).
If v be an arbitrarily chosen positive number, we can choose e such that
e < 1/P; thus n can be so chosen that S,+,, - S,, < r, and hence S,l has a
definite limit as n is increased indefinitely.
INTEGRABLE NULL-FUNCTIONS AND EQUIVALENT INTEGRALS.
257. If f (x) be integrable in (a, b), and be such that its integral in
every interval contained in (a, b) is zero, then f (x) is said to be an integrable
znull-Jfnction.
The necessary and stfficient condition that a limited fanction f (x) may be
an integrable null-function is that the set of points for which f (x) i> k, when
the set is closed by the addition of its limiting points, shall be of content zero,
whatever positive value k may have.
To prove that the condition stated is sufficient, let us suppose the interval
(a, b) to be divided into sub-intervals by a system of sub-divisions; at any
stage, let 28' be the sum of those intervals which contain in their interiors or
at their ends points at which / f(x) I k. The sum, of which the limit is
jfi (x) dx, is in absolute value < P8' + (b - a -  ') k, where P is the upper
limit of If () in (a, b). If the content of the closed set obtained by adding
to the set for which  f (I>) -k its limiting points, have content zero, then
28' has the limit zero, as the number of sub-divisions of (a, b) increases indefinitely; hence the absolute value of  f (x) dx is c   k (b - a); and as k is
arbitrarily small, this shews that the integral vanishes. The same argument
applies to any interval (a, /) contained in (a, b). To shew that the condition
is necessary, let us assume that f (x) has, in every interval contained in (a, b),
an integral which vanishes. At any point cx,, at which f (x) is continuous,
f (x,) must be zero. For let f(x,), if possible, have a positive value A; then a
neighbourhood (x, - h, x, + h) can be found such that at every point in it
f (x) lies between A - e and A + e, where e is any assigned positive number
<A; now the integral of f(cw) through this interval (xc,-h, xl + h) is
>(A -e)2h, and thus cannot be zero, contrary to hypothesis. It is therefore impossible that f (x,). can have a positive value; and that it can have a
negative value can be shewn, in a similar manner, to be also impossible; and
thus f (x) vanishes at every point at which it is continuous. Considering




348


Integration


[oH. v


next a point at which I f(x) I  i _c, or a point which is a limiting point of the
set of such points, we see that the saltus at the point is   kic; for every
neighbourhood of the point contains points of continuity at which f (x)
vanishes. The condition of integrability consequently ensures that the set
of points at which I f (x) { > k, when closed by adding the limiting points, has
content zero.
The condition may also be stated in the concise form that:A limited function is an integrable null-function, if it vanishes at all points
of a set of which the measure is equal to that of the whole interval for which
the function is defined.
Two integrable functions f (x), b (x) have the same integrals in every
interval contained in their domain, provided they differ from   one another
by an integrable null-function.
If f(x) be an integrable point-wise discontinuous function, and if the
function f (x) be defined, as in ~ 191, by extension of that function which
is defined only at the points of continuity o f(x), and has at those points
the same functional values as f(x) itself, then fi (x), although it is in general
multiple-valued at the points of discontinuity of f(x), is an integrable
function.  It has been explained in ~ 255 that Riemann's definition is applicable to such a function.  That if (x) is integrable, follows from the fact
that it is continuous at all the points of continuity of f(x), and these form a
set of points of which the measure is equal to that of the whole interval in
which /(x) is defined. The difference of the two functions is zero at the
points of continuity of f(x), and is discontinuous only at the points of discontinuity of f(x), which form a set of points of zero measure.  The function
f(x) -f; (x) is accordingly an integrable null-function, and the two functions
f(x), J (x) have equal integrals in any interval for which f(x) is defined.
It has therefore been shewn that an integrable function f(x) is equal to
the sum of an integrable null-function and of the function obtained by extension
of the function defined by the values off(x) at its points of continuity.
EXAMPLES.
r) (2x)      (nx)
1. Riemann's function f(x)  12 + 22   +..., where (x) denotes the positive
or negative excess of x over the nearest integer, and (x) = 0 when x is half-way between
two integers, has been shewn in Example 2, ~ 190, to be point-wise discontinuous, with all
its discontinuities ordinary ones, and everywhere-dense in the interval (0, 1). Since all the
discontinuities are ordinary ones, and the function is limited, f(x) is integrable in (0, 1).
2. Letf(x) be defined for the interval (0, 1) as follows:-If x be irrational, let / (x)= 0;
if x=p/q, where p/q is in its lowest terms, let f(x)= l/q; also let f(0) =f(1) =0. This
function is an integrable point-wise discontinuous null-function; thus f f (x) dx = 0.
J n




257, 258]            The fundamental theorem                        349
There are only a finite number of points at which the functional value exceeds an
assigned positive number.
3. Let f (x) = 0, for all rational values of x; and f (x)=, for all irrational values of x.
This function is not integrable in any interval, for it is totally discontinuous.
1
4. Let f(x) be defined* for the interval (0, 1) as follows: —For 2  x <  1, letf(x)= 1;
1       1; f  1                 1
for 2< x, let f (x); for <, let / (x) =2; and generally, for
2n, i<v-     =  'letf (x) =; andf(0)=0.
This function is integrable, and f| (x) dx= 2_l -. 22.-2 where x. is between
1      1
t and
29.   2~ - 1'
THE FUNDAMENTAL THEOREM OF THE INTEGRAL CALCULUS.
258. The fundamental theorem of the Integral Calculus asserts that the
operations of differentiation and of integration are in general inverse operations.
Before we proceed to consider the conditions under which this is the case, the
following theorem will be established:If f (x) be a limited function which is integrable in the interval (a, b), then
J f (x) dx is a continuous function of x, for the whole interval (a, b), and
it is a finction of limited total fluctuation in (a, b).
It has already been shewn that  f/ (x) dx exists, for any point x of the
interval (a, b); denoting its value by F(x), we have
F(x   h) -F() =F()   / (x) dx;
hence by (7), of ~ 256, F (x + h) - F (x) < Ph, where P is the upper limit of
j f (x) in (a, b). If e be any arbitrarily chosen positive number, and we take
hA < e/P, then for all values of h which are = h1, we have [ F (x + h) - F (x) < e;
but this is the condition of continuity of F(x) at the point x. In case x be
one of the end-points of (a, b), h must be restricted to have one sign only.
To prove that F(x) has limited total fluctuation in (a, b), let (a, b) be
divided into n sub-intervals by the points a, x,, x2... Xn,_, b. The sum of the
absolute differences of the values of F (x) at the ends of these sub-intervals is
Ij' f(x))dx +|    f f(xs)dx +... + fr   l(x) dx 
rb
and this is, in accordance with the theorem (7) of ~ 256, -  | f(x) I dx; and
therefore the sum is less than a fixed positive number. Since the total
* Dini, Grundlagen, p. 344.




350


Inztegrationo l


[CH. v


variation of F(x) in (a, b) is limited, it follows from the theorem of ~ 196,
that the total fluctuation in the interval is also limited.
When f(x) is integrable in (a, b), the function  f (z) dx, which has been
shewn to be continuous, and of limited total fluctuation in (a, b), is said to
be the integral function corresponding to f(x).
If f(x) be any function defined in (a, b), a function ( (x) which, at every
point x of the interval, possesses a differential coefficient equal to f (x), is
said to be an indefinite integral of f(x).
The definition is, however, extended to cases in which  '(x) either does
not exist, or is not equal to f(x), at points belonging to an exceptional set;
the condition  ' (x) =f(x) being satisfied at all points not belonging to the
exceptional set.
Taking the function F (x) =     f(x) dx, as the integral function corre-at
sponding to f (x), the following properties will be established:(A) Under certain restrictions, F(x) possesses a differential coefficient
which is equal to f (x), and thus F(x) is an indefinite integral of f (x).
(B) Also it will be shewn that, if b (x) be a function which possesses a
differential coefficient f (x), then f (x) has in general an integral F (x), in an
interval (a, x), which integral differs from p (x) by a constant only; and thus
that the indefinite integral of f(x) is determinate except for an additive
constant.
It will appear that there are cases of exception to both theorems. When
F(x) is an integral function, it happens in certain cases that F (x) does not
possess a differential coefficient; and when k (x) is a function which possesses
a differential coefficient, it is not always the case that the latter is integrable,
and when integrated yields the function q (x) except as regards a constant.
259. If f (x) be continuous in the interval (a, b), and F(x) denote the
integral function  f (x) dx, then, at every point in (a, b), F (x) possesses a
diferential coefficient which is equal to f (x).
For since f(x) is continuous, an interval (x- hA, x + h,) can be found such
that!f (x + Ohh)-f (x) < e, for all proper fractional values of 0. It follows
rx+h
that F (x + h) - F (x)  f (x) dx, lies between h [f(x) + e] and h [ (x) -],
ided lb < hl. Hence    Fx-,x + lb)
provided h <,. Hence since  (        F (x) lies between f(x) + e, f(x) - e,
h
for h < h,, it follows that f (x) is the differential coefficient of F (x). At
the points a, b, the function F (x) possesses derivatives on the right and on
the left respectively, and their values are f (a), f (b).




258-260]


The fundamental theorem


351
a 0 I.


If qb (x) be a function which at every point of (a, b) has a differential
coefficient, which is a continuous fanction f (x), then
(,x) -  (a) =  f (x) dx.
For let  f (fx) dx be denoted by F(x), then the function ( (x)-F (x)
has at every point a differential coefficient which is zero, and therefore by the
theorem of ~ 206 the function ) (x) - F(x) is constant; it is clear that this
constant must be qb (a); and thus the theorem is established. In this theorem
and elsewhere, a derivative at a on the right, and a derivative at b on the
left, are included in the term differential coefficient.
260. If a given limited integrable function f (x) be not everywhere
continuous in the integral (a, b), the proof given above is applicable to
prove that, at any point of continuity of f (x), the function  f (x) dx has a
differential coeficient equal to f (x).
At a point of ordinary discontinuity of f (x), the same proof, when
modified by taking only positive values of h, or only negative values of h,
and using f (x + 0) or f (x- 0), in the two cases, instead of f (x), will shew
that F (x) has at such a point derivatives on the right and on the left, and
that these are f (x + 0), f (x - 0) respectively. At a point at which f (x)
has a discontinuity of the second kind, the proof fails altogether; at such a
point therefore F (x) need not possess a differential coefficient, nor definite
derivatives on the right and on the left, but may have all its four derivatives
D+F(x), D+F(x), D-F (x), D_F (x) of different values.
If f (x) be an integrable point-wise discontinuous function, and * (x) is
the function formed by extension of the functional values of f(x) at its
points of continuity, as explained in ~ 257, we have f(x) =X (x) + + (x),
where X (x) is an integrable null-function; and therefore / (x) and r (x) have
the same integral function F(x). The derivatives of F (x) are independent
of the function x (x), and depend only upon ~ (x), which is determined by
the values of f (x) at its points of continuity.
Since F (x + ) - F (x) = f +   (x) dx, and since the values of ~ (x) in the
interval (x, x + h) all lie between g (x + 0) + e1, and ~ (x + 0) - e, where
F' (x + h) - F (x)l
el,, 6 converge to zero as h does so, we see that   +  -  () lies between
- (xc + 0) + e1 and - (x + 0) - e2, hence D+F (x), D+F (x) both lie between'*
" It is stated by Schonflies, see Bericht iiber die Mengenlehre, p. 208, that the derivatives of
F (x) are equal to ip (x +0), ' (x+ 0), (x - 0),  ( (x -0). This, however, is not necessarily the
case, It has been shewn by Hahn, Monatshefte der Math. u. Physik, vol. xvi, p. 317, that f (x)




352


Integration


[OH. V


(x +    0),  (x + 0).  By taking h negative, we see that D-F(x), D_F (x)
both lie between  r (x - 0) and  (x - 0). In case * (x) be continuous on the
right, F (x) has a derivative on the right, r (x + 0); and in case * (x) is continuous on the left, F (x) has a derivative - (x - 0) on the left. It may
happen that r (x) is continuous at a point of discontinuity of f(x); at such
a point F(x) has a differential coefficient equal to the value of + (x).  Even
when    (x) has a discontinuity of the second kind, it is possible that F(x)
may have a differential coefficient, or a derivative on the right or on the left,
or both.
If f (x) be integrable, and F(x) be the corresponding integral function, any
one of the four derivatives DF (x) of F (x) is integrable, and has F (x) for its
integral function.
For DF((x) differs from f (x) only at a point of discontinuity of f (x), and
at a point of discontinuity DF(x) lies between the upper and lower limits of
s (x); thus f(x) - DF(x) is an integrable null-function. Therefore
f(x) dx -   D+F (x) dx =   D+ F(x) dx =  D-F (x) dx
j a        ^ ci^a          a              a.
=   D_F (x) dx =             F(x )
It has been shewn that the integral function of an integrable point-wise
discontinuous function has a differential coefficient at the everywhere-dense
set of points of continuity of the discontinuous function; there may however
also be an everywhere-dense set of points at which this continuous function
does not possess a differential coefficient.
261. It has been shewn that, if the continuous function   (x) possesses
everywhere a differential coefficient f(x) which is everywhere a continuous
function, then
(x) (-    )    f /() dx = F (x)
This is a particular case of the following more general theorem:If ( (x) be a function continuouts in the interval (a, b), and if one of its four
derivatives D+ (x), D+ (x), D-~ (x), D_^ (x) be a limited integrable function
in (a, b), then each of the other three derivatives is also limited and integrable
in (a, b), and ( (x) - c (a) is the integral of any one of the four derivatives
through the interval (a, x).
If (a, x) be divided into a number of parts (a,. x), (x,, x,),... (xa-,:x), it
may be so chosen that the corresponding integral function has, at a particular point, derivatives
on the right having arbitrarily given values lying between, or equal to, the values of / (xz+0),
ip (x + 0) at the point; and in particular that f (x) may be so constructed as to have, at the point,
a definite derivative which has an assigned value between the two limits.




260-262]


The fundamental theorem


353


has been shewn in ~ 217, that  (x.-   r   lies between the upper and
xr - Xr-I1
lower limits of any one of the four derivatives Dc) (x) in the interval (x,.-, x,).
It follows that p (x)- c (a) lies between the two sums
(x1 - a) U(a, x1, D) + (x2 - xi) U(x1, x2, D) +... + (b - x,_-) U(x_, b, D),
(x1 - a) L (a, x,, D) + (x2- x1) L (x1, x,, D) +. + (b - x_,) L (x,n_, b, D),
where U(x._1, x,., D), L (xl, x,., D) are the upper and lower limits of DO (x)
in the interval (xr-, x,.); and it is known that these are the same for all four
derivatives. The limits of the above sums, when the intervals are diminished
indefinitely, so that the greatest of them converges to zero, are the upper and
lower integrals of any one of the four functions Db (x). If it be known that
any one of these derivatives is integrable in (a, x), then the upper and lower
integrals are equal, and the other three are also integrable, the common value
of the integral being  (x) - ~( (a). Thus
C (x) -  (a) = D    (x) dx = D+ (x) dx = fD- (x) dx =  D_  (x) dx.
It should be observed that, as has been shewn in ~ 219, the four derivatives
are all equal to one another at a point at which one of them is a continuous
function; and thus at such a point there is a differential coefficient. If one
of the derivatives be integrable, there is therefore a set of points of measure
equal to that of the interval (a, b), at which all four derivatives have equal
values, and at which therefore a differential coefficient exists.
262. In case Do (x) be a limited function which is not integrable, the
above proof shews that  ( (x) -  (a) lies between the upper and lower integrals in (a, x) of any one of the four functions Do (x). This includes the
case in which ~ (x) has a differential coefficient which is limited but not inrx           rx
tegrable; in that case < (%x)- S (a) lies between   ' (x) dx and J  (x) dx.
x+h,~o /
Since;    '(x) dx is in absolute value less than h. U, where U is the
upper limit of i' (x) in (a, b), it follows as in ~ 258, that j '(x) dx is a cona
tinuous function of x; similarly it may be seen thatf  ' (x) dx is a continuous
Ja
function of x. At a point of continuity of b' (x), both f' (x) dx, 0 ' (x) dx
have the differential coefficient +'(x), as may be seen by a process precisely
similar to that in ~ 259. Thus the upper and lower integrals of 0' (x) possess
properties similar to those of the integral of 0' (x) when it exists.
The function D( (x) when not integrable, may be a non-integrable pointwise discontinuous function, or it may be totally discontinuous.


H.


23




354


Integration


[CH. v


If f(x) be any non-integrable limited function, the following theorems
may be established by proofs similar to those in ~ 258 and ~ 259:The upper and lower integrals  f (x) dx,  f (x) dx are continuous, and
of limited total fluctuation in (a, b).
At any point of (a, b) at which f(x) is continuous, the upper and lower
integrals  f(x) dx,  f(x) dx each possess a differential coefficient which is
equal to f(x).
263. An important general class of continuous functions for which the
four derivatives are not integrable, even when a differential coefficient exists,
or when derivatives on the right and on the left always exist, is the class of
everywhere-oscillating functions. Those functions which become everywhereoscillating functions when a linear function is added have the same property.
If a derivative DF(x) be such that in every interval it has no finite
upper limit or no finite lower limit, it is certainly not integrable; it is therefore only necessary to consider an interval in which the function DF(x) is
limited. Let (a, x) be such an interval, and let us suppose that F (x) - F (a)
is not zero.
In every interval (x,.-, x,.) contained in (a, x), U (x,_1, x,., D) the upper
limit of DF(x) is positive, and L (x._i, Xy, D) the lower limit of DF (x) is
negative; thus the two sums
(x, - a) U (a, x,, D) + (x2 -  ) U (x,, x,, D)+...
+ (b- x_,) U (x_-,, b, D),
(x - a) L (a,,1, D) + (2 - x,) L (x,, x, D) +...
+ (b - xn-) L (x,,_,, b, D),
are such that the first is essentially positive, and the second essentially
negative, the non-vanishing number F(x) - F(a) lying between them.
It follows that the limits of these two sums, as the number of subdivisions of (a, x) is increased indefinitely, must be different from one another,
since they cannot have zero as their common value; thus
JDF (xc) dx, JDF (x) dx
are distinct from one another.
It has thus been proved that a continuous function which is everywhereoscillating in (a, b) cannot have a derivative which is integrable in (a, b), even
if it have everywhere a differential coefficient, or definite derivatives on the right
and on the left.




262-264]


The fundamental theorem


355


The function DF (x), or f(x), in the case of such function, may be a
point-wise discontinuous function such that the measure of the set of points
of discontinuity is greater than zero, or it may be a totally discontinuous
function.
A continuous monotone function, which is not reducible to a function with
an infinite number of oscillations by the addition of a linear function, has
at every point definite derivatives on the right and on the left, each of
which is either continuous or is an integrable point-wise discontinuous
function, since either derivative has only ordinary discontinuities. Thus such
a function has integrable derivatives, provided these derivatives are limited
in the interval.
In case the continuous function F(x) have a differential coefficient, or a
derivative which is not everywhere finite, or is not limited in the interval,
this derivative is not integrable in the sense in which we have hitherto defined
integration. This case will be considered in connection with the theory of
improper integrals.
264. The preceding investigations provide answers to the questions
which arise as regards the validity of the two propositions (A) and (B)
of ~ 258, which together constitute the fundamental theorem of the Integral
Calculus asserting that the operations of differentiation and of integration
are in general reversible. The definition of a definite integral has hitherto
been restricted to that of Riemann, and is applicable to limited functions
only. The extensions of that definition to the case of unlimited functions,
which will be considered later, and also a more general definition of integration due to Lebesgue, of which an account will also be given, will lead
to corresponding extensions of the scope of the fundamental theorem.
As regards ththeorem (A), that the integral function F(x)=  ff (x) dx
of a limited integrable function possesses a differential coefficient equal,
at a point x of (a, b), to f(x), it has been shewn that the theorem holds
without restriction in case f(x) is a continuous function; but that, if f(x)
be not continuous, the theorem still holds as regards every point of continuity
of f (x). It follows that the points of (a, b) at which F (x) either possesses
no differential coefficient, or possesses one which is not equal to f(x), form
a set of zero measure, which may however be everywhere-dense in (a, b).
The theorem (B) that, if'  (x) possess a differential coefficient f(x),
then the corresponding integral function F (x) f- x)dx differs from < (x)
a
only by a constant, holds iff(x) be a continuous function, and more generally,
if f(x) be limited and integrable. In case q (x) do not at all points possess
a differential coefficient, the more general theorem is applicable that, if any
23-2




356


Integration


[OH. V


one of the four derivatives of + (x) be limited and integrable, then the
integral function corresponding to that derivative differs from  (x) by a
constant only. The theorem fails either in case ( (x) be not a function with
limited derivatives, or in case it be a function with limited derivatives, but
those derivatives do not satisfy Riemann's condition of integrability.
The problem of the determination of a continuous function which shall
have a given function f(x) for its differential coefficient, at every point at
which f(x) is continuous, may be here considered in the case in which f(x) is
restricted to be limited in the interval (a, b) for which it is defined. This
problem is regarded as having a determinate solution provided functions
exist which satisfy the condition, and further provided any two such
functions differ from one another by a constant only, that constant having
one and the same value for the whole interval (a, b). In the first place, the
problem cannot be determinate unless f(x) be integrable; for either of the
rx       rx
two functions Jj (x) dx,  f (x) dx satisfies the condition of the problem,
and these functions do not differ from one another by a constant, as they
both vanish at the point c, and are elsewhere unequal. Next, if f (x) be
integrable, the function  f(x) dx satisfies the condition of the problem,
but the solution is not necessarily determinate. In case however the points
of discontinuity of the integrable function f(x) form an enumerable set,
the theorem of ~ 206 shews that the solution is determinate; for any two
functions which have equal finite differential coefficients at all points of
(c, b) except those of an enumerable set, differ from one another by a
constant. In this case f f(x)dx+C is the function required. When the
points of discontinuity of the integrable function f(x) form an unenumerable
set, although that set must have zero measure, the problem has not a
determinate solution.  For, although  f(x)dx is a function which has
the required property, another solution is obtained by adding to it any
continuous function which has all the intervals complementary to the perfect
component of the unenumerable set as lines of invariability; that such
functions exist has been established in ~ 208. There exists however only
one function, viz. ff () dx, with limited derivatives, which satisfies the
condition of the problem; for it has been shewn in ~ 224, that any two
functions which have limited derivatives, one of which derivatives is
prescribed at all points not belonging to a certain set of measure zero, differ
from one another by a constant.
Similar remarks apply to the more general problem of the determination
of a function which shall have one of its four derivatives, say the upper




264, 265]             The fundamental theorem                          357
one on the right, equal to a given function f(x) at every point of continuity
of f(x).  This problem has a solution whenever f(x) is limited; in virtue of
the theorem  of ~ 206, the solution is determinate when f(x) is integrable,
and the points of discontinuity of f(x) form an enumerable set.  When f(x)
is integrable, and the set of points of discontinuity is unenumerable, there
exists, in virtue of the theorem  of ~ 224, only one solution for which the
derivatives are limited.  As before, if the restriction, that the required
function is to have limited derivatives be not imposed, the solution of the
problem is indeterminate.
EXAMPLE.
Let G be a perfect non-dense set of points in the interval (a, b), and such that its
content is greater than zero. Let (a, 3) be an interval complementary to the set G, and
1                                    1       1
let  (x, a)=(x -a)2sin, and therefore (' (x, a)=2 (x-a)sin -— cos -. The
X- a                                  — a    X —a
function (' (.x, a) vanishes at an infinite number of points in (a, /); let a +y be the greatest
value of x which does not exceed  (a + /), for which p' (x, a) vanishes. Let F (x) =0 at
every point of G, and in each interval (a, j) complementary to G, let F (x)- (x, a), for
values of x such that a S x  a+y; let F(x)=  (a+y, a) for values of x such that
a+yx< _S3-y; and let F(x)= —(x,,y) for 3-y xE b. The function F(x) is
continuous, and has everywhere a finite differential coefficient which is limited in the
interval (a, b). It is easily seen that F' (x) vanishes at every point of G. The function
F' (x) has a discontinuity of measure 2 at each point of the set G which is not of zero
content, and therefore F' (x) is not an integrable function. This example was given* by
Volterra, as the first known example of a continuous function possessing a non-integrable
limited differential coefficient.
FUNCTIONS WHICH ARE LINEAR IN EACH INTERVAL OF A SET.
265.  The existence of continuous functions which are linear in each
interval of an everywhere-dense set of intervals has been already referred
to in ~ 213.   It has been shewn in ~ 208 how a function f(x) can be
constructed which is continuous, and has as lines of invariability the intervals
complementary to a non-dense perfect set of points. It is clear that the
integral function   f (x) dx is linear in each of the intervals, and being also
continuous, it is a function of the type referred to. A more general function
which is continuous, and is linear in each interval of the set, may be obtained
by adding to     f(x) dx any continuous function for which the intervals
of the set are lines of invariability.


* Giorn. di Battaglini, vol. xix, 1881,




358


Integration


[CH. V


MEAN VALUE THEOREMS.
266. It is frequently of importance to be able to assign upper and
lower limits between which the value of a definite integral lies, in cases
where the exact determination of the value of the definite integral is not
required. Such estimates of the value of a definite integral may frequently
be made by means of theorems known as mean value theorems; the most
important of these will be here given.
Iff (x) be a limited integrable function defined for the interval (a, b), it is
clear from the definition of the integral as the limit of a sum, that if U, L
denote the upper and lower limits of f(x) in the interval (a, b), then
L(b - a) =  f(x) dx _- U(b - a);
rb
it follows that            f (x) dx = (b - a) M,
where M is some number which satisfies the condition U - M _ L.
In case f(x) is a continuous function, there must be some value or
values of x in (a, b) for which f(x) = M; if then such a point be denoted by
a + 0 (b - a), we obtain the following theorem:If f(x) be continuous in the interval (a, b), then
f (x) dx= (b- a)f a + 0 (b - a)},
where 0 is some number such that 0 -< 0  1.
Next, letf (x), 4 (x) be integrable functions defined for the interval (a, b),
the function  (x) being positive, or zero, for the whole interval; we then
have immediately from the definition of the integral
ff(xc) b (x) dx,
L  f(x)          dx f (x)U ( )d  4 (x) d,
where, as before, U, L are the upper and lower limits off(x) in (a, b).
It follows at once, that
rb                 rb
f(x)    (x) dx = M,   (x) dx,
where M1 is some number such that
L.
In case f (x) be a continuous function, we obtain the following theorem



266, 267]


Mean value theorems


359


If f (x) be continuous in (a, b), and p (x) be a limited integrable function
which has the same sign throughout (a, b), except where it may be zero, then
f (x)    (x) dx =f {a + 0 (b- a)} f   (x) dx,
where 0, is some number such that 0 _ 0,  1.
This theorem, including also the more general case in which f(x) is not
continuous, is known as the First Mean Value Theorem.
An extension to the case in which b (x) is not necessarily everywhere of
the same sign in the interval (a, b), may be obtained by applying the theorem
to + (x)+ C, where C is so chosen that this latter function is of invariable
sign in the interval.
267. If the limited function f(x) be everywhere positive in the interval
(a, b), and never increase as x increases from a to b, and is consequently
integrable, and if b (x) be limited and integrable in (a, b), then
f (x) 0 (x) d =f (a) f    (x) dx,
where   is some number such that
a   _-b.
Also if f (x) be everywhere positive, and never diminish as x increases
from a to b, then
ff (x)   (x) dx =f (b) f  (x) dx,
where ~ is some number such that a < _  b.
This theorem was first given* by Bonnet, and applied by him to the
theory of Fourier's series.
Another form of the theorem was obtained by Weierstrass, and also byfDu Bois Reymond, and is generally known as the Second Mean Value
Theorem. This is as follows:If f (x) be monotone and therefore integrable in (a, b), and if q (x) be
limited and integrable in the same interval, then
fi /()   (x) dx =f(a)f j (x) dx +f/(b)   (x) dx,
where           i is some point in the interval (r, b).
The theorem in this form is deducible immediately from Bonnet's theorem,
by writing in that theorem f(x) -f(b), or f(x) -f-(a), in the two cases,
* See Mem. Acad. Belg. vol. xxIII (1850), p. 8; also Liouville's Journal, vol. xiv (1849),
p. 249.
t Crelle's Journal, vol. LXIX (1869), p. 81.






r

360                           Integration                           [CH. v
instead off(x); Bonnet's theorem is however not* immediately deducible
from the second theorem.
It is clear that, since the value of
fb
f /(x)(x) dx
is unaltered by changing the values of f(a) and f (b), we may in the above
statement instead of f(a), f(b) take any two numberst A, B which are such
that the function  r (x) defined by
r (a)=A,    F(b)= B, q (x)=f(x),
for a < x < b, is monotone. We have thus the following generalized form of
the theorem:f (x)    (.)   =OA     () d + B      (x) dx,
where                   A - f (a + 0), B   f (b - 0),
if f (x) never decreases; or
A _f(a+O), B       f(b-0),
if f (x) never increases.
The value of ~ depends in general upon the chosen values of A and B.
In this generalized form, the theorem includes Bonnet's theorem as a
particular case. For we may take A = 0, B=f(b), if f(x) is positive and
never decreases, as x increases from a to b; or A = f(a), B = 0, in case f (x) is
positive and never diminishes, as x increases from a to b.
Various proofs  + ofth the theorem of Weierstrass and Du Bois Reymond~ have
been given.  In these proofs the function b (x) is usually restricted to change
its sign only a finite number of times; but a proof free from that restriction
was given by Du Bois Reymond. A proof has been given by Pringsheimll in
which + (x) is not restricted to be a limited function, but may be any
function such that it has an absolutely convergent integral, or in certain
cases it may have an integral which does not converge absolutely.
The following proof, in which C (x) is restricted to be a limited function,
is due to Holder T.
* This was pointed out by Pringsheim, Miinchener Berichte, vol. xxx (1900), where an account
of various proofs of the theorem is given.
+ Du Bois Reymond, Schlomilch's Zeitschrift, vol. xx (1875); Hist. Lit. Abtg. p. 126.
+ For example, by Hankel, Schlmnilch's Zeitsc7hrift, vol. xiv (1869); by Meyer, Math.
Annalen, vol. vi (1872); by C. Neumann, Kreis- iKugel- und Cylinderfunctionen, Leipzig, 1881,
p. 28.
~ Crelle's Journal, vol. LxxIX (1875), p. 42. See also Kronecker's Vorlesungen, vol. i.
11 Loc. cit.
~ Gittinger Anzeigen, 1894, p. 519. Another proof has been given by Netto, Schlomilch's
Zeitschrift, vol. XL (1895). See also Iowalewski, Math. Ann. vol, Lx, 1905,




Mean value theorems


361


Let a, b be denoted by aO, a, and let a, a2, a,... ac_ be points in (a, b)
in order from left to right. We consider the sum
3 —1
2  (a+, -  av)f(c,),b(c),........................(1)
v=O
where c, is any point in the interval (ar, a,+1); the limit of this sum defines
the integral
rb
f    /() (x) dx.
This sum may be transformed into
n-2                 v
I {[f/(c) - / (c+)] Z (a,+, - a,^) c (C)}
v=O               /x=o
n-i
+ f(c)      (aC+,- a) (, )  (c)...... (2)
/x=0
By increasing the number of points in the interval, we may obtain a convergent sequence of sub-divisions, then
n-1
f(cn-) S (a,+ - a,) b (c/)
U0=O
rb
has as limit              f(b - 0)    (x) dx.
We have now to examine the sum
n-2                V
e Il[f(c) -f(c+l))] S (a(,+1- a,) (c)}.
v=O              z/=O
Since f(x) is monotone, this is equal to
n-2
M  E [f(cO)-f(c~+A)],
v=o
where M is between the greatest and least of the numbers
v
(a.+, -a) (c/,).
/z=o
We have now, from (1) and (2),
n-i
Z (a,+- a) f (c,) 0b (c,)
v=0
n-I
-f(cn-l)   (a,+1 - a)             ( (c,) = M {f(Co) -f(Cn-,)}....(3),x=0
In order to estimate the value of M, we have
(a    - a)   (c) -   +   (x)dx= v            ) - ()};
fk=0                Ja a          /Ah=O
the absolute value of this difference is at most
v                n-l
(a+,h -   )A-_  2 (aa+,-0)A,
tk=0             t/=0




362


Integration


[cH. V


where A, is the difference between the upper and lower limits of q (x) in the
interval (a,, c,+,). We have therefore,
2 (at1, - f), (-ca )  J  ( ) dx +   (aAl - a,) A,,
t=O0                 a             t~=0
where                      -1   0   1.
If G be the greatest, and H the least value of the continuous function
f < (x) dx,
of I, in the interval (a, b), we have now, from (3),
n —1 n-1
H - '2 (a,,+ - a,)/a,  M  G +    (a,.+ - a,).
We now obtain from (3), the inequalities
n-l
{H -    (a,+,, - a,) A,} [f(co)-f(cnl-)]
n-1                           n-1
- E (avl - av)f(c,) b (cr) -f(cn-i) E (a,+l - a,) (c,)
v=O                          tx =O
n —1
- {G + E (a,, - a,) A,} [/(Co) -/(c-l)],
=o0
provided that f(x) does not decrease as x increases in the interval (a, b); in
the other case, the signs < and > must be interchanged.
We obtain now, by proceeding to the limit,
H { f (a +  )-f (b-O)}   f (x) () dx
-f(b-O)) f   (x) dx _ G {f(a + o)-f(b - 0)};
rb                    Cb
hence     f(x) (x) dx-f(b -0) O   (x)dx =   ' {f(a+0)-f(b    0)},
where M' is between H   and G.   There must be a value of X in the
interval (a, b), such that
M'= f    (x)dx;
we therefore obtain the equation
f (x) 4 (x) dx =f(a + 0) f (x) dx +f(b-0) f    (x) dx.
268. The theorem thus obtained is not identical with the theorem of
Du Bois Reymond and Weierstrass in its original form, since f(a + 0),
f(b- 0) take the places of f(a), f(b); it is however a particular case of the
generalized theorem which includes the original one. We therefore proceed
to deduce the generalized theorem from the special form obtained in ITolder's
proof,




267, 268]             Mean value theorems                       363
Let the monotone integrable function F(x) be defined by the conditions
F(x)= A, for a< x: <a + e,
F(x)=f(x), for a+< x< b-e,
F(x) =B, for b     -e _ x _b;
we can then apply the foregoing result to F(x) instead of f(x); it is clear that
F(a + )=A, F(b-O)=B.
We have now
fF(x);(x)dx=A       O ( )dx+B B  (x)dx;
fb              rb
but     f () f  (x ) d-x-  F(x) (x) d
roe                    rb
a                      b e
I[f(x) - A] (x a + f    [/(x) - B]. ( d.
The absolute value of the expression on the right-hand side is less than
(P + Q), where P, Q are the upper limits of
I {f(x) - A} 0 (x) 1,
I {f(x) - BA  f (x) I
in the interval (a, b). Now e is arbitrarily small, hence
f(x) (x) dx - A      () dx - B  ( (x) dxo,
where 5 depends on e, is < (P - Q) e; hence
f (x)   (x) dx=A     (x) d + B f   (x) dx,
where ~x is the limit of 4, when e converges to zero. The condition that
F(x) is monotone is satisfied, if
A <f(a+O), B     f(b-O),
when f (x) never decreases as x increases from a to b; and if
A _f(a + ), B    f(b - 0),
in case f(x) never increases.
It will be observed that the sole conditions which are attached to the
functions f (x), p (x), are that they be both limited, and integrable,
and that f(x) be monotone; both functions may have any set of discontinuities which-is consistent with integrability, and no assumption is made
as to their differentiability. The proof of the theorem given by Weierstrass
depended upon an integration by parts, in which the existence of a differential
coefficient of'one of the functions is a necessary assumption; this assumption
would place a restriction on the validity of the theorem which would unduly
limit its applicability to investigations such as those connected-with the
theory of Fourier's series.




364


Integration


[CH. V


269. If the function f (x) be not monotone, but be such that the interval
(a, b) can be divided into a finite number of portions in each one of
which f(x) is monotone, the second mean value theorem may be applied
to the integral taken through each of these portions separately; we then
have
rb                      XI                  a2
j f(x) 4 (x) dx = / (a -- O)  P (x) dx +f (a2 - 0)J  (x) dx
J a                   J a                        aq
+f(a, + o )  +P () dx +f(a, - ) U () dx
a2                 S2
+...............
where (a, a2), (a2, a3)... are the intervals in each of which f (x) is monotone.
When f(x) is a function with an infinite number of oscillations, and is of
the first species, a function f(x)- lx can be found which is monotone in
(a, b); we then have
f(x)         f()  x =f(a) j   (x) dx +f(b)  (x) dx
{^            rib
- laf   (x) dx - lb   (x) dx
rb
+ I  xC(x) dx, where a _   c b.
rb            re           rb
Again,         xcp (x) dx = a f  (x) dx + b f  (x) dx,
where a c ' < b; we thus find that
rb                  rt
\f (x) ' (x) dx =f (a) f ' (x) dx
rb                  rw
+f () f     (x) dx - I (b -,a)  (x) dx,
where I is, in accordance with ~ 214, any number which does not lie between
the upper and lower limits of the derivatives of f(x); and the values of I, '
will in general depend upon the value of I chosen.
IMPROPER INTEGRALS.
270. The definition of a definite integral becomes nugatory if, in the
interval (a, b), there exists any sub-interval in which the upper limit or the
lower limit of the function is indefinitely great. In such a case the function
is not limited, and the sums whose limits are the upper and lower integrals
become in one case or in both cases indefinitely great; and thus the function
is not integrable in the sense defined.
Let us suppose that a point c, where a< c < b, is such that in its
arbitrarily small neighbourhood the function has no upper or no lower limit,




269, 270]


Improper integrals


365


and let us suppose further that c is the only point of this kind, so that the
function f(x) is integrable in any sub-interval of (a, b) which does not
contain c in its interior or at an end. The two integrals
rc-e         rb
f    (x) dx,     f (x) dx
both exist, whatever sufficiently small positive values be assigned to e, e'.
It may happen that, as e, e' are diminished independently so as to converge
in each case to the limit zero, the two integrals also converge to definite limits;
if this be the case we define the sum
rc-e            rb
lim    f (x) dx + lim  f (x) dx
e=oa           e'=O c+
to be the improper integral of f (x) in the interval (a, b), and we denote
this improper integral by
rb
f (x)dx,
using the same notation as in the case in which f (x) is integrable in (a, b).
The condition that
lim  f /(x) dx
e=o0 j a
should exist is that, corresponding to each arbitrarily small number 8 which
may be chosen, a number e1 can be found such that
f (x) dx < 8,
J C-el     i
whatever value 0 may have, subject to the condition 0 < 0 < 1.
A similar condition must be satisfied in order that
fb
lim    f (x) dx
e'-O +J ce
may exist.
It may happen that, although the two limits
f  f(x) dx, 1   f (x)dx
do not exist, yet if we take c' = e, the sum
f(x) a +        f (x) dx
may have a definite limit; when that is the case, this limit defines Cauchy's
principal value of the integral of f(x) in (a, b).
It thus appears that a principal value may exist when the function
possesses neither an integral nor an improper integral in the interval (a, b).




366


Integration


[CH. V


In case the point a itself be a point of infinite discontinuity, then the limit
f (x) dx,
for e = 0, when it exists, is defined to be the improper integral
rb
f f() clx
off (x) in the interval (a, b).
A similar definition applies in case the point b is a point of infinite
discontinuity of the function.
If in the interval (a, b) there are two points of infinite discontinuity
C1, c2 where (a < c, < c2 < b), then take any point c between c, and c2. In case
the four improper integrals
f  ( )dx,    f()dx     f f(x) dx,  ff(x )dx
all exist, their sum is defined to be the improper integral of f(x) in (a, b),
and is denoted by.f'(x) dx;
and it is clearly independent of the value of c. The definition in case one or
both of the points c,, c2 are end-points of the interval (a, b) is of the same
character. If c, = a, c2 = b, then if the two improper integrals
ra         rb
f (x) dx,  Jf (x) dx
exist, their sum defines the improper integral
rb
f (x) dx.
The definition of an improper integral is now immediately extensible to
the case in which there are any finite number of points of infinite discontinuity
in the interval. If these be c1, c2, c3... c, taken in order from left to right,
then if the improper integrals
ff(x) dx, ff(dx,... (f(x)dx
all exist, their sum is defined to be the improper integral
rb
f: t(x) dx.
271. The definition of an improper integral was extended by Du Bois
Reymond and by Dini to the case of a function with an indefinitely great
number of points of infinite discontinuity forming a set of the first species.




270, 271]                 Improper integrals                          367
The definition has, however, been further extended by Harnack to the
more comprehensive case in which the set of points of infinite discontinuity
is any set of zero content. The set is closed, since any limiting point of
points of infinite discontinuity is also such a point.  It is sometimes convenient to include in such a set, points at which the functional value is
regarded as indefinitely great. Unless the upper or lower limit of the
function for an arbitrarily small neighbourhood of such a point is also, when
the functional value at the point is disregarded, indefinitely great, such
a discontinuity is a removable infinite discontinuity.
The general definition of an improper integral is obtained by extending the principle which has been applied to the case in which the
number of points of infinite discontinuity is finite, namely that the neighbourhoods of all the infinities are excluded in taking the integral. If the
integral is to be considered as in any sense belonging to the whole interval,
the sum of the excluded parts of the integral should have the limit zero; and
thus the case in which the content of the closed set of infinite discontinuities
is zero indicates the extreme extension which can fairly be given to the
meaning of an improper integral through a given interval.
Let the points of infinite discontinuity of the function f(x), defined for
the interval (a, b), form a non-dense closed set G of zero content; and further
let f (x) be integrable in any sub-interval of (a, b) which contains no point
of G either in its interior or at an end.
Let the set G be included in a definite number n of sub-intervals
81, 82)... 89)
each interval 8 containing at least one point of G in its interior, so that
the remaining part of (a, b) consists of a number of sub-intervals
71)i,,... D%,
which are free in their interiors and at their ends from points of G. Denote
by Sn the sum of the integrals of f(x) taken through all the sub-intervals r as
these intervals are diminished and their number increased.
The number n can be so chosen that L8 is arbitrarily small; and therefore
1
as n increases indefinitely Z 8 approaches the limit zero. Let a series of
1
values of n be so chosen that 28 has a sequence of diminishing values e,, e2,...
which converge to the limit zero, and let 1a,, n,3... be the corresponding
values of n.
If* the numbers S, Sn2, S3... form a convergent sequence of which S is
* See Harnack, Math. Annalen, vol. xxiv, p. 220, where this definition is given in substance.
See also Jordan, Cours d'Analyse, vol. II, p. 50, where a similar definition is given, except that
the condition, that the set of points of infinite discontinuity should have zero content, is omitted.




368


Integration


[OH. V


the limit, and if S be independent of the particular choice of the intervals 6,
then the number S is defined to be the improper integral of f(x) in (a, b), and
is denoted by j  f(x) dx.
The condition for the existence of the improper integral thus defined is
that, corresponding to any arbitrarily small number e, it must be possible to
find a number ' such that, if 81, 82, *. 3 and 8/,, 2',... ',,, be any two sets of
intervals whatever of the type defined above, and such that
1        1
then the absolute difference of the corresponding sums SG, S'i,' is less than e.
In case the improper integral  f (x) dx exist, it is the limit of the sum
of the improper integrals of f(x) through the set of sub-intervals complementary to the set of points G.
It is easily seen that the general definition of the improper integral is
consistent with the definition which has been given for the case in which
the number of points of infinite discontinuity is finite.
272. A definition of the improper integral has been given by de la ValleePoussin* which depends on a principle different from    that employed in
Harnack's definition.
Let JM, M2,... Us... and N1, N2,...  s... be two independent sequences of
positive numbers each of which consists of continually increasing numbers which
have no upper limit. Let a sequence of new functions be defined as follows:If f(x) be the given function, which has points of infinite discontinuity in
the interval (a, b) for which it is defined, let f, (x) be defined so that
fs () =f( )
for all values of x which are such that
Ms -_f(x)   -Ns; but fs(x)= Ms
for all values of x for which  f(x) > Ms;
and                           fs (x) = - Ns
for all values of x such that  f (x) < - Ns;
if f(x) be such that the integrals
rb          rb               b
f (x) dx,   f2 (x)dx,...  fs (x) dx...
all exist, and  e such that the fo   a seqence which converges to  inite
all exist, and are such that they form a sequence which converges to a definite


* Liouville's Journal, ser. 4, vol. vIII, p. 427.




271-273]


Improper integrals


369


limit independent of the particular sequences {M}, {N], then that limit is defined
to be the improper integral off(x), and is denoted by
J f (x) dx.
It will be observed that, whereas in Harnack's definition the improper
integral is defined as the limit of a sequence of integrals of the same function
taken through different domains, in de la Vallee-Poussin's definition the
improper integral is defined as the limit of a sequence of integrals of different
functions all taken through the same domain. It will however be seen later
on, that this distinction is an unessential one.
ABSOLUTELY AND CONDITIONALLY CONVERGENT INTEGRALS.
rb
273. An improper integral   f (x) is said to be absolutely convergent if
rb
the improper integral  f(x) I dx also exist; otherwise it is said to be conditionally convergent.
It has been seen in ~ 256, from the definition of a proper integral, that all
such integrals are absolutely convergent, and therefore the distinction between
the two classes of integrals has reference to improper integrals only.
Every function f(x) can be exhibited as the difference of two functions
f+ (x) and f- (x), defined so that f+ (x) = f () when f (x) > O, f+ (x) = 0 when
f(x) < 0; and f- (x) =-f(x) when f(x) < 0, f- (x) =  when f(x)  0.
We have then f(x) =f+ (x) -f-(x), and f(x) I=f+ (x)+f- (x). In an
absolutely convergent improper integral both the functions f+ (x), f-(x)
possess improper integrals, but not so in the case of a conditionally convergent improper integral. For, the points of infinite discontinuity of the
two functions f(x), If(x) I being the same, the improper integrals of these
two functions are the limits of proper integrals taken over sets of intervals
which are the same for the two functions. It follows that If(x) +f(),
If(x) -f(x) have improper integrals, provided If(x), f(x) both have
improper integrals.
The improper integrals defined in the manner of de la Vallee-Poussin
fb
are allabsolutely convergent. For, since  fs (x) dx has a limit as the two
numbers Ms, Ns are independently increased indefinitely, and the integral is
the sum of two parts, one dependent on Ms, and the other on N, it follows
that each of these parts has separately a limit. Therefore the improper
rb                               rb               rb
integral ff (x) dx exists, and similarly also f   (x) dx; hence  If f(x) dx
[b
exists in any case for which j f(x) is defined as an improper integral.
J a


H.


24




370


Integration


[CH. v


The definition of improper integrals in accordance with the method of
Harnack applies both to absolutely, and to conditionally, convergent integrals,
and is thus wider than the definition of de la Vallee-Poussin.
It has been pointed out by Schonflies that the condition that the set of
points of infinite discontinuity must be of zero content, is deducible, in the case
of de la Vallee-Poussin's definition, from the condition, contained in the
definition, for the existence of the integral.
Since the existence of
f f(x)I dx
[b
follows from that of         f f(x) dx,
there will be no loss of generality if we suppose f(x) to be everywhere
positive. Now the condition for the existence of the integral as a finite
number is that
rb
J[ f+ (x)- f(x)} dx
should, as s is increased indefinitely, have the limit zero. Considering any
convergent set of sub-divisions of (a, b), let 81, 2,... an be the sets of subintervals at any stage, and let a be the sum of those 8's which contain points of
infinite discontinuity of f(x). In all these latter sub-intervals values offs (x)
equal to Ms, and values offs1+ (x) equal to M+1,, occur; thus, in the sum whose
limit defines the integral, the upper limit of f,+, (x) -f (x) is MA+, - Ms, and
the sum of the products of the intervals into the upper limits of the function
in those intervals is _ (M,+, - Ms) a: hence the integral cannot have the limit
zero unless a converges to zero as the number n of intervals is increased
indefinitely. Therefore the points of infinite discontinuity must form an
unextended set, if the integral is to exist as a finite number.
274. In the case of absolutely convergent integrals it can be shewn
that the definition of Harnack and that of de la Vallee-Poussin are in
complete agreement.   Since, in accordance with either definition, the
existence of the absolutely convergent integral of f(x) involves that of
the improper integrals of f+ (x), and f- (x), it is clearly sufficient to consider
the case in which f(x) is positive or zero at every point of the interval (a, b).
Let us then assume that the function f(x), which is never negative, has an
improper integral in accordance with Harnack's definition. The set of points
of infinite discontinuity of G is enclosed in a finite set of sub-intervals
{8}, and the remaining part of (a, b) consists of a set of sub-intervals {X}.
The sum E8 can be chosen so small that the integral of f(x) through the
intervals t{} is less than Harnack's improper integral by less than an
arbitrarily chosen positive number 1. Let N be a positive number not less
than the upper limit of f(x) in all the intervals {I}, and let. f,(x) be the




273-275]


Improper integrals


371


function, corresponding to N, employed in de la Vallee-Poussin's definition.
Let another set of sub-intervals {['} all interior to intervals of {8}, enclose
all the points of infinite discontinuity of f(x), and let {q'} be the intervals
complementary to these. The integral of f(x) through f{'} lies between
the value of the integral through I[}, and that of Harnack's improper
integral, and therefore differs from the latter by less than ~.
It follows that the integral of f(x) through the intervals obtained by
removing the set {('} from the set {8} is also < '; and since f,,(x) -f(x),
we see that the integral of f (x) through the same set of intervals is < 4.
From this we deduce that f,,(x)dx taken through the intervals {8} is less
than 4+ NES'; and since this holds for an arbitrarily small value of E8',
N being fixed, we see that f, () dx, taken through the intervals {3j, is< 4'.
It now follows that (n (x) )dx-  f (x) dx - 4; and since ' is arbitrarily
fb
small, n being sufficiently increased, it follows that  f,, (x) dx has a definite
limit when n is indefinitely increased, and that this limit is Harnack's
rb
improper integral  f (x) dx. It has thus been shewn that a function which
has an absolutely convergent improper integral in accordance with Harnack's
definition, has one also in accordance with the definition of de la ValleePoussin, the integrals having the same value in the two cases.
To prove the converse, we assume that
fn (x) dx  or    f(x) dx +    f (x) dx
has a definite limit as n is indefinitely increased and ES indefinitely
diminished. Since both the integrals are positive, it follows that f  (x) dx,
which increases as E8 is diminished, is less than a fixed finite number, and
therefore has a definite upper limit. Therefore Harnack's improper integral
exists, and it has been shewn above that it must then have the same value
as de la Vallee-Poussin's. The two definitions have thus been shewn to be
completely equivalent to one another, so far as they both apply to absolutely
convergent integrals. The definition of Harnack is the wider, in that it
applies to the case of non-absolutely convergent integrals.
EXISTENCE AND PROPERTIES OF ABSOLUTELY CONVERGENT
IMPROPER INTEGRALS.
275. A definition of the improper integral of a function with infinite
discontinuities having been given, it is necessary to investigate whether the
limit employed in the definition really exists; and it is further necessary to
24-2




372


Integration


[CH. V


discuss whether, in the case of the existence of the improper integral in (a, b),
that improper integral shares the fundamental property of integrals that it
exists for any and every sub-interval whatever of (a, b), and also exists for
every set of such sub-intervals, and is in particular such as to satisfy the
relation
f    /() dx =  f (x) dx +   f (x) dx.
It will be proved that the limit really exists in the case of absolutely
convergent integrals, and that the improper integral then possesses the fundamental properties just specified. In the case in which the convergence is
conditional, it appears that the improper integral when it exists possesses
some but not all of the fundamental properties; in view of this fact, doubt
has been thrown by some writers upon the appropriateness of regarding
improper integrals with conditional convergence as really entitled* to the
name of integral.
Consider first the case of a function f(x) which has no negative values; so
that in (a, b), f(x) > O. In this case it can be shewn that the numbers
&1m) SnP, Sn3
either increase beyond all limit, or have a limit S which is independent of
the mode of formation of the successive sub-intervals.
Take sets of intervals
s81(), a  82(I),   -.. ^l(1)
81(2),  82(2),...  n  (2),
81 (i), 82 (in)  (in).*.  ~2 o..... ***.o *.. 
each set of which conaints all the points of infinite discontinuity; and let the
corresponding sets of intervals which are free in their interiors and at their
ends from the points of infinite discontinuity be
1  (1), ) 72(1) 
Y]  (2)  97  (2)  (2)
1 (mi)  (m)...   (nm)
Moreover, let the system of sets be so chosen that all points contained in
the n(m) are also contained in the  (mi+1).  Since f(x) is never negative, the
sums S-,, S1,... Snw... form a constantly increasing sequence; and thus the
* See Stolz, Sitzungsberichte der kais. Akad. Wien, vol. cvII, IIa (1898), p. 207, and
vol. cvII, IIa, p. 1234, also vol. cvii, p. 211. See also Stolz's work Grundziige der DZff. U.
Integralrechnung, part iii, p. 273. In these writings there is a systematic treatment of the
absolutely convergent improper integrals in accordance with Harnack's definition.




275]        Absolutely convergent improper integrals            373
sequence must have a definite upper limit S, unless its terms 'increase indefinitely, in which case the improper integral is certainly not convergent.
When S exists, we have to shew that it is independent of the particular
sets of intervals used in obtaining it. The number m may be chosen so
large that S - Sn,, < e, where e is a fixed arbitrarily chosen number.
Next, take any other set of successive sets of intervals which enclose the
points of infinite discontinuity, leaving corresponding free intervals, and let Sm
be the sum, at any stage, of the integrals taken through these free intervals:
then compare Sn,,i with Sm,. The two sums of integrals contain a number of
integrals in common, namely integrals taken over those pieces of the interval
(a, b) which are common to the sets of intervals belonging to Sn', and Sm';
Sn,, may contain parts that do not belong to Sm', these all forming parts of the
intervals r(m2); also Sm' may contain parts that do not belong to Sn,,' Now the
difference Sm, - S,, is less than the sum of the integrals taken over these
latter parts, all of which lie within the intervals 8(m), and in all these parts
the function f/() is limited; it follows that
s5n - &  < P8(m)
where P is a number which does not increase as m is increased. Now m may
be chosen so great that
28m) < e 
and then                   Sm' - S    < e,
hence                         < S +e;
and since e is arbitrarily small, we have,, _<, S;
thus Sm, cannot be greater than S. We have again, by similar reasoning
S      5M-S' < Qg8/
where Q is a number which does not increase as the number m' is increased,
depending as it does on the upper limits of f(x) for intervals all of which lie
in the 1('n); hence when m' is sufficiently great
Snm-Sn '<, or S, >Sn,,-e>S-e.
It thus appears that the second system of divisions can be so far advanced
that Sm, lies between S - e and S + e, where e is arbitrarily small; and hence
Sm, has S for its limit.  The existence of the improper integral is now
established for a function f(x) O 0, provided there be no divergence.
The theorem proved may be immediately extended to shew that the
rb
integral f f(x) dao exists, provided
f (x) I dx
exist, that is provided the convergence be absolute.




374


Integration


[CH. V


Replace f(x) by the difference of the two functions f+ (x) and f-(x);
then both the integrals
f +()dx,        f- (x) dx
exist, unless the sums of integrals of which they are the limits increase
indefinitely.
Now           f If(x)ldx=f f+(x)dX Jf-(x)d=x;
hence if ff(x)   have an integral, both the functions f+(x), f-(x) have
integrals, and thus f(x) has also an integral.  The existence of absolutely
convergent improper integrals has thus been established.
276. If c be a number such that a < c < b, and if f (x) have an absolutely
convergent improper integral in (a, b), then it has also such integrals in (a, c)
and in (c, b), and the sum   of the two latter integrals is equal to the
former one.
Consider a system of intervals {[8, [{} as in ~ 275. If the point c lies in
an interval 8 it does not affect the sums of the proper integrals through the
intervals r; but if, at any stage of the limiting process, the point c comes to lie
within an interval v, it divides it into two parts. Clearly, the sum of the
integrals of I f(x) I taken through those intervals V which are on the left of
c, and through that part of the interval containing c which lies on the left
of c, is less than the sum of the integrals of If(x) I through all the intervals ij.
The same is true of the sum of the integrals of If(x) I through all those
intervals X which lie on the right of c, and through that part of the interval
containing c which lies to the right of c.
Thus, since the integral of f (x) I through all the intervals XV lies below a
fixed limit, the same is true of the two parts into which the integral is
divided by the point c; and thus the two integrals


exist; and therefore also
J
exist.


If (x) I dx,  If (x) dx
f (x) dx, Jf(x) d
a        Jc


The splitting up of the sum of the integrals of f(x) through the intervals
sV into two parts does not affect that sum; hence also in the limit we have
f (x) dx=    f (x) dx +bf(x) dx.
A corollary from this theorem is that if f(x) have an absolutely convergent
improper integral in (a, b), it is also integrable in any interval (a', b') which
forms part of (a, b).




275-278]   Absolutely convergent improper integrals


375


277. If A,,,... A,,,... be a sequence of non-overlapping intervals
contained in (a, b), in descending order of length, the sum of the integrals of
f(x) taken through Ai, A,... A,, converges to a definite limit as n is
increased indefinitely, provided
rb
f (x) dx
converges absolutely.
Consider the two functions f+ (x), f- (x) defined as in ~ 273. The function
f+ (x) is integrable in each of the intervals A, and the sum of the integrals
off+ (x) through the intervals Al, A2,... A, is positive and does not diminish
as n is increased; also this sum never exceeds the integral of f+ (x) through
(a, b). It follows that the sum of the integrals of f+ (x) through
A1, A2,... An,
converges to a definite limit as n is increased indefinitely. The same is true
of f- (x); and hence f(x), which is f+ (x) -f (x), is such that the sum of its
integrals converges to a definite limit when the number of the intervals A is
increased indefinitely.
It has thus been shewn that, if (x) have an absolutely convergent improper
integral in (a, b), it has also an improper integral through any portion of
(a, b) which consists of a finite, or of an infinite, number of continuous intervals.
278. If H be a set of points in (a, b) of zero content, so that the points of
H can be enclosed in intervals 01, 02,... 0p whose sum is arbitrarily small, then
the integral off(x) taken through the intervals 01, 0,... 0p has the limit zero,
as the sum of the intervals converges to zero, their number being indefinitely
increased, f () having an absolutely convergent improper integral in (a, b).
The points of infinite discontinuity off(x) can be enclosed in a set of
intervals 81, 8,... An such that the integral of If(x) taken through these
intervals is < e. Let b0, 02,... br7 be the parts of the intervals 01, 0,... Op
which are common with the intervals 1,,2,... -l, which remain in (a, b)
when the 8's are removed. The absolute value of the integral of f(x) taken
r              p
through the intervals 01, 02,... O is less than e + U2O, or than e + UO, where
1              1
U is the finite upper limit of f( x)l in all the intervals i7.  Having
fixed the intervals 81,   8,.. n, we can choose the intervals 0 so that
f0 < e/ U; thus the absolute value of f(x)dx taken through the intervals 0
is then < 2e, which is arbitrarily small. The theorem is therefore established.
rb
It follows that, in the definition of the improper integral f f(x) dx as the
limit )f the sum of the proper integrals through the intervals q7, we may
suppose the neighbourhoods of the points of H to be removed from (a, b),




376


Integration


[CH. v


these neighbourhoods being so chosen that their sum converges to zero as the
sum of the proper integrals converges to its limit, the improper integral of
f(x) in (a, b).
The theorem may also be stated in the form that, if H be any set of points
of zero content, then in applying Harnack's definition, the set H may be added
to the set G of points of infinite discontinuity, without altering the value of the
integral.
From this theorem we may deduce that if f(x), f (x) have both absolutely
convergent improper integrals in (a, b), their sum f (x) + 4 (x) has an absolutely
convergent improper integral in (a, b).
rb
In relation to   f(x)dx, the points of infinite discontinuity of +(x)
form a set H such as is contemplated in the foregoing theorem; thus in the
fb
definition of  f(x) dx we may exclude the neighbourhoods of the points H.
[b
A similar remark applies to f  r (x) dx. The points of infinite discontinuity
of f(x) + * (x) consist in general of the sets for f(x) and for f (x) together.
rb
It therefore follows that f{ f(x) +  (x)} dx exists, and is identical with
rb         rb
f(x)dx +     *(x)dx.
The following theorem has been incidentally established:If 2e be an arbitrarily chosen positive number, then a positive number r,
dependent on e, can be determined, such that, for any finite set of non-overlapping
intervals whatever, whose sum is less than rV, the absolute value of the sum of the
integrals of f(x) through the intervals is less than 2e; it being assumed that
f (x) has an absolutely convergent integral in (a, b).
279. If f(x), f (x) have both absolutely convergent improper integrals
in (a, b), and if the two functions have no points of infinite discontinuity in
common, then the product f(x) J (x) has an absolutely convergent improper
integral in (a, b).
The infinities of f(x) may be included in intervals 3,, 2,... an, and those
of r (x) in intervals 81', 8',... 8m', such that no interval 8 encroaches on any
interval 8'. Let U be the upper limit of f(x) in all the intervals 8', and let
U' be the upper limit of f (x) in all the intervals 8; thus U, U' are definite
numbers. In the intervals 8, f(x) 0 (x) I never exceeds U' I f(x) 1, and hence
since f(x) i is integrable in the intervals 8, having an improper integral in
each of these intervals, in accordance with the definition in ~ 271, it is clear that
If(x) -x (x) is also integrable in these intervals 8; and  I f(x)() () dx




278-281] Absolutely convergent improper integrals


377


taken through the intervals 8 is _ U' f f(x) dx taken through the same
intervals. Similarly it may be shewn that I f(x) (() ) is integrable in the
intervals 8'. It follows that If(x) r(x) I is, under the conditions in the
enunciation, integrable in the whole interval (a, b); and that therefore
f(x) * (x) has an absolutely convergent improper integral in that interval.
280. If f(x) have an absolutely convergent improper integral in (a, b),
then the improper integral  f f() dx is a continuous function of the upper
limit x.
This theorem is the extension of that of ~ 258 to the case of absolutely
convergent improper integrals.
Let F(x) denote  f f(x) dx, which has been shewn, in ~ 276, to exist for
rx+h
a c x _ b. We have F (x + h) - F (x) =  f(x) dx; if then x is not a point
of infinite discontinuity of f(x), h may be so chosen that (x, x + h) does not
contain any such points, and in that case IF (x + h) - F (x) \L h Ih U, where U
denotes the upper limit of I f(x) I in the interval (x, x + h); since U does not
increase as h is diminished, it follows that, corresponding to a fixed number e,
a value of h, say h, can be found such that F(x + h) -F(x) <e, for
]h   Ah; hence F(x) is continuous at x.
In case x be a point of infinite discontinuity of f(x), we may enclose all
such points in a finite set of intervals such that the integral of I f(x) I taken
x+h          rx+h
through all of them is < e. Since  xf(x) dx <  j f(x) dx, if we choose
h so small that (x, x + h) is entirely within that interval of the set which
contains the point x, we see that l F(x + h) - F(x) < e; and hence the point
x is a point of continuity of F (x).
NON-ABSOLUTELY CONVERGENT IMPROPER INTEGRALS.
281. It has been shewn that an absolutely convergent improper integral,
when it exists, possesses the fundamental properties which belong to a proper
integral, viz. that the function is also integrable in any part of the interval
which is either continuous or which consists of a finite or infinite number of
rb         rx         rb
continuous portions, and that  f(x) dx =  f (x) dx +  f (x) dx.  Further
it has been shewn that  f(x) dx is a continuous function of the upper
limit x,




378


Integration


[CIi. v


If we apply the definition of ~ 271, to the case in which j If(x) I dx does
not exist, it has not been shewn that the sum of the integrals of f(x) through
the intervals r which remain when the points of infinite discontinuity are
enclosed in a set of intervals 8, necessarily either converges to a definite limit,
rb
or increases indefinitely. Further, if the limit which defines  f (x) dx in
any particular case actually exists, it has not been shewn that f(x) is
necessarily integrable in (a, x), or in general in every interval contained in
(a, b), the proof in ~ 276, depending essentially upon the assumption that
If(x) I is integrable in (a, b). It is thus a matter for further investigation
whether a non-absolutely convergent improper integral defined in Harnack's
manner necessarily possesses the fundamental properties which would justify
us in regarding it as an extension of the conception of a proper integral. The
definition of de la Vallee-Poussin, in ~ 272, is applicable only to absolutely
convergent improper integrals.
Under these circumstances the definition of a non-absolutely convergent
improper integral has, by some writers*, been restricted to the case in
which the set G of points of infinite discontinuity of the function is enumerable and of the first species. The mode of definition usually applied in this
case will first be briefly considered, before the more general definition of
Harnack is considered.
First let us suppose that G consists of a finite number of points
C1,  C2,... Cl *
then as in ~ 270,            j      f(x) dx
Cr-l
is defined to be the limit of
fo  /~ f(x) dx,
r —1+e
as e, E independently of' one another converge to zero, on the assumption that
this limit exists.
If the integrals
rci          rv?2           rb
f /(x) dx,     f(x) dx,...    f(x) dx,
all exist in accordance with this definition, their sum is defined to be the
improper integral
rb
f (x)dx.
* See Du Bois Reymond, Crelle's Journal, vol. LXXIX, pp. 36 and 45, also Dini, Grundlagen,
p. 404,




281]    Non-absolutely convergent improper integrals


379


This definition is applicable, whether If(x) I be integrable in accordance
with it, or not; and it thus defines non-absolutely convergent integrals in the
case considered.
Next, let us suppose that G is of the first species and of the first order.
In this case G' consists of a finite number of points e1, e2,... e..
If all the improper integrals
f(x) dx,       f (sx) dx,  /f() dx,..,
cona      J" ei +e'      J e2q-e2'
each of which falls under the last case, have each a definite limit as
61, 61 2, e,,,... converge, independently of one another, to the limit zero,
then the sum of these limits is taken to define the integral
I f(.x) dx.
J.
It is clear that this definition admits of extension to the case in which G
is of the first species and of any order. It is also clear that an integral
in (a, b), which exists in accordance with this definition, entails the existence
of the integral in (a, x), and in any continuous interval contained in (a, b); and
further the truth of the theorem
rb          rX          rb
j f(x) dx =     / (x) dx +  f (x) dx
is assured.
The definition has been extended by Schonflies * to the case in which G
is enumerable but possesses derivatives of transfinite order.
In the case in which f(x) is absolutely integrable in accordance with
Harnack's definition, and in which G is of the first species, it can be easily
shewn that Harnack's definition reduces to the one here given.
It should be observed that, in the case of a non-absolutely convergent
improper integral which has an infinite set of points of infinite discontinuity,
the theorem that the function is integrable through any set of intervals
contained in the interval of integration, does not in general hold; so that such
improper integrals are not in this respect on a parity with proper integrals.
For it may be possible to choose an infinite set of intervals so that f(x) is
everywhere positive in them; and then the sum of the integrals off(x) through
these intervals does not in general converge to a finite value, the existence of
the integral in (a, b) depending essentially on the cancelling of the integrals
through those parts of (a, b) in which f(x) is positive, with the integrals
through those parts in which f(x) is negative. The two integrals
fJ+(x) dx,     f   () dx
* See his Bericht, p. 185; a similar definition has also been employed by de la Valle-Poussin,
loc. cit., p. 453,




380                            Integration                          [CH. v
have no finite values, although
J {f+(.)- -(x)} dx
may have a definite finite value.
That F (x)= f(x) dx is a continuous function of the upper limit x, in
the case when the integral is a non-absolutely convergent improper integral,
in accordance with the definition here given, can be shewn as follows:rx+h
Since             F(x + h) - F(x) =      ff(x) dx;
if x be a point of infinite discontinuity of f(x), we know that the integral on
the right-hand side has the limit zero, when h is indefinitely diminished
either through positive or through negative values; and hence a value h, can
be found such that     I F (x + h)- F(x) < e, for h I < h.
In case x be not a point of infinite discontinuity of f(x), the proof is
identical with that which has been given for the case of a proper integral.
EXAMPLES.
1. Let f(x) denote a function which is integrable in every interval (a, b), where
0 < a < b; and let f(x), be in the neighbourhood of the point 0, of the form,k (, where
k is positive, and p (x) is a limited function of constant sign.
We have
If   k dt < A J     <'d [A-k    -
where A is some positive number.
If 0 < k < 1, it is clear that | ~ () dx is arbitrarily small for a sufficiently small
value of ' (> e), and therefore the improper integral f| (x) dx exists, being convergent
at the point x=O. If k > 0, the improper integral does not exist.
2. Let f(x) be, in the neighbourhood on the right of the point 0, of the form
[lo  ]1, where p is positive, and f (x) satisfies the same condition as in Ex. 1.
x [log x]1 + '
We have
i e'  l (x) +dx <   {[log E] — [log '] -};
x [logX]14p    p
and thus the improper integral  (x) dx exists, being absolutely convergent.
3.  It   x dx, taken through any interval which contains a point of infinite discontinuity of tan x, does not exist,




281, 282] Non-absolutely convergent improper integrals  381


7r
f2 etan       2 1 sin '
For                      2  tan dx > - log sin 
xT      x  >l      sin E
and this is arbitrarily great for a sufficiently great value of e/e'; thus the integral does not
converge at the point x = 7r. The integral possesses however a principal value at the
point Tr. For the sum of the integrals taken through the intervals (~ - e, ~  r - e') and
(7 r+E', r. +e) is
Ot. -2     dx < 2e ( 7r2- E2)-1 (1-6e2)-1,
and this converges to 0, with e.
1   1      1
4. The function cos (ex) + - ex sin (ex) oscillates between indefinitely great positive and
negative values, in the neighbourhood of the point x =0. For every value of x except
d        1
x =0, the function =  - {x cos (ex)}.
CE d        1           1        1
Also             | d {x cos (ex)} dx = e cos (e) - e' cos (e')
where e > e' > 0. It thus appears that the integral of the function converges at the point
x=0; and therefore the function is integrable in an interval containing that point.
282. The general theory of improper definite integrals, both those
which converge absolutely, and those which converge non-absolutely, defined
according to Harnack's definition, has been treated by E. H. Moore*, who
has also considered other definitions of such improper integrals.
It has been shewn in ~ 276, that if J(x) have an improper integral in
(a, b), in  accordance with    Harnack's definition, and    such  integral be
absolutely convergent, then f(x) is also integrable in any interval (a', b')
which is part of (a, b). It will now be shewn that this holds whether
the improper integral converges absolutely or not+.
Let {8} denote a finite set of intervals enclosing all the points of infinite
discontinuity of '(x), each interval of the set enclosing at least one such
point; we may denote by fs (x) a function which is zero at all points interior
to the intervals {8}, and is at every other point of (a, b) equal to f(x).
The corresponding function for any other such set of intervals {8'} may be
denoted by fs, (x).
The condition stated in ~ 271, for the existence of the improper integral
fb
f(x) dx, may be expressed in the form that, corresponding to an arbitrarily
fixed positive number e, it shall be possible to fix a number ', such that, for
any two sets of intervals {8}, {8'}, such that Z8 <,    <S' < C, the condition
fa (x) dxf- fs, (x) dx < 6
may be satisfied.
* Trans. Amer. Math. Soc., vol. II, 1901, p. 296, and a second paper, p. 459.
t This is contrary to a statement made by Stolz; see Grundziige, vol. III, p. 277.




382


Integration


[CH. V


Assuming that this condition is satisfied for every value of e, it will be
shewn that, for every pair of points a', b' in (a, b), the condition
s (x) dx -    fs, (x) dx < e
is satisfied, provided {8}, [8'} are any two sets of intervals of the prescribed
kind, and such that Z8, Y8' are each less than I.
Let it be assumed that, if possible, a', b', {8}, {8'} can be so determined,
subject to the conditions 28 <     <[, S'< <, that
f: (x) dx -  ff, (x) dx  -e.
It will then be shewn that finite sets {8(2)}, {8(3)} can be determined, each
of total length less than I, and each containing the set of points of infinite
discontinuity, for which
b            rb
f 2) () d -  f a ( x) d x  e;
and since this is contrary to the hypothesis, the impossibility of the above
assumption will have been demonstrated.
To define {S(2)}, {(3)}, we take any interval of {3} within (a', b'), as an
interval of [1(2)}; and any interval of {['} within (a', b'), as an interval of
{8(3)}. Further, we take for the parts of {8(2)} and {8(3)} within (a, a')
and (b', b) the set of those intervals which are common to the parts of
{8} and {8'} that lie in (a, a') and (b', b).
In case a' is contained in intervals (a, /3), (a', 3'), of {8} and of {8'}
respectively, we take (a', 8) and (a', /3') as intervals of {}(2) and of {8()}
respectively, where a' >a. A similar specification will refer to b'.
It is now clear that, in accordance with these definitions of {8(2)} and
{8)}, we have f (x)=f(2) (x), and fa (x) =f-) (x), if x is within (a', b');
and A/(2) (x) =f,3) (x), if x is within (a, a'), or within (b', b).
It follows that
rb           fb            rbl         l'b'
f 2f (x) dx -  f( (x) dx = Js (x) dx -  f, (x) dx,
a   (a       c/ 05'         Cl'J         Ct
and thence that
rb           rb
f() (x) dx-   fA( (x) dx;
-  a         J a
moreover it is clear from the mode of construction of {8(2)} and {8(3)}, that
S(2) <,   (3) < '.  The impossibility in  question  has therefore been
demonstrated.
Since for every pair of numbers a', b' such that a   '< ' a b'  b, corre



282]    Non-absolutely convergent improper integrals


383


spending to any arbitrarily chosen number e, a number I-' can be found
such that
fa (x) dx -  fs, (x) dx < e,
for every pair of sets of intervals {8}, {S'}, enclosing the points of infinite
discontinuity, at least one such point being contained in each interval of
rb'
either set, and such that Z8 < -, S8' <, it follows that  f(x) dx exists.
Moreover, since ~ is independent of a', b', we have established the
following theorem:If   f (x) dx exist as an improper integral, in accordance with Harnack's
rb'
definition, then  f (x) dx also exists, where a', b' are such that a - a' < b  b;
and the convergence of this integral is 'uniform for all values of a' and b'.
The last part of this theorem expresses the fact that
\|f(x) dx-    fs,(x) dx  6,
provided Z (8) < 2,} for every value of a' and b'; the number t depending on
the arbitrarily chosen number e.
The theorem,     f(x) dx +   f'(x) dx==  f(x) dx
is valid.
This follows from the corresponding theorem  for the proper integrals
of f (x); for it appears that the expressions on the two sides of the
equation differ from one another by 2e at most; and since e is arbitrarily
small, their equality is established.
Since the existence of the integral off(x) in any sub-interval (a', b')
of (a, b) has been shewn to be a necessary consequence of the existence of
the integral in (a, b), it is clear that the integral of f(x) taken through
any finite set of non-overlapping intervals contained in (a, b) also exists;
being the sum of the integrals taken through the separate intervals. However, if a non-finite set of non-overlapping intervals be taken in (a, b), it is
not in general true that the sum of the integrals of f(x) through these
intervals converges to a definite number, unless the integral of f(x) is
absolutely convergent, which case has been treated in ~ 277. It will in
fact be shewn, by means of an example, that the property in question,
that f(x) is integrable through a non-finite set of intervals in (a, b),
does not appertain to non-absolutely convergent integrals, and must be
regarded as peculiar to absolutely convergent integrals. This does not
however seem a sufficient reason for refraining from  applying the term
"integral" to non-absolutely convergent improper integrals.




384


Integration


[CH. v


283. The following theorem contains the necessary and sufficient conditions for the existence of the improper integral of a function f(x) in
an interval (a, b), in which the set G of points of infinite discontinuity
of f(x) exists. 
The complementary intervals of G being denoted by (a,, b,), the necessary
rb
and sufficient conditions for the existence of  f (x) dx are
(1) that all the integrals    f(x)dx shall exist, each such integral
1a
being defined as the limit of  v f (x) dx, when e, e' converge independently to
CbV+e
the limit zero, and
(2) that (0 + oa +...+  I c shall converge to a definite number, as v is
indefinitely increased; where to denotes the fluctuation of  f (x) dx int the
al
interval (a,, bv).
Moreover, when the conditions (1) and (2) are satisfied, the sum
v rbV
E1   f (x) dx
v=l J a
is convergent, and its limit, as v is indefinitely decreased, is  f (x) dx.
a
For the proof of this theorem, which is due to E. H. Moore, reference
must be made to the original memoir*.
It will be observed that in Harnack's definition of an improper integral,
the set of intervals {8} which are of arbitrarily small sum, and which enclose
the points of infinite discontinuity of the function, have been so chosen that
each interval 8 contains at least one of these points. If this latter condition
were omitted from the definition, the amended definition would admit only
of the existence of absolutely convergent improper integrals. An integral
thus defined has been named by E. H. Moore a broad integral, in contradistinction to the narrow integral as given by Harnack's original definition.
It is unnecessary here to shew that a broad integral is necessarily absolutely
convergent, because the corresponding definition for double integrals will be
fully considered below. The broad integrals are a special case of the narrow
ones; those narrow integrals which are not broad ones are the non-absolutely
convergent integrals.
284. A method will now be given of constructing a function f(x) which
is continuous at every point of the interval (a, b), except at the point b at
* loc. cit., p. 324.




283, 284]  Non-absolutely convergent improper integrals


385


which the function has an infinite discontinuity of such a character that
rb
f f(x)dx converges non-absolutely.
Let a sequence of intervals (a,, b,), (a,, b,),... (a,, b)... be defined in the
interval (a, b), such that no two of the intervals overlap, and that b is the
limiting-point of each of the sequences (a1, a,... a,,...), (b, b2,... bi,...).
Let U, + u2 +... +u,M +... denote a non-absolutely convergent arithmetic
series (see Chap. VI). In (an, b,,), let f(x) be defined so as to be continuous
in that interval, and everywhere of the same sign, and let f(x) vanish at
a,, and bu. Further, let f(x) be so chosen in the interval, that
f(x) dx ='.
At all points of (a, b) exterior to all the intervals (a,, b,,), let f(x)= 0.
The function f(x) so defined is continuous in (a, b), except at b.
In (an, b,,), the function If(x) I has a maximum greater than n,,I l/(b, - a),
and therefore f(x) has indefinitely great positive and negative values in every
neighbourhood of the point b.
We have now
f (x) dx =    E tn + On,,+,,
vJf( a)d~.6 ~,r=l
if x lies in the interval (b,, b,+,); where 0 is some proper fraction.
Now the improper integral   f(x) dx is defined by lim  f(x) dx, and
J a                  ix-=b
its value is therefore the limiting sum of the series u, +u +... + un....
rb
It is further clear that  f /(S ) ldx does not exist, since the series
|'l,  +  I U2+... +  1+ I  1+  **.
is not convergent.
This case may be employed to illustrate the fact that the non-absolutely
convergent improper integral is not necessarily the limit of the sum of
the integrals taken through a set of intervals which in the limit converges
to the whole interval of integration; and thus that such an integral is not a
broad integral.
Let the integral of f(x) be taken through the intervals
(a, b,n), (al, Eb.), (ap,  p2).. (aG,,', b.,'),
where p, p2,,... p, are increasing numbers all > nm, such that t,,, ut,2... tt
are all of the same sign. It is clear that in may be so chosen that / (x) dx
H.                                                       '25
H.                                                        25




386


Integration


[c. v


rb
is arbitrarily near to f (x) dx; then, for such a fixed value of m, the numbers
p1, p2,... pr may be so chosen that uP, + tP +...  is,. is as large as we please,
since the series 2ua is non-absolutely convergent. As in is increased indefinitely, the set of intervals (a, bm), (ap, bp)... (a,., b,.) converges to the
whole interval (a, b), the total length of the complementary part of (a, b)
diminishing indefinitely, and yet the sum of the integrals of f(x) taken
through the set of intervals is divergent.
This example may be used to illustrate the fact that the theorem
established in ~ 278, for absolutely convergent integrals, does not hold for
non-absolutely convergent integrals. It is not in fact true that, in defining
rb
f (x) dx, the set of points a, a2,... al,... b,, b2,... b,,... b, which is of zero
content, may be excluded by enclosing these points in a set of intervals
of arbitrarily small sum. For we may include all the points ai, a2,... am,
bl, b,... bm which occur in (a, b.,,) in a finite set of intervals, so that when
fbin
these are excluded from  the domain of integration of     f (x) dx, that
integral is altered by an arbitrarily small amount.
Again we may shorten each of the intervals (ap, bpl)... (c,., b,.) at each
end, so that the sum   of the integrals taken through these intervals is
diminished by an arbitrarily small amount. All the points a1, a2,...,
b,, b,... b are now included in a finite set of intervals, such that the integral
of f(x) taken through the complementary intervals has an arbitrarily great
sum. These complementary intervals consist of those intervals which have
been obtained by shortening (p,, bp,)... (c,., bp,), and of the parts of (a, b,,)
which remain when the points a,, b1, a2, b2,... bM. have been included in a
suitable set of intervals.
Let +b(x) be an improper integral for which all the points ac, a2,..,
rb
bi, b2,... b are points of infinite discontinuity; and thus j  (x) dx may exist
in accordance with Harnack's definition. Also  f (x) dx exists, as defined
above, having its single point of infinite discontinuity at b. It appears
[b
however that   {/(x) + < (x)j dx does not exist, because f(x) + + (x) has
infinite discontinuities at all the points ac, b1, and at b, and its existence
rb
would imply that in defining  f(x) dx we could employ sets of intervals
which exclude not only the point b, but also all the points an, b,.




284, 285] The fundamental theorem for improper integrals 387


THE FUNDAMENTAL THEOREM OF THE INTEGRAL CALCULUS FOR THE
CASE OF IMPROPER INTEGRALS.
285. The theorem of ~ 260, that if f (x) be integrable in (a, b), and
F(x) be the corresponding integral function, any one of the four derivatives
DF(x) of F(x) is integrable in (a, b), and has F(x) for its integral function,
is applicable to the case in which the integral off(x) is improper in the sense
in which an improper integral has been defined above in the two cases of
absolute and of non-absolute convergence.
Let F(x), r (x) be two functions which are both continuous in (a, b), and
let us suppose that one of the four derivatives DF (x) is finite and. equal to
the corresponding derivative D# (x), at every point of (a, b) with the exception of a set of points G, non-dense in (a, b), and such that the content of the
closed set H, obtained by adding to G all its limiting points, is zero. At the
points of H, the derivatives DF(x), D# (x) may be supposed not to be finite,
or not to be equal.
The function F(x) - r(x) is then constant throughout any one of the
intervals complementary to H; and it has been shewn in ~ 206 that
F(x)-   (x)
is constant throughout (a, b), in case H be enumerable, but that it need not
be constant if H be unenumerable. In the latter case the complementary
intervals of H are lines of invariability of F (x) - r (x), and the function
DF (x) - D# (x)
is a null-function with an improper integral in (a, b).
If p (x) be a continuous function, and one of its four derivatives
Do (x)= f (x),
have a set of points of infinite discontinuity which is enumerable, and Do (x)
have an improper integral in (a, b), then
f (x) d=()     - =( ) (a).
For the set of points of infinite discontinuity is non-dense and closed, and
has zero content, since  f(x) dx exists.  If q (x) =  f(x) d, the two
functions f (x), b (x) are both continuous, and have the derivatives Djr (x),
DO (x) everywhere identical with f(x), except at that set of points, of zero
measure, at which f (x) is discontinuous. Hence (see ~ 224), in any interval
containing no points of infinite discontinuity of f (x), the functions p (x), (x)
differ by a constant. Since f(x) -b(x) has as lines of invariability the
intervals complementary to an enumerable closed set, it is constant throughout (a, b); and it is clearly equal to ( (a).
If the set of points at which Dc (x) =f(x) is infinite be of the power
of the continuum, we can no longer conclude that f (x), p (x) differ by a
constant. In this case we have the theorem:25-2




388


Integrat?"'Wil


[OH. v


If b (x) be a continuous function, such that one of its derivatives
D4^ (x) f (x)
possesses an improper integral in (a, b), and if the set of points of infinite
discontinuity of Do (x) be unenumnerable, then
Xf (x) dx = q (x) - c (a) + U(x)- U(a)
where U (x) is a function with an everywhere-dense set of lines of invariability.
Accordingly, in this case, the fundamental theorem  of the Integral
Calculus.does not hold, in its original form.
The following definition of the definite integral of a function f(x),
which in the interval (a, b) possesses an enumerable set G of points of infinite
discontinuity, has been given by Holder*.
Let F(x) be any function which is continuous in (a, b), and is such that,
for any two points x1, x2, such that no point of G lies in the interval (xi, x2),
the relation
/12
F (x,) - F (xi) =I f(x) dx
holds; the function J(x) being assumed to be integrable in every such
interval. Then the definite integral off(x) in any interval (a', b') whatever,
rbf
contained in (a, b), is defined by  f (x) dx = F(b') -F(a').
That F(x) is unique, except for an additive constant, has been shewn
above. If G were unenumerable, this definition would not suffice to define
the integral, because F(x) would not be unique.
GEOMETRICAL INTERPRETATION OF INTEGRATION.
286. Let f(x) be a limited function defined for the interval (a, b), and of
which all the values are positive or zero. This function may be considered to
define a two-dimensional set of points (x, y) which consists of all the points
of which the coordinates satisfy the conditions a _- x  b, 0 _ y < f(x). In
accordance with Jordan's theory of the measure of sets of points (see ~ 84),
this set has an exterior extent, and an interior extent, and the set of points is
measurable when the two have the same value. The extent of a twodimensional set of points may be regarded as a generalization of the conception of area; thus in the present case, the exterior extent and the interior
extent may be spoken of as the exterior area and the interior area of the
space bounded by the axis of x, the two straight lines x= a, x = b, and the
" curve" defined by y =f(x). 'This set of points G has an area, in the ordinary
* Math. Annalen, vol. xxiv, 1884.




285, 286] Geometrical intepretation of integration


389


sense, when the exterior area and the interior area are equal. The frontier
of the two-dimensional set G consists of those points of G which are limiting
points of the complementary set C (G), and of those points of C(GS) which are
limiting points of G. Those points of G which do not belong to the frontier
are said to be interior points of G. If a rectangle be drawn on (a, b) as base,
and of height greater than the upper limit of f(x) in (a, b), and if this
rectangle be divided into a number of rectangular portions by drawing
straight lines parallel to the axes of coordinates, then if the number of these
rectangles is increased indefinitely, in such a manner that the maximum of
the diagonals has the limit zero, then the interior extent of the given twodimensional set of points is the limit of the sum of those rectangles every
point of each of which is an interior point of G. The exterior extent is the
limit of the sum of those rectangles, each of which contains at least one point
which is either an interior point or a point of the frontier of G.
If, at any stage of the subdivision into rectangles, those sides which are on
(a, b) be 81, 82,... n, the two sums just referred to are:8u(s), S8 L(),
1         1
where U (8), L (8) are the upper and lower limits of f(x) in the interval 8;
and the limits of these sums are the upper and lower integrals off(x) in the
interval (a, b).
It thus appears that the upper integral  f (x) dx is the exterior extent
of the set of points defined by a < x < b, 0 < y  f(x); and the lower
rb
integral fJ (x) dx is the interior extent of the same set. If f(x) be integrable,
the upper and lower integrals are equal, and the set of points is measurable
in accordance with Jordan's definition of measure. Thus the integral represents the area defined as the measure of the set of points, when that set is
measurable.
In case f(x) be limited, but not always positive or zero, we may take
f(x) =fi (x) - fi(x), where f (x) = f(x) when f(x) is positive or zero, and
f (x) = 0, when f(x) is negative; with a corresponding definition of f (x). In
case the two sets of points (x, y) for which a < x < b, 0 < y < f (x) and
a c x <  b, 0  y  f2 (x) are both measurable, the integral  f(x) dx is the
excess of the measure of the first of the two sets over that of the second; and
this may be interpreted as the excess of that part of the area defined
by x = a, x = b, y = 0, y =f(x) which is above the x-axis over that part which
is below it.
If the two sets of points be not measurable, the exterior and interior




390


Inztegrationn


[CH. V


extents of the first set are  f (x) dx,  fi (x) dx; and those of the second
b           b
set are  ( f (x) dx,  f2 (x) dx respectively.
The upper integral f (x) dx is then the excess of the exteri  extextent of
the set a _ x c b, 0 _ y <   fi (x), over the interior extent of the set a cx S b,
0 < y < f2 (x); whilst the lower integral  f(x) dx is the excess of the
interior extent of the first of the sets, over the exterior extent of the
second set.
The condition of integrability of the function f(x) is that the frontier
which consists of the set of points a - x S b, y =f(x) when closed by adding
the limiting points, shall be a set of zero measure; this measure being that
which is applicable to two-dimensional sets. This is the condition for the
existence of the area in the ordinary sense of the term, and is equivalent
to that of the existence of the corresponding integral.
If a linear set of points G be defined on the x-axis, which is limited and
lies in the interval (a, b), then a function f(x) may be defined by the rule that
f(x) = 1, if x be a point of G, andf(x)= 0, if x be not a point of G. This set
G has always an exterior extent, and an interior extent, which are given by
7b         rb
f(x) dx,   f(x)dx
respectively, as may be seen by referring to the definitions. For it is easily
seen that the exterior or the interior linear extent of G is numerically
identical with the corresponding extent of the two-dimensional set, defined
by the function y =f (x).  When G is measurable in accordance with
Jordan's definition of a linear measure, the function f(x) is integrable in
fb
(a, b), and f f(x) dx is the measure of G. This measure may be regarded as
a generalization of the notion of length of a linear interval. The condition
that a linear set G be measurable is that its frontier, which consists of
those points of G which are limiting points of C (G), and of those points
of C(G) which are limiting points of G, have the linear measure zero.
LEBESGUE'S THEORY OF INTEGRATION.
287. A definition of integration has been developed* by Lebesgue which
is applicable to a more extensive class of functions than those which are
integrable in accordance with Riemann's definition. The theory depends
essentially upon the employment of the conception of the measure of a set of
* See his memoir "Int6grale, Longueur, Aire," Annali di Mat., series IIa, vol. VII, 1902;
also his Leeons sur l'intgration, Paris, 1904,




-286, 287]


Lebesgue's theory of integration


391


points, in the sense in which the term is employed by Borel and Lebesgue. It
has been shewn in Chapter III, that a set which is measurable in accordance
with the definition employed by Jordan is also measurable in accordance
with the definition employed by Borel and Lebesgue, but that the converse
does not hold.
A fuinction f(x) defined for the interval (a, b), is said to be summable, if
the set of points x of the interval (a, b), for which A <f(x) < B, is always
emeasurable, whatever numnbers A and B may be.
A function f(x) which satisfies the condition stated in the definition may
or may not be limited.
The set of points of (a, b) for which f(x) has a fixed value k is measurable,
if f(x) is a summable function. For this set is the set of points common to
the measurable sets for which k -  < f(x) < k + 8, where 8 has a sequence of
values converging to zero; hence, by a theorem of ~ 82, the set for which
f(x) = k, is measurable.
Let f(x) be a summable function which is limited in (a, b), and is never
negative. Let the interval (L, U) of variation of f(x) be divided into any
n parts (a,, a,), (a,, a),... (an_, an), where a0 = L, a,, = U. Let e, be the linear
set of points in (a, b) for which f(x)  a,, and let e,' be the linear set of points
in (a, b) for which a, <f(x) < a,+1; let E denote the two-dimensional set of
points for which a < x: b, O <y _f(x). Those points of the set E for
which the values of x belong to e, form a two-dimensional set, of which the
measure is a,m (e,); and those points of E for which the values of x belong to
e,' form a set which contains a set of measure a,m (e,'), and is itself contained
in a set of measure a,+m n(e,').
The set E contains a set of measure
E a,m (e,) + 2 aml(e',_) = M;
L=0 O=l
and it is contained in a set of measure
==O        6=1
The interior measure of E is > M, the measure of the set which E contains; and
the exterior measure of E is _ M', the measure of the set in which E is contained. The measures M and M' of the two sets differ from one another by
not more than (b - a) a, where a is the greatest of the numbers a, - al. If
n be increased indefinitely, and the sub-division of (L, U) be such that the
greatest interval a converges to zero, we see that the limits of
am (e,) + E a-m (e,-,)
1=0         i=1
n           ftn
am (e,) + X am(e)
=0         C=1
both exist, being identical in value, and that E is measurable, its measure
being the common limit. It is easily seen, by superimposing two sets of




392


Integration


[cu. V


sub-divisions, that the common limit is independent of the particular sets
of sub-divisions of (U, L). It has thus been proved that:If f (x) be a limited summable function, which is never negative in the
interval (a, b), then the two-dimensional set E of points (x, y) defined by
a _ x  b, 0 _ y  f(x) is measurable, and its measure m (E) is the common
limit of the sums given above.
The value of the Lebesgue integral of f(x) in the interval (a, b) is defined
to be the measure m (E).
It may be shewn that:The Lebesgue integral off(x) lies between the upper and lower integrals of
f(x), and is identical with the Riemann integral in case the latter exists.
For it is clear that m (E) is not greater than the sum Em employed in ~ 252,
rb                          rb
in defining the upper integral  f (x) dx, and hence m (E) - ff(x) dx.
In a similar manner it can be hewn that  f()  
In a similar manner it can be shewn that m, (E) ' J f(Z) dx.
If G be any measurable set of points contained in (a, b), and e,, e,' now
denote those measurable sets of points of G at which f(x) = a,, a, <f(x) < a,1+
respectively, then the common limit of the two sets of numbers M, M', formed
as before, defines the Lebesgue integral of the summablefunction f (x) relatively
to the measurable set G.
288. If f(x) be limited and summable, but be not restricted to be
positive, we can express f(x) as the difference of two summable functions
fi (x), f2 (x) each of which is positive or zero. Thus fi (x) =f (x), when f(x) _ 0,
and f, (x) = 0, when f (x) < O; also f2 (x) = -f(x), when f (x) _ 0, and f2 (x)= O,
whenf(x) > O. The two sets of points a < x < b, 0 < y  f, (x), and a = x _ b,
0   y  f2(x), being measurable, their measures may be denoted by mn(El)
and nm (E2). A similar statement holds when the interval (a, b) is replaced
by a measurable set G contained in that interval.
The Lebesgue integral of a limited summable function f(x) is defined to
be m (E,) - m (E2), where E1, E, are the two sets of points above defined.
The measures of the two-dimensional sets El, E2 are the areas in the
extended sense of the term, which are defined by the parts of the function
f(x) which are respectively above and below the axis of x.
The measures m (E1), m (E2) are identical with Jordan's measures of the
same sets, in case the latter measures exist; and thus Lebesgue's value of the
integral is in agreement with the value according to Riemann's definition, when
the latter is applicable. Sets of points which are not measurable according to
Jordan's system are in general measurable in accordance with the BorelLebesgue definition; accordingly functions which are not integrable according
to Riemann's definition may be so according to Lebesgue's definition. It is




287-289]


Lebesgue's theory of integration


393


not known whether sets exist which are not measurable; but, as we have seen
in Chapter III, all the sets which are defined in the various modes usually
employed, are measurable. Thus Lebesgue's definition of a definite integral
has the advantage over that of Riemann, in that all summable limited
functions are integrable in accordance with it.
The essential distinction between the two definitions of an integral as the
limit of a sum is that, in Riemann's definition, a system of successive subdivisions of the interval (a, b) of the variable is taken as the basis, whereas
in Lebesgue's definition, a system of successive sub-divisions of the interval
(L, U) of the function is fundamental.
The relation of Lebesgue's integral to the fundamental theorem of the
Integral Calculus, and to the problems which arise in connection therewith
will be dealt with in Chap. VI. It will there be shewn that if the limited
function f(x) possess a limited differential coefficient f(x) in the interval
(a, b), then f' (x) is always integrable in accordance with Lebesgue's definition,
and   f'(x) dx =f() -f(a). It has been seen in ~ 264 that this does not
' a
always hold when the definition of Riemann is employed.
ra                                rb
289. If b > a,   f(x) dx may be defined to be -    f(x) dx. It is..b...           va
clear that if f(x) is integrable in (a, b) it is integrable in any part (a, x) of
(a, b), and that
f (x) dx =    f(x) dx +  f(x) dx.
J a       J a          x
If 01, 02,... f, be limited summable functions, and if
F(01...,=X)-( = X),
be a function which is continuous with respect to (01, 02, a...  n), and is limited
in the interval (a, b), it is also a summable function.
If L1, U1 are the lower and upper limits of ~0 in the interval (a, b), we
may divide the interval (L1, Ux) into parts (L1, y,), (y, 2),... (y_-2, y-_,),
(yn-i, U1), each of which is less than a prescribed number e. Let, (x) be
defined as follows:-*- (x)= L1, for all values of x such that L1 =< 0 (x) < yl;
1 (x)= y, for all values of x such that y, _ +j (x) < y2; and generally
~x (x) = y, for all values of x such that y. <.b (x) < y,.+. It is clear that the
function 5 (x) is summable, and it is such that ij 0 (x) -  1 (x) I < e. Let the
functions *2 (x), *3 (),. -,, (x) which correspond to O2 (x), O3 (x),.... ((O )
respectively, be defined in a similar manner. Since F is a continuous function
of 01, 2o,... 2 m, therefore, corresponding to a fixed positive number r, a
number e can be found such that
I F(D02(,,  2, ** (,I) - F(^,... 3 ) | <,)
when *1, *2,... *r, are defined as above for the value of e which corresponds to A. Thus if F(#1, *2,...  b) be denoted by X (x), we have
x (X) - x () < }:




394


Integration


[CH. v


also, if L, U be the lower and upper limits of X (x), in (a, b), we have
L-   < X (x)< U +,
for the whole interval (a, b). Now let, s have successively the values of the
numbers in a decreasing sequence q7, 12,... q,,... which converges to the
limit zero, and let X, (x), X, (x),... X, (x),... be the corresponding functions
X (x). If A, B be any two numbers in the interval (L, U), the sets of points
for which A < X, (x) < B, A < X, (x) < B, &c., are all measurable; and the set of
points for which A < X (x) < B is such that each point belongs to an infinite
number of the sets for which A < X (x)< B, and is therefore, by a theorem
of ~ 82, measurable. It thus appears that the function X (x) is summable,
and therefore has an integral in accordance with Lebesgue's definition.
In particular, the sum, or the product, of a finite number of summable
functions is summable.
If fi (x), 2 (x) be summable in the interval (a, b), then
[f (x)+ f 2()]   =    f(x) dx +   f, (x) dx.
It laving been shewn above that f, (x) +f2 (x) is a summable function, we
observe that if s1 (I), qF (x) be two functions defined as above, eachl of which
has only a finite number of values in the interval (a, b), and be such that
f1(x)-1(x) I< 6, | f2(X)-qf2(X) <,
then                        [f (x) + f2 ()] dx
rb
differs from               [ [  (x) +  2 (x)] dx
by less than 2e (b - a).
rb                    rb          rb
Also        f[I (x) +   2) (x)] dx) d 1, (x) dx + I  2 (x) dx,
^ a                 */ a          c a
each integral being an integral in accordance with Riemann's definition;
rb
hence since                   I   (x) dx
differs from                    p ~ (x) dx
by less than e (b - a),
rb
and                           J  r2 (x) dx
rb
differs from                    q 2 (x) dx
by less than e (b - a),
(b
it follows that             [fJ (x) +f2 (x)] dx
fa




289-291]


Lebesgue's theory of integration


395


differs from              fi (X) do +  f2 (x) dx
by less than 4e (b - a). Since e is arbitrarily small, the equality contained
in the theorem is established.
Thrat
That                          f (x) dx,
for a function f(x) which is summable in (a, b), is a continuous function of x,
is established in the same manner as in the case of a function integrable in
accordance with Riemann's definition.
290. A general theory of integration has been developed by W. H. Young
independently* of the work of Lebesgue, in two memoirs. In the second
of these memoirs, the theory there developed is brought into relation
with the work of Lebesgue. The domain of the independent variable,
for which the function is defined, is taken to be any set of points, and
the following definition of integration of a function with regard to such a
set is formulated:
Let the fundamental set be divided into measurable components in any
conceivable way, and let the measure of each component be multiplied by the
upper (lower) limit of the values of the function at points of that component,
and the sum of all such products be formed; then the outer (inner) measure
of the integral is defined to be the lower (upper) limit of all such summations.
If it be assumed either (1) that all sets are measurable, or (2) that
all functions are summable, then the outer and inner measures of the
integral are equal to one another, and their common value defines the
integral of the finction with respect to the fundamental set of points.
291. Lebesgue has extended his definition so as to afford a definition of
an absolutely convergent improper integral. It is clearly sufficient to take
the case of an unlimited summable function f(x) which is nowhere negative
in the interval (a, b) for which it is defined.
Let a, a,, a2,... a,...  be a sequence of increasing numbers, such that
a0 = 0, and that an has no upper limit as the index n is indefinitely increased:
also let the differences a, - a0, a - a,... an+ - an,... be limited, having v as
their upper limit. Consider the two series
co         sGo
* e=  i aam (er.) +  le ari (e ),
r=O         r=O
r=    a 0m (e,,) + 2 am (em),
where em., e>.' have the same meaning as in ~ 287.
* See his papers " On upper and lower integration," Proc. Lonid. Math. Soc., ser. 2, vol. II;
also " On the general theory of integration," Phil, Trans., vol. ccIv, 1905,




396                          Integration                      [CC. V
Since the difference of the two series is E (a,.+ - a,.)m (er'), which is less
-0,co
than   X mn (e,'), it is clear that the two series are either both convergent, or
r=O
are both divergent. Let us suppose that the function f(x) is such that both
series are convergent; it can then be easily shewn that they are still
convergent when further numbers are interpolated between each consecutive
pair of the numbers ao, a, a2,..., and the corresponding new series are
formed; for by this process a- is increased, and a' is diminished. Therefore
as the process of further sub-division of the interval (0, o ) proceeds, in any
manner consistent with the continual diminution of 7 to the limit zero, the
sums ar, a' both converge to one and the same number. By superimposition
of different systems of sub-division it can also be directly shewn that the limit
to which o- and a' converge is independent of the particular system of subdivision chosen. The common limit of a- and a', when it exists, is then
b
defined to be the value of the improper integral ff(x) dx. In order that an
improper integral of the function f(x) may exist, it is necessary, though not
sufficient, that f(x) be a summable function; and also that the measure of
those points (x) at which f(x) is > an arbitrarily great number N shall be
arbitrarily small for a sufficiently great value of N. For it is a necessary
00
consequence of the convergence of the above series, that X {m (e,.)+ nn (e,.)},
X r=n
which is the measure of that set of points at which f(x) _ a,, should have a
value which converges to zero, as n and a,, are indefinitely increased. It is
however not necessary that the content of the set K~, of all the points of
infinite discontinuity should be zero; in fact it is even possible that the
improper integral may exist whilst every point of (a, b) is a point of infinite
discontinuity.
It will now be shewn * that Lebesgue's definition can be replaced by one
which differs from that of de la Vallee-Poussin only in the one respect, that
the convergent sequence of proper integrals f,(x)dx consists of Lebesgue
integrals, which are not necessarily Riemann integrals.
From the condition of convergence of the second series, it follows that,
corresponding to an arbitrarily chosen positive number e, we may determine
s so that
r=S         3r=s-1
a'=, arn (e,.) +  E  a,.+,m (e,,') + R,
r=O          r=0
where R < e; whilst at the same time   l may be chosen so small that a'


" Hobson, Proc. Lond. Math. Soc. ser. 2, vol. iv, p. 144,




291]             Lebesgue's theory of integration               397
differs from  f(x) dx by less than e. Now let ac.= N, and let J;, (x) be
that function which =f( ), for f(x)< N, and = N, for f(x) NW.
The Lebesgue proper integral   f (x)dx is then the limit, when nv
converges to zero, of the sum
r'=s        '=s —1              X            oO
2 acr,.(e~.) +  2  ac,.+tm (er') +, 2  ml(e,.)+c, C 2 tn(,.');
-r=0         =O r r=s+1l                     =s
and this sum is equal to
r=s         s-l
2 a,,.m (e,.) + 2 a,.~+m (e,') + S,
r =O        r=O
where S < R < e. Keeping a. = N fixed, we may now, if necessary, diminish
v by interpolating further numbers between the    pairs  of numbers
ao, aC, a,...., until we have the new sum which corresponds to
r ='=S      s-l
2 c,.m (e,.) + E m (e,.') + S,
r=O         r=O
differing from  j f(x) dx by less than e; the part S not having been
increased by any diminution of?. We thus find that a-' differs from
fb
f,(x) dx by less than e, when N is sufficiently great, and q} sufficiently
rb
small. Also a' has been taken to differ from  f(x)dx by less than e,
r having been chosen sufficiently small. Since e is arbitrarily small, it
[b                     [b
is clear that  f,(x) dx converges to   f(x) dx, as N  is increased indefinitely.
It has now been shewn that de la Vallee-Poussin's definition of an
improper integral may be extended to the case in which the integrals
j f,(x)dx exist only in the sense defined by Lebesgue. This definition
is then equivalent to that of Lebesgue. It is clear that Harnack's definition
is only capable of extension, in the case in which K, has zero content. If
the condition, that K,~ have zero content, be satisfied, the reasoning in ~ 274
is applicable without essential change; and in that case Harnack's definition
of an improper integral can be extended to the case in which the proper
integrals employed in that definition exist only in the sense defined by
Lebesgue. Thus, in this case, all three definitions are equivalent to one
another.




398


Integration


[CH. V


INTEGRALS WITH INFINITE LIMITS.
292. The definition of the integral of a limited integrable function
given in ~ 251 is applicable only to the case in which both the limits a, b are
definite points, and in which therefore the interval of integration is finite.
Let xI, x2,... x,-,... be a sequence of increasing numbers which has no
upper limit; it may then happen that the sequence of integrals
x,         rX2           rxn
f() d,       () dx,...  f (x) dx,...
has a definite limit A, independent of the particular sequence {xj} chosen.
When this is the case f(x) is said to have an integral
f /() dx,
in the unlimited interval (a, o ), the value of this integral being A. It has
been presupposed that, in every interval (a, x), the function f(x) is limited
and integrable.
If the integrals
b          rb            nb
f (x) dx,  f (x) dx,...  f (x) dx,.
I Wi     J ^2            It
where x, x,,... x,... is a sequence of descending values of x which has no
lower limit, all exist, and have a limit B independent of the particular
sequence chosen, the limit B is denoted by
f(b
f (x) dx.
If the two integrals
f (x) dx,    f(x) dx,
as thus defined, both exist, their sum is denoted by
f (x) dx.
The three numbers
J (x) dx,   f(x) dx,  f f(x) dx,
(a         -J -
being the limits of integrals, and not themselves in the proper sense of the
term integrals, belong to the class of improper integrals.
In each case it is necessary, but not sufficient, for the existence of these
improper integrals, that f(x) be integrable in every finite interval contained
in the intervals (a, oo ), (- oo, b) or (- oo, oc ); and it will be at first assumed
that f(x) is limited in every such finite interval, and thus has therein a
proper integral.




292, 293]         Integrals with infinite limits                 399
In case the integral j  f(x) dx have a definite limit, as c is indefinitely
increased, that limit is said to be the principal value of
f(x) dx.
This principal value may exist, even when the integral
f f(x) dx,
as defined above, does not exist; but in case the latter do exist, its value is
equal to its principal value.
The necessary and sufficient condition for the existence of the integral
f( f(x)dx,
is that (1) the integral exist in every interval (a, x) where x > a, and (2) that,
corresponding to every arbitrarily chosen positive number e, a value ~ of x
can be found such that
f f(x) dx < e,
for every value of ' such that:'>:.
A similar condition applies to the case of
f   f(x) dx.
293. It was shewn in ~ 253 that the necessary and sufficient condition
that f(x) be integrable in the interval (a, b) is that, if 81('2), S,2(),...,(nm)
be a particular set of in'tervals whose sum is (a, b), and of which A,, is the
greatest, NDS shall converge to zero, as m is increased without limit, the
1
system being subject to the condition that A,, have the limit zero; the
fluctuation of f(x) in 8 being denoted by D. If this condition be satisfied
for any one such succession of sub-divisions, then it is satisfied for every
other such system. We have to enquire how far a corresponding condition
applies to the case of an integral through an infinite interval.
Go
Since   f(x) dx, when it exists, is the limit
a
li      (x)dx,
=im j f (x) du,
b= co a
P00
we see that                     f (x) dx
is given as the repeated limit lim lim U8. The question then arises whether,
b=oo A=O




400


Integration


[cH. V


and under what conditions, the order of the repeated limits may be reversed,
so that
f    (x) dx = liin  U US =im  a E
a  A=O    A=0
where i U8 or EL8 denotes the limit of the sum of the US, or the L8, taken
through a finite number of intervals 8, as the number of these intervals
is increased indefinitely owing to continual increase in b.
In the first place, it is clearly necessary for the truth of this proposition
GO   GO                                      00
that the limits Y US, sL8 should exist, and that their difference ED8 should
converge to zero when A does so.
Let us consider a sequence of intervals (a, x), (x,, x2),... (xI-_, x)...,
where x,, has no upper limit as n is increased indefinitely. Assuming that
f(x) has a proper integral in each of these intervals, a system of successive
sub-divisions of (a, x) can be found such that SD8 converges to zero as the
greatest of the 8 does so; and thus a set of intervals 8 exists in (a, x1) such
that
EDS< -e,
where e is a fixed arbitrarily chosen number. Similarly a set of intervals can
be chosen for (xi, x2) such that ZD$ < - e, and so on; thus for (x-_,, Xn) a set
of sub-divisions can be found such that EDS < I e. A set of sub-divisions can
accordingly be found for the unlimited interval (a, oo ), such that ED8 for (a, x,)
is <e (I-   ). The sum ZD8 thus found converges, as n is indefinitely
increased, to a value which is < e. By taking a sequence of values of e,
converging to zero, we now see that a system of successive sub-divisions of
00
(a, o ) can be found such that ZD$ converges to zero, as the greatest of
the 8 converges to zero. Conversely, if a system of sub-divisions of (a, oo )
00
exist such that S D8 converges to zero as the greatest of the 8 does so,
it follows that f(x) is integrable in any finite interval contained in (a, o ).
co
For let the successive sub-division be so far advanced that ED8 < v, where vj is
some fixed number, and consider any interval (a, /3) in (a, oc ). Then XDZ
taken for those 8, finite in number, which either lie wholly inside (a, /3), or
have one end inside (a, 3), is less than; thus we have a set of sub-divisions
of (a, /3) such that SD8 < r. By letting D decrease through a sequence of
values which converges to zero, we see that a system of successive sub-divisions
of (a, S3) exists, such that  D)8 converges to zero, and hence that f(x) is
integrable in (a, 3).
It has thus been established that the necessary and sufficient condition
that a limited function f(x) defined for (a, oo) be integrable in every finite




293, 294]          Integrals with infinite limits                  401
interval contained in (a, oo ) is that a system of successive sub-divisions of (a, o )
should exist such that 2S)8 converges to a value Z, which itself converges to zero
as the greatest of the intervals 8 converges to zero.
The convergence of ED8 to zero, for a particular system of successive
sub-divisions, is not sufficient to ensure the existence of the improper integral
f f() dx,(
fb
but only that                  J    f(x)dx
shall exist for every value of b. In this respect an integral through an infinite
interval differs from one through a finite interval; since, in the latter case, the
convergence of the finite sum IDS to zero, for a particular system of subdivisions, is sufficient to ensure the existence of the integral.
294. In the case of a finite interval of integration, the convergence of
2 US, EL8, where U, L are the upper and lower limits of the function in the
interval 8, to one and the same definite limit, follows as a consequence of the
convergence of ZD3 to zero; but in the case of the integral through (a, o ),
00  00                                       00
the convergence of S US, 2L8 does not necessarily follow from that of D83.
It will however be shewn that, iffor a functionf(x), limited in the interval
(a, o ), a system of successive sub-divisions of (a, oo) exist, such that E U3,
00
EL8 have the same definite value E which converges to a definite number A, as
the greatest interval s converges to zero, then the integral  (x) dx exists, and its
value is A.
From the existence, and convergence to the same value, of ZUS, ZL:, the
convergence of 2D3 to zero follows, and therefore f(x) is integrable in any
finite interval of (a, o ).
a+18
Again 2 U - J     f (x) dx c 2D8, for any finite value of n; and hence, if
1     a            1
all the 8's are so small that 0D8 < v, we have,.a+2
1     J a
Since S U8 converges, as n is increased indefinitely, to a definite value X,
a number m can be found such that, if n > mr,  U8 < rj; hence
n+i
U- U-j    f(x) dx < 21, if n  m.
Ha


H.


26




402                          Integration                       [CH. v
Now we may suppose the successive sub-division to be so far advanced that
0o
i U  -A I< r; we then have
n        \
[a+Y.S
I  1
A-       f (x) dx <.3, for n  mn.
It follows that A -  f(x) dx < 37, for all values of  such that = a + E8,
J~~~~~~a 1.~~~~1
n _- m. If then X be any number greater than the least of these values of I,
two values of ~ exist between which X lies; and if g be the smaller of these, we
have f (x) dx < Uo, where a is the greatest of all the intervals 8, and U is
the upper limit off (x) in (a, o ). It now follows that
A - f(x) dx < 3 +Uo
for all values of X which exceed a fixed number. Since q and a- are both
[x
arbitrarily small, it follows that  f(x) dx converges to the limit A, as X is
indefinitely increased, and thus that the integral f(x) dx exists.
a
295. It will now be proved that, if f(x) dx have a definite finite value,
a particular system of successive sub-divisions of the unlimited interval (a, co )
0o
can always be found, such that Z U3 exists for each set of the system, and converges to the value of the integral as the greatest of the intervals 8 converges
to zero.
For in (a, x,) we can find a set of intervals such that
U- Jf (x) dx <      e;
IX2        1
in (xa, x2) we can find a set such that  U3 -   f (x) dx <   e; and generally
in (x,_-, xn) a set can be found such that E U - j f(x) dx <  e  Thus a
n-i1
set exists in (a, xn) such that  U-  f (x) dx < e 1 - 2. Now x7 can be.taken so great that      f (ax) dx -  f (x) dx <.
Thus an infinite set 2 US exists such that
oo    Xn      1
I U6 - ( U6 < e,
and that              i US- f () dx < r      + e.
I @



294-296]


Integrals with infinite limits


403


00                     r~0
Therefore Z US converges to a definite limit S U3 which differs from  f (x)
J a
by less than v + e, or by not more than e, since r is arbitrarily small. By
taking a sequence of values of e, we obtain a sequence of sub-divisions of
(a, o ) such that E US converges to the value / (x) dx.
It may also be possible to define a system of successive sub-divisions,
such that the greatest of the intervals converges to zero as the successive
Xn
sub-division advances, but such that 2 U  does not converge to a value
X US; and thus the theorem
f  (x) dx = lim  US =  lim EL8
Ca     -A=0       A=0
0o
only holds provided the sub-divisions of (a, o ) be such that 2 US exists for
each successive set of intervals, after some fixed one.
It can also be shewn that when the integral  f(x) dx has a definite finite
value, then, for every system of successive sub-divisions of (a, oo ) which is such
00
that 2D8 exists and converges to zero as the greatest 8 converges to zero, the
co
set of numbers Z U3 exists and converges to the value of the integral.
For since, for every value of n,
n
n      ra+i8        nf
E U  <     f (x) dx + rDa,
a
we see that S UB converges, as n is indefinitely increased, to the value
f(x) dx +    D8;
00
and hence, since YD8 has the limit zero, as the successive sub-division proceeds
oo                rp0
indefinitely, it follows that, U8 has the limit  f(x) dx. For a system of
o00                   n
successive sub-divisions which is not such that  D38 converges to zero, Y U
does not in general converge to a definite finite value as n is increased
indefinitely.
296. The definitions of  f(x) dx,   f (x) dx may be extended to the
J a      J - oo
case in which f(x) has points of infinite discontinuity. If the improper
integral  f (x) dx exists for every value of X which is > a, and if it converges to a definite limit as X increases indefinitely through a sequence of
values, independently of the particular sequence, then that limit defines
00
f (x) dx.
26-2




404


Integration


[CH. V


The integrals  f () dx,   f (x) dx, when they exist, possess many of the
X         -00
properties of a proper or improper integral  f(x))dx. These integrals are
continuous functions of the finite limit x. For
f (x) dx=      f (x)dx +    f (x) dx,
where a < x; and since  f(x) dx is a continuous function of x, so also is
f f(x) dx.
If a function F (x) be such that the sequence F (x), F(xz),... F (x),...
where x,, x2,... Xn,... is a sequence of values of x having no upper limit, has a
definite limit, independent of the particular sequence {cx}, then that limit may be
denoted by F (o ), and the function F (x) is said to be continuous at x = o.
When the integral  f (x) dx exists, the integral ff(x) dx is continuous
for all values of x in the interval (a, co ), including x = oc.
If the integral  f (x) dx exist for every finite value of x in the interval
(a, oo ), and if 6 (x) be a function which is finite and continuous for every such
value of x, and be such that
4(x) - 4 (a)=   f (x) dx,
a
then, provided 4 (x) be continuous for x = oo, the function f(x) is integrable
in (a, oo ), and: (oo) - ) (a)=  f (x) dx. If the function ) (x) have a derivative, say D+4 (x), which is integrable in every interval (a, x) of (a, co ); then
if the integral be a proper one, or be such an improper one that the relation
(x) -   ) (a) =  D+ (x) dx
subsists, then, provided the limit ) (oo) exist, we have also
(oo ) -  (a) =  D+ (x) dx.
A similar statement applies to each of the other derivatives of 4 (x). In the
case in which 4 (x) - 4) (a) differs from J D+f (x) dx by an integrable nullfunction, this holds also for the limit x = co.
297. An integral    f(x)dx is said to be absolutely convergent when
the integral  If(x) dx exists; otherwise it is said to be conditionally or
a:




296-298]


Integrals with infinite limits


405


relatively convergent. If  l f(x) dx exists, then also  f(x) dx exists; for
f(x) dx _j     f (x) dx, and hence the convergence of the latter integral
follows from that of the former one.
If* f(x) and f(x) b (x) be both integrable in every interval (a, x) contained in
(a, o ), and if  f (x) dx be absolutely convergent, and if 4) (x) be, from and
after some fixed value of x, numerically less than some fixed number, then the
integral  f (x) qb (x) dx exists, and is absolutely convergent.
For since  f(x) dx is absolutely convergent, we can find, corresponding
rt+A
to a fixed positive number o, a number f > a, such that j    f(x) Idx < a,
for all positive values of h; we have then
f (x) (x)dx _K        If(x) dx _ Ko-,
where K is the upper limit of I (x), and is by hypothesis finite. It is
thus seen that  f(x) 4 (x) dx is convergent. Also since
J a
'1+h                   t h
If(x) 4  (x) dx  K   ff( I f dx - Ka,
we see that the convergence is absolute.
298. If f(x) and f (x) 4 (x) be both integrable in every interval (a, x), and
if   f (x) dx exist, and if further, from and after some fixed value of x, () (x)
be monotone, and If(x), I (x) I be each less than some fixed number, then
rt
jf (x) 4) (x) dx has a definite finite value.
If o be a fixed positive number, a value f of x can be so chosen that
f(x) dx < a, for every positive value of h; also
re+h                   rt+Oe;e+k
f (x) 4 (x) dx = 4 ()J   f(x)dx + 4( +h)       f(x) dx,
where 0 is in the interval (0, 1). If ] (x) I < K, for every value of x concerned, we have
f +   (x) (x)dx < 2aK;
hence, since a- is arbitrarily small, the integral  f (x) 4) (x) dx is convergent.
Ja


* Riemann's Werke, p. 229; also Pringsheim, Math. Annalen, vol. xxxvII, p. 591.




406


Integration


[CH. V


299. An important set of tests of the absolute convergence of an
integral   f (x) dx is the following:If f (x) be integrable in every interval (a, x), then  f (x) dx converges to
a definite finite value, provided f(x) converge to zero, as x is increased indefinitely, in such a manner that one of the expressions f(x). xl-t+, f(x) x (log x)l+k,
f(x) x log x (log log x)+k,...... f (x) x log x. log log x...... (log log... log x)1+k,
converges to zero as x is indefinitely increased, k denoting some fixed number
greater than zero.
The integral  f (x) dx is not convergent, in case f(x) be of invariable
sign, from and after some fixed value of x, and provided also any one of the
above expressions do not converge to zero as x is increased indefinitely, when
k has the value zero.
rx+h
We see that in the first case,   f(x) dx is numerically less than one
Jx
of the expressions
[Y+h dx       fX+h    dx          f Y+h        dx
JX    xl+k +  J)x      (log x)k'  Jx   x log x (log log x)+k'"
where C is a constant dependent on X, which converges to zero as X is
indefinitely increased. These expressions have the values
c          1    1   a     1          1     1
fk [    (X + h)k' i    (log X)k  (log X + h)  '
C       1             1       -1.
k (log log X)k  (log log X + h)k '
hence, k being positive, it is clear that X may be so chosen that
f       +f(x) I dx
is less than an arbitrarily fixed number, and thus
f: (x) dx
is convergent. In the second case, k being now zero, we see that
rx+h
j    f (x) dx
is numerically greater than one of the expressions
fX+ dx   C   +h   dx      f+h       dx
x    X      x      slog fX      Xlog Xolog'
or than one of
C lo X +_ h     log (X + h)    log log log (X + h)
0  X     l     log X                log X




299-301]          Integrals with infinite limits               407
and these expressions increase indefinitely as h is increased. It follows that
/f (x) d is in this case divergent.
300. An important case of convergent integrals, which do not necessarily
converge absolutely, is that of the integrals
(x) sin x dx,     (x) cos x dx,
when b (x) is monotone, from and after some fixed value of x, and converges
to zero as x is indefinitely increased.
We have   f   (x) sin x dx =  (x)f sin x dx + h (x2)f sin x dx,
where x, is so great that b (x) is monotone for x _ xj, and x2 > x,. From
this we have
j  (x) sin  x   2 K2  1 (h1) 1 + 2 1 (X2)  4  1 (x1) i;
and hence, if x, be taken sufficiently great,  (X) sin  dx is, for all values
of x2, less than an arbitrarily fixed number e. The convergence of the integral
has thus been established. The case of the second integral may be treated
in the same manner.
INTEGRATION BY PARTS.
301. Let f(x), ( (x) be two functions which are continuous in the interval (a, b); also let Df(x) be one of the four derivatives of f(x), and Do (x)
one of the four derivatives of b (x). If the two derivatives Df (x), DO (x) be
both limited integrable functions in the interval (a, b), the relation
f (x) DI (x)dx= f/(^)         (     x) -  () (x) dx
is satisfied, where
L(f)   (Ix)]  denotes f (b) (b) - f(a) b (a).
If U be a function which differs from Df(x) only by an integrable nullfunction, and V differ from Do (x) only by an integrable null-function, the
above formula may be written in the form
fV{   Uddx) dx= [(fr Udx) (  Vdx)] -    U (  Vdxl) dx,
where a, 3 are arbitrarily fixed points in the interval (a, b). This general




408


Integration


[CH. V


theorem in integration was first * obtained by P. Du Bois Reymond, and is a
generalization of Leibnitz's formula for integration by parts,
bdv                b du
u -dx= uv     -       dx.
J   dxa               dx
To prove the theorem, let (a, b) be divided into n sub-intervals 8,, 8,... -n;
then, if xr-, x,. denote the end-points of the sub-interval 8,., where x, =a,
xn = b, we have
(f) (  (x)  =  e  (xr) tf(x,.) - f(x-)} + f (x) {f (r, ) - b (X,_-)}] 
a  r=l
Now   (Xr) -f(x. 1) lies between the upper and lower limits of Df(x) in the
sub-interval 8r, and similarly ' (  )  (x"-r) lies between the upper and
8a2
lower limits of Db (x) in the same interval; therefore it follows that the sum
on the right-hand side of the above identity may be written in the form
8r, [f (X,) X% +f/(Xr-) %r],
r=l
where X, is between the upper and lower limits of Df(x), and.r between
those of Df (x) in the interval,..
Let Xr = Df(Xr) + Er, f,. = D() (xrl) + br, where  er.I cannot exceed the
fluctuation of Df(x) in the interval 8,., and \r\ cannot exceed that of
Df (x) in the same interval.
We have now
f(x) 4 (x)     = ~ 8, [4 (x,.) D)f(xr) + f (x.-) Do) (x,_-)]
_  _a r=1? =n
+ E 8s [ (x,.) Er +f(Xr-) ~r];
r= 1 —
and the absolute value of the second sum on the right-hand side cannot
exceed
r=n f         r=n ( 
\LrEr, }+F 2s 8|.l,
~~r=l~ [r=l 
where <, F are the upper limits of I \(x), If(x) in the interval (a, b).
Since Df (x), D( (x) are by hypothesis integrable in (a, b), the set of subintervals may be so chosen that
r=n          r=A
E       Ir  1 er   8r  I  Ir  I
r=l          r=l       J
are arbitrarily small. Since ()(x) Df (x), f(x) D(f (x) are by hypothesis integrable, we see that, if the number n be made to increase indefinitely, and
* Abhandlungen d. Miinch. Akad., vol. xii, p. 129.




301, 302]


Integration by parts


409


the intervals 8 be so chosen that the greatest of them converges to the limit
zero, the above identity assures us that
Lf()   (X)     f () Df (x) dx +  f () D  (f) d;
and thus the theorem is established.
302. The theorem may be extended to the case in which Df(x), Df (x)
are not restricted to be both limited functions, but may have points of
infinite discontinuity forming an enumerable non-dense closed set. It has
been shewn above that, in any interval (c, x) contained in (a, b), in which
b (x) Df (x) +f(x) Do (x) has a proper integral,
[f(x) c (x)  = {b () Df () +f(x) Do (x) dx.
It follows from the theorem of ~ 285, that
\f(x <f (x)] =, {k (x) Df (x) + f (x) D  (x)} dx,
provided that the improper integral on the right-hand side exists. If now
the two functions b (x) Df (x), f (x) Dc (x) possess absolutely convergent
improper integrals in (a, b), it follows that their sum has the same property,
and that
() (x)] = f     (x) Df(x) dx +  f (x) D  (x) dx.
a. a. a
The theorem has now been proved under the suppositions that f(x), ( (x) are
continuous in (a, b), that Df (x), Df (x) have at most points of infinite discontinuity forming a closed enumerable set, and that the functions f(x) D(P (x),
4 (x) Df(x) possess absolutely convergent improper integrals in (a, b).
Iff (M) ) (x) have a finite limit for x = oo, and be continuous at x = oo, in
the sense defined in ~ 296; and if further the conditions be satisfied, that the
functions f(x) Dc (x), p (x) Df(x) are integrable in the infinite interval
(a, oo ), then the formula
L                () (x  Df (x) dx +  f (x) Do (x) dx
holds. If the formula for integration by parts holds for (a, b), whatever
finite value b may have, but if f(x) 0 (x) be not finite and continuous for
x = o, then one at least of the integrals
(  (x) Df (x) dx,  f (x. D( (x) dx
is infinite, or does not converge to a definite value.




410


Integration


[OH. V


CHANGE OF THE VARIABLE IN AN INTEGRAL.
303. Let f(x) be a limited function, integrable in the interval (a, b).
We now assume that x is a continuous function * (y) of another variable y,
defined for the interval (a, P), where a = J (a), b =  (/3), and that p (y) is
monotone in the interval (a, /). If we further assume that k(y) has no
lines of invariability in the interval (a, /), then y can be regarded as a singlevalued function 4 (x) of x, defined for the interval (a, b) of x; and also f (x)
can be regarded as a function f {, (y)} of y, defined for the interval (a, /).
If Dr (y) be one of the derivatives of + (y), then, provided D# (y) be
limited and integrable in (a, /3), the function f {t (y)} Dqf (y) is integrable in
(a, /3), and
jf(x) dx =   f f  (y) D (y)   dy.
This result is a generalization of the well-known formula of substitution
f (x) dx =j   f{ (y)} ' (y) dy,
the particular case of the theorem which arises when + (y) is differentiable
throughout the interval.
Let the interval (a, /) be divided into n parts s,',,',... Sn'; then the
interval (a, b) is divided into corresponding parts 81, 2,... Sn,
Since + (y) is a continuous function of y, tile intervals 3t'} can be so
chosen that in each of them the fluctuation of + (y) is less than a fixed
arbitrarily chosen positive number e, and hence so that each of the intervals
8 is less than e. If then a sequence of diminishing values of e be taken,
which converges to the limit zero, we obtain a convergent system of sets {8'},
and corresponding to them a convergent system of sets {t} of sub-intervals of
(a, b). We may assume that f (x) is positive; for if it is not so throughout
the interval (a, b), it may be made so by the addition of a properly chosen
constant c: then, if [c +f{N t(y)}] Dr(y) be integrable in (a, B), it follows,
since D  (y) is integrable in that interval, that f {t (y)} Df (y) has the
same property. Let U,., L,. denote the upper and lower limits of f(x) in a,,
or of f{  (y)} in r/; and let u., 1,. denote the upper and lower limits of
D# (y) in 8/'. The fluctuation D.' off {# (y)} D*4 (y) in 6r' is - U,.u, - L1Z,.,r
or _ U, (u,r- r) + 1. ( U. - Lr). Hence, since 1,r,.' =<,., it being assumed
that D# (y) is always positive, or zero, so that,. is not negative, we have
Dr/' 8./  U (,. - 1.) 3,/ + 8r ( Ur - L),
where U is the upper limit off(x) in (a, b).
We have now
XD,.'fs, u_ 2 (u, - lr) 8,/ + E (Ur - Lr) 8.;
1          1            1




303-305]   Change of the variable in an integral


411


and it follows, from the conditions of integrability of D f(y) in (a, /3), and of
f(x) in (a, b), that f{r (y)} D# (y) is integrable in (a, 8).
Next, we have
n   n                n
S,rf () - E s,'f!{+ (yr)J DA (yr) =  8.'f { f (yr)} {xr - Df (y,)},
1         1                   1
where Xr lies between the upper and lower limits of D* (y) in the interval
8,'. The absolute value of the expression on the left-hand side is conn
sequently less than USr'(U~ - I), which is arbitrarily small, on account of
1
the condition of integrability of D\r(y) in (a, /3). It has thus been established, that
f (x) dx =   f { (y)} Df (y) dy.
304. The theorem can be extended to the case in which D# (y) has
points of infinite discontinuity forming an enumerable closed set, provided that
D# (y) have an absolutely convergent improper integral in the interval (a, /).
For in this case, the equality
f (x) dx =  f{# (y)} D+ (y) dy
holds for any interval (c', y) which contains none of the points of infinite
discontinuity of DA (y). The integral  f (x) dx is a continuous function of
the upper limit x, and the integral  ff{ (y)} Dj (y) dy is a continuous
function F(y) - F(c') of the upper limit y. Since f{f(y)} is limited in
the interval (a, /3), f {f (y)} Df (y) has an improper integral in that
interval; and the relation F (y) - F (c') = ff{  (y)} D ((y)   dy holds in
intervals (c', y), which contain points of infinite discontinuity of D't(y). It
thus appears that the substitution formula holds for the interval (a, b) together
with the corresponding interval (a, /3).  Precisely similar considerations
suffice to shew that the theorem still holds when f(x) has a set of points of
infinite discontinuity forming an enumerable closed set, provided f(x) have an
absolutely convergent improper integral in (a, b).
305. The theorem of substitution is still valid when f (y) is no longer
monotone in the interval (a, /), but has a finite number of maxima and
minima, or, under certain restrictions, an infinite number of maxima and
minima. The function f (y) may also have lines of invariability. In this
case x has a single value for each value of y in (a, 3) defined by the continuous function f (y), but the inverse function p (x) is not single-valued.




412


Integration


[CH. V


It may happen that the values of x given by * (y) do not all lie within the
interval (a, b). In this case it is convenient to conceive the function f(x) to
be defined for all such values of x, so that it is integrable in every interval
of x, such interval extending so far as necessary beyond (a, b). The result of
the substitution will consist of two or more integrals with respect to y.
Let us assume that the points y at which   r (y) has a maximum   or
minimum, or is the end-point of a line of invariability, form a set of points
in (a, /3) with the content zero; they can then all be enclosed in a finite set
of intervals of which the sum is less than an arbitrarily chosen number
e. If it be assumed that D# (y) has finite upper and lower limits in (a, 3),
then the sum of those intervals in (a, b), produced if necessary, which correspond
to the finite set of intervals constructed in (a, /), is less than e multiplied by
the upper limit of ] D# (y) ] in (a, /); and this may be made as small as we
please by choosing e small enough. To each of the remaining intervals of
(a, 8/), when the finite set of intervals is removed, the theorem of ~ 303 is
applicable, it being remembered that along a line of invariability Dr (y) = 0.
In such an interval (a,., /,.) we have
jf i# (y)} D  (y) dy =    f (x) dx,
where (a,., b,.) corresponds to (ar, /g,), since tr(y) is monotone in the interval.
The sum of the integrals on the left-hand side, taken for all the finite
number of values of r, differs from
f     {* (y)}) 0   (y) dy
by less than e multiplied by the upper limit of f { pf (y)} Dr (y) I in the
interval (a,,3); and the sum of the integrals on the right-hand side, taken for
Ob
all the values of r, differs from  ff (x) dx by less than e multiplied by
the upper limit of I Df (y)  in (a, /), and by the upper limit of f (x) for
all the values of x for which f(x) has been defined. Since e is arbitrarily
small, the theorem  has been proved for the case in which    r (y) has
maxima and minima at a set of points of zero content, and may also have
lines of invariability of which the end-points form a set with zero content.
The theorem here established can be extended, as in ~ 304, to the case in
which D F(y) has points of infinite discontinuity forming an enumerable
closed set, provided Dr (y) have an improper integral in (a, /3). An extension
can be made to the case in which f(x) has a set of points of infinite
discontinuity, with zero content, on the assumption that f(x) possesses an
improper integral in (a, b).




305-307]   Change of the variable in an integral


413


306. The theorem of ~ 303 may be generalised so as to apply to the case
when the limited function f(x) is not necessarily integrable in the interval
(a, b). It being assumed as before that r (y) is monotone in (a, /3), and has
no lines of invariability, and that D# (y) is limited and integrable in (a, /3), it
will be shewn that the upper integral of f(x) in (a, b) is equal to the upper
integral off {r (y)} D4 (y) in (a, /3).
Denoting by u' the upper limit of f {* (y)} Dr (y) in the interval 8r', we see
that u,' lies between Urur and Url. Also 3. lies between ur3' and l,.8', thus
we may write Sr =,' {Ur - Or (u, - 4.)}, where 0 A 0r, 1. We have to prove
that S Ur and 2u'3/.' converge to the same limit. Writing CUr -U, -.'
in the form
r' [Ur {u,-Or    -.  )(  - lr)}-uj,
or,.' ( UU,., - Ur) -,' Or. U (u, - 14),
we have     28' (Ur(Ur - ur/) -  8/rr U, (- -  =,) _ U38r, (Ur -.),
and also           ( -         'U (U,. - 1r) (   U:S8r,' (,. - r-).
Since Df (y) is integrable in (a, 3), it follows that 83,' (ur - l,) is arbitrarily small, when the number of intervals 3' is increased; it thus appears
that S Ur3r. - zUr'8.' is arbitrarily small, and thus E U,r,., u,./38' converge to
the same limit. It has now been established that:If Dt (y) be one of the derivatives of * (y), and if Df (y) be limited and
integrable in (a, /3), the continuous function r (y) being monotone and without
lines of invariability in (a, /), then
f     (x) dx = f J{* (y)} D* (y) dy.
The corresponding theorem for the lower integrals may be established in
a similar manner.
It follows from these theorems that, under the conditions stated, if either
of the integrals  f (x) dx,  f {Sf (y)} Dfr (y) dy have a definite value, then
the other one has the same definite value. The method of substitution may
fb
accordingly be applied to  f(x)dx without assuming that this integral
has a definite value, and may thus be employed to decide the question of
the integrability off(x).
The extensions in ~ 304 and ~ 305 may be applied to the case of the
upper or the lower integrals off (x).
307. It may happen that, when x and y are connected by the relation
x=r (y), an infinite interval for y corresponds to the interval (a, b) of x.
For example, y = oo may correspond to x = b: in that case we assume that




414


Integration


[CH. V


Ar (o ) is continuous, i.e. that it is the limit of 4 (y) when y is indefinitely increased. If the conditions of the theorems in ~ 303-305 be satisfied for every
interval (a, b - e) of x, with the corresponding interval (a, 1') of y, we have
f /(x) dx =       f  f  (y)} D) (y) dy;
and since this holds for every e, we have, on proceeding to the limit e= 0,
jff(x) dL-x = f { (y)} D# (y) dy,
in accordance with the definition of the improper integral on the right-hand
side. Similar considerations apply to the case in which y =- oo corresponds
to x = a.
308. If one or both of the limits a, b be indefinitely great, it may happen
that finite values of y correspond to. x = a, x= b, or else that one, or both, of
the limits of y may be infinite.
The method of procedure adopted above suffices to establish the theorem
that, if f(x) be integrable in the finite or infinite interval (a, b), then the
fb
integral f (x) dx can be transformed by means of x = r (y) into the integral
ff NS (y)} D#r (y) dy, in which the interval of integration is finite or infinite,
a
provided that r (y) be finite and continuous in (a, 13), or have at most in every
part of (a, 3) points of infinite discontinuity forming a set of zero content; with
the further conditions that DS (y) be integrable in (a, /), and that in every part
of this interval its maxima and minima and end-points of lines of invariability
form a set of points of zero content.
It will be observed that a large class of those improper integrals which
have an infinite interval of integration may be transformed into proper
integrals.  It has been suggested by Kronecker that every such integral
through an infinite interval can be transformed into a proper integral.
fb
309. If it be desired to transform the integral  f (x) dx, by means of
the relation y= (x), where + (x) is a single-valued function of x, then,
unless 0 (x) be monotone in the interval (a, b), the inverse function f (y) will
not be everywhere single-valued.
If it be assumed that ( (x) is monotone in (a, b), and that a= b (a),
/3 =  (b), a derivative Do (x) of p (x) is reciprocal to a derivative Dr (y)
of +(y). If it be assumed that D   (  is integrable in (a, /3), and that
Dqf (x)
the same holds for   () considered as a function of y, or else that f (x)
is integrae di (x)
is integrable in (a, b), then we may use the transformation




307-310]  Change of the variable in an integral


415


ff(x)dx=f   /^         dy.
| / (a) v doaDob (x))y= (x)
If 4 (x) be not monotone, f (x) dx can not in general be transformed into
- a
a single integral in y. If, for example, + (x) increases from x= a to  = k,
and then diminishes from x = k to x = b, we must take
f (x) dx =     {    x)       dy + (k) {Dx()}    dy,
Do (x),j y=, x   ( (k) Do (x*)}y,:.
and the integrals on the right-hand side cannot in general be amalgamated
into one integral through the interval (a, /9), because in the two integrals
the integrand has different values for the same value of y.
Thus, for example, if y = sin x,
f    ==f [  x] dy + f I-(x)           ddy
Cos X        LOS x
of (sin-'f y)    2(o   -   y2  )
1 -"'~{) fdy +~  V1        d- 
the value of cos x in the second integral on the right-hand side of the
first equation being negative, and the values of sin-ly, cos-ly being in
the interval (0, w7r).
DOUBLE INTEGRATION.
310. Let G denote a set of points in two-dimensional space, entirely
contained in a rectangle with sides parallel to the x and y axes; the
set G is accordingly a bounded set. It has been pointed out in ~ 286 that, if
the fundamental rectangle be divided into any number of parts by means of
straight lines parallel to the sides, and the sum of those rectangles be taken
each point of which belongs to G, then the sum of the rectangles has a
definite limit S1, the interior extent of G, when their number is increased
indefinitely in any manner, subject to the condition that the diagonal of the
greatest of the rectangles have the limit zero. Also the sum of those rectangles each of which contains at least one point, either of G or of the
frontier of G, has, under a similar condition, a definite limit S2, the exterior
extent of G. When S = S2, the set of points G is measurable in accordance
with Jordan's definition, and consequently also in accordance with the
definition of Borel and Lebesgue; and the set G then has a single definite
extent or area, the measure of the set.
Let a function f(x, y) be defined for the bounded set G, which we shall
assume to have a definite extent in the sense just explained. We further
assume that If(x, y)  has a definite upper limit for the set G, so that




416


Integration


[CH. v


f(x, y) is a limited function.  It will be convenient to assume that f(x, y)
is extended to the whole rectangle in which G is contained, by providing
that f(x, y) vanishes at every point of the rectangle which belongs to the
complementary set C (G).
The definition of the double integral of f (, y), with respect to the set
G, is now similar to Riemann's definition, in ~ 251, of a single -integral of'
a function with respect to a linear interval.
Let the fundamental rectangle be divided into ni rectangular portions
n
Al), 2(),... Sn,), so that E 3r() = A, the area of the fundamental rectangle;
r =1
and let Ai denote the greatest of the diagonals of the rectangular portions.
Let these rectangles be further sub-divided in any manner, so that the whole
area A is divided into n2 parts, the greatest diagonal being A2; the rectangular portions of A now being   l(2), 2(2),... n2(2). Let this process of subdivision of A be carried on indefinitely, so that at any stage of the process
A is divided into nm rectangular portions A1(m), 8S(m),... nm (m, the greatest
of the diagonals of these portions being Am. If this system of sub-division
of A be made in any manner whatever, which is such that the sequence
A1, A2,... Am,... has the limit zero, it will be spoken of as a convergent
system of sub-divisions of the area A.
Let M(8(m)) denote any number whatever which is so chosen as to be
not greater than the upper limit, and not less than the lower limit, of f (x, y)
in 3s ); and consider the sums
S, =, () M(Sl(1)) +  2  ( )M (8  +  + n,() M(Sn(),
S2 = 81(2)  ( (2)) +  (2) M(2)) +...+ S  M (22)),
Sm= 81 ( M(i(n ) ( + 8() 2)M (32()) +...+ n (m (m) M( (8, (m)).
If the sequence S,, S2,... Sm,... be convergent, and have the same number
S for its limit, whatever convergent system of sub-divisions of A be employed,
and however the numbers M(i8(m)) be chosen, subject only to their limitation in
relation to the upper and lower limits of f(x, y) in the rectangle s, m), then the
function f (x, y) is said to be integrable in the set G, and the number S defines
the value of the double integral. This double integral*, when it exists, may be
denoted by J  f (x, y) (dxdy).
It will be observed that the double integral has been defined as the
single limit of a sum, and accordingly the sign of integration is here employed
* The extension of Riemann's definition to double integrals was given by H. J. S. Smith,
Proc. Lond. Math. Soc., vol. vi, p. 152, 1875, and by Thomae, Einleitung in die Theorie der
bestimmten Integrale, p. 33, 1875; also Schlomilch's Zeitschrift, vol. xxi, p. 224.




3102 3111


Double integration


417


only once in the notation adopted for a double integral. The term " double,"
in the traditional name double integral, must be taken to have reference to
the two-dimensional set of points for which the function is defined.
311. As in the case of the single integral, the investigation of the condition for the existence of the double integral is facilitated by the definition
of the upper and the lower integrals* of the function.   If the number
M (s("m)) be identified with U (8(m)), the upper limit off(x, y) in the rectangle
5s(m), and we denote by Em, the corresponding sum
l (m U(1(m)) ( +  (82+ U (n3()) +.+, (m) U (8nM1)
it will be proved that E,m converges, as m is indefinitely increased, to a
number E, which is independent of the particular convergent system of subdivisions chosen. This limit E is termed the upper integral of f(x, y), and
may be denoted by f    f(x, y)(dxdy). A similar definition applies to the
G)
lower integral f  f(x, y) (dxdy), which is obtained by identifying M(8(m))
- G)
with L (8,(n)), the lower limit of f (x, y) in s(,).
The upper and lower double integrals always existing, the necessary and
sufficient condition for the existence of the double integral consists in the
equality of the upper and lower integrals.
To establish the existence of the limit X, we observe in the first place
that, as the process of sub-division proceeds, Em may diminish, but cannot
increase; moreover Em > AL, where L is the lower limit of f (x, y) in A; it
thus follows that the sequence E, 2,... E,..., for a definite convergent
system of sub-divisions, has a definite limit E. It remains to prove that this
limit X is independent of the particular system of sub-divisions employed.
Let us suppose, if possible, that with another convergent system of subdivisions the limit were 5', different from Z; we may suppose, without loss
of generality, that $' < S. Let e e, 2,... e be a set of this second system of
sub-divisions, where s is so great that the sum for this set is <  ' + ', where
~ is so chosen that  ' + g < E.  We assume that m is taken so great that
A,n is less than the least of the diagonals of the rectangles e, so that no e is
entirely in the interior of a B. Then let the two sets of rectangles,(m),
82(m,...;n (m-) and el, 6,... e be superimposed, the rectangle A being thus
divided up into a set of rectangles which may be regarded as a continuation of either set of sub-divisions. The sum of the rectangles of the new set,
* See Jordan's Cours d'Analyse, vol. I, p. 34. An elaborate treatment of double integration
has been given by Stolz, Grundziige, vol. II, where the triangle or polygon is employed in relation
to the measures of sets of points, instead of the rectangle. On this point, see Schonflies, Ber.,
p. 179.


H.


27




418


Integration


[CH. V


each multiplied by the corresponding upper limit of the function, is obtained
by diminishing Em by less than U - L multiplied by the sum of those of
the b's which are not interior to rectangles e. Those of the rectangles 8 which
are interior to a rectangle e are also elements of the superimposed system.
If we consider any one of the rectangles e, the sum of the areas of those
rectangles 8 which encroach on this rectangle e, but are not contained in
its interior, is less than the perimeter of this rectangle e, multiplied by A,
the greatest diagonal of all the rectangles 8. Therefore the sum of the areas
of all those rectangles 8 which are not entirely interior to some rectangle e
is less than AmP, where P is the sum of the perimeters of all the rectangles e.
Hence the sum Em is diminished in the new system, obtained by superimposition, by less than (U-L)APP. Since Am diminishes indefinitely as m
is indefinitely increased, we may choose m so great that (U-L) AmP is less
than an arbitrarily chosen positive number J. Hence the sum for the superimposed system  is >Eam-, or >H —.      But this new sum is certainly
< I'+ 4. Now y   can be chosen so small that the conditions that the new
sum is < I'+ 4 and >   -   are incompatible with one another. It thus
appears that E and Z' cannot be unequal.
The existence of the lower integral follows from the fact that it is the
upper integral of -f(x, y), if its sign be changed.
The condition for the existence of f  f (x, y) (dxdy) is then that thefundavmental rectangle can be divided into a number of rectangular parts such that
the sum of the products of the area of each part into the fluctuation of f(x, y)
in that part is less than an arbitrarily chosen positive number.
By reasoning precisely similar to that in ~ 254, the intervals and neighbourhoods being replaced by rectangles, it can be shewn that the condition
for the existence of the double integral is reducible to the following form:The necessary and sufficient condition for the existence of the double
integral of a limited function defined for a domain G contained in a rectangular area is that those points at which the saltus of the function is - k,
form, for each value of the positive number k, a set of points of zero content.
It will be observed that those points of A which, without being points of
G, are points of the frontier of G, will in general be points of discontinuity
of the finction f(x, y) extended to the whole domain A by attributing zero
values to the function for all points of A which do not belong to G. It has,
however, been assumed, in assuming that the frontier G has zero measure,
that those points of A which are limiting points of G, without belonging
to G, form a set with zero measure. Therefore the content of the points
at which the saltus of the function is - k, is unaffected by the points of the
boundary of G which do not belong to it; and, in the above statement of




311, 312]


Double integration


419


the condition of integrability, the set of points at which the saltus is _ k
may be taken to be points of G only.
As in the case of single integrals, the necessary and sufficient condition
for the existence of the integral may be expressed in the form, that the set of
points of discontinuity of the function in the domain for which it is defined
must form a set of which the two-dimensional measure is zero.
It is clear that the definition, and the condition for the existence, of
an integral of a function of any number of variables, called a multiple integral,
are of the same character as in the case of a double integral. For simplicity,
the investigations will be here restricted to the case of double integrals;
and it will then be easy to extend the results to triple or to n-fold integrals.
312. In defining the double integral, successive sub-division of the
fundamental rectangle into rectangular portions has been employed; it will
however be seen that this mode of sub-division is not essential, but is a
special case of a more general mode. Let us consider a closed connex set e
of points in the fundamental rectangle. The distance between a pair of
points of e has an upper limit, when every possible pair is considered; this
upper limit we may speak of as the diameter of the closed connex set e.
Let the fundamental rectangle be divided into a definite number m of closed
connex sets e, e2,... er, the frontier of each of which has zero content; let
Ue,, Ue,,... Uer denote the upper limits of the limited function f(x, y) in the
various sets e; and let Le1, Le,,... Le,. denote the corresponding lower limits;
also let d denote the greatest of the diameters of the sets e.
Let us now consider an indefinite succession of such sub-divisions of the
fundamental rectangle, subject to the condition that, as the number r of the
parts of the rectangle is increased indefinitely as the successive sub-division
proceeds, the number d converges to zero.
r         r
The two sums E Uem(e), Lm (e), where m(e) denotes the measure of
1         1
the set e, converge to two definite numbers. For E Um (e) cannot increase
as the successive sub-division proceeds, and it is not less than L multiplied
by the area of the fundamental rectangle; and therefore it converges to a
definite limit. The proof is precisely similar for the second sum. It will be
shewn that these limits are independent of the mode of successive subdivision of the fundamental rectangle, provided the conditions stated above
are satisfied. It follows that, in the definition of the double integral, any
mode of successive sub-division, of the type specified, may be employed;
the sub-division into rectangular parts being merely a special case of the
general mode. The division of the rectangle into curvilinear portions by
means of two families of ordinary curves is a special case of the mode of
sub-division specified above.
27-2




420


Integration


[CH. V


In the first place, since 2 Uem(e) is greater than a fixed number, it follows
that there exists a lower limit E of all the numbers, s Uem(e), for every
1
value of r, and for every possible system of sub-divisions of the type
considered.
It will next be shewn that any system of rectangular sub-divisions, as in
~ 311, is such that X is the lower limit of the sum 2m for such sub-divisions,
i.e. that $ = 2.
Consider the rectangles 8,(m), 82('),... 6,(m), for which,= -S (m) U(8(m)).
We observe that a system of sub-divisions el, e2,... e8 may be so chosen that
2 Uem (e) is less than E + I, where v is an arbitrarily chosen positive number.
We may suppose m to be fixed so large that Am is less than the least of the
diameters of el, e2,... es. Some of the rectangles 8 will be interior to one or
other of the e's, and others will contain points of the frontier of two or more
of the e's. We may divide   8(m) U(8(8)) into two parts, corresponding to
this distinction relatively to the rectangles 8. Denoting these two parts by,mi, 2m2 respectively, we have
En =  Ml + m 2:X Uem (e) + (U - L) 2,
where 2 denotes the sum    of all those of the 8's which are not interior
to one of the e's.
Now, since the frontiers of all the e's have zero content, it follows
that, when m   is sufficiently large, (U-L) E is less than an arbitrarily
chosen number '; and therefore Em <   +E  + +. Also Em _; hence, since
vr and ' are arbitrarily small, it follows that 2, the limit of m,, is equal to E.
Lastly, it will be shewn that for any system of successive sub-divisions e,
r
of the type defined above, the limit of X Uem(e), when the sub-division is
continued indefinitely, is the same as for a rectangular system  of subdivisions. The general theorem will then have been established, that, whatever system of sub-divisions be taken, of the type defined above, the limit of
r
2 Uem (e) is one and the same number, which is the upper integral of f(x, y)
in the fundamental rectangle.
Taking a fixed set of rectangular sub-divisions, for which the sum
Z (m) U(3(m)) is less than X + X, we may suppose the successive sub-divisions
into parts e to be so far advanced that d is less than the least of the sides of
the rectangular sub-divisions. Some of the parts e will then be interior to
a rectangle 8, but none will contain such a rectangle in its interior.




312, 313]


Repeated integrals


421


We have then
Uem (e) E 5m + Pd (U - L),
1
where P denotes the sum of the perimeters of all the rectangles. Since d is
arbitrarily small, and Em < E + V, we see that
Ue"?, (e) <, +,
r
where ' is arbitrarily small. Hence E Um(e) lies between S and  + ';
r
and therefore the lower limit of E Uem (e) is E. The corresponding theorems
1
for the lower integral may be deduced by changing the sign off(x, y).
If we denote the measure of a portion e of the fundamental rectangle by
se, we may denote the double integral of f(x, y) over this rectangle by
ff(x,y)de, where dxdy is written, as in ~ 310, for de, in case a rectangular system
of sub-divisions of the fundamental rectangle be supposed to be employed.
The definition of the integral of a summable finction, due to Lebesgue
(~ 287), can be immediately extended to the case of summable functions with
two or more variables. In this case the formal theory is exactly similar to
that developed in ~~ 287, 288, for the case of summable functions of a single
variable. The sets e,, eL' there employed, must be interpreted to be sets of
points in two or more dimensions, the measures m (e,), mn (e,') denoting twodimensional, or multi-dimensional, measures.
REPEATED INTEGRALS.
313. The actual evaluation of a double integral over the fundamental
rectangle, of which the sides are x = x0, x = x1, y = Y0, y = y1, is usually made
to depend upon the evaluation of successive single integrals taken first
with respect to one of the variables, and then with respect to the other.
The expression
jdx   f (x y) dy,
in which f(x, y) is supposed to be integrated first with respect to y, for
a constant value of x, and then with respect to x, is called a repeated
integral.  Similarly, the expression
ery  rC i
y dy   f (x, y) dx,
i wso                              caeo
in which the integrations are performed in the reverse order, is also called a




422


Integration


[COH. V


repeated integral. The question of the existence of these repeated integrals,
and in any given case, their relation with one another, and with the double
integral, will be here investigated.  It will be observed that the double
integral has been defined as a single limit; whereas the repeated integrals
are each, when they exist, obtained as the results of repeated limits. We
have then to investigate whether, or under what conditions, a double integral
is capable of representation as a repeated limit of one of the forms indicated.
It cannot be assumed a priori that the existence of the double integral
necessarily implies the existence, for each value of x, of the single integral
J f(x, y)dy
J go
as a definite number.  Neither is the existence of this single integral,
as a definite number, necessary for the existence, as a definite number,
of the repeated integral
jdxj f(x, y) dy.
In fact, if we assume that the upper and lower integrals
f f(x,y)dy,   J f(,y)dy
have different values for some of the values of x, it may happen that the two
repeated limits
Jdxj f (x, y) dy,     dxj f (x, y) dy
o,' yo              o. _yo
have identical values.
The repeated integral will consequently be regarded as, in this case,
existing; and thus it may be defined as
dxj f(x, y) dy,
o _2o0
where the upper or lower integral with respect to y is to be taken indifferently, provided the repeated limit exists as a definite number.
In a similar manner      dyJf (x, y) dx,
Y 2/o  JZO
when it has a definite value independent of whether the upper or lower
integral with respect to x be used, will be regarded as the repeated integral,
first with respect to x and then with respect to y.
314. It was first established by P. Du Bois Reymond* that, when the
limited function f (x, y) has a double integral in the fundamental rectangle,
then the two repeated integrals exist and are each equal to the double integral.
* Crelle's Journal, vol. xcIv, 1883, p. 277.




313, 314]


Repeated integrals


423


We shall first give a proof* of this theorem which exhibits its relation with
the theory of sets of points.
The following preliminary theorem will be first established:If If(x, y) (dxdy), taken through the fundamental rectangle, have a definite
value, then the set of values of x for which the single integral f  (x, y) dy, taken
with x constant, has a definite value, defines a set of points on a side of the rectangle, of linear measure equal to the length of that side.
It follows from this theorem, that the set is everywhere-dense in the
interval, and of cardinal number c. Moreover, the points at which
jf(,y)dy,     f (, y) dy
differ from one another form a set of measure zero.
On the assumption of the existence of the double integral, the set K of all
the points at which the saltus off(x, y) is _ k, where k is an arbitrarily chosen
positive number, is a closed set of plane content zero. If a straight line be
drawn parallel to the y-axis through the point x of the side of the rectangle,
then the component of K on this straight line will be denoted by K~, and its
linear content by I (K.).  It has been shewn in ~108, that, o denoting a
prescribed positive number, the linear content of that set of points x, on the
side of the rectangle, for which I (Kxa)  ar, is zero; and thus that I (Kx),
considered as a function of x, is an integrable null-function, for each value
of k.  The function x (x), = lim I(Kx), is also an integrable null-function;
kc=
for the set of points at which X (x) does not vanish is made up of those sets
of points at which I (K(')), I (K;(2)),... I (Kx()),... do not vanish, where
K(l), K(2),... K(),... correspond to a diminishing sequence of values of k
converging to the limit zero: and since each of these sets has zero measure,
it follows that the set of points at which % (x) does not vanish has zero
measure. At any point xl, at which % (x) vanishes, I(K,) vanishes for every
value of k.
It should be observed that, at a point of IK, it is not necessarily the case
thatf (x, y), considered as a function of y, with x constant, has its saltus ' k;
in fact, this saltus may be less than k, or may be zero. However, all the
points at which the saltus of f(x, y) taken with x constant, is _ k, are
certainly included in the set Ka.
For any fixed value of x, the upper and lower integrals
j f(,y) dy,    f(, y) dy
* The investigation is founded on that of Schonflies, Bericht ilber die Mengenlehre, p. 193.




424


Integration


[CH. V


both exist, and the two have equal values at any point of the everywheredense set of points x, at which X (x) vanishes. The preliminary theorem
above stated has thus been established.
Let F(x) denote ff(x, y) dy, where F(x) consequently has a single
determinate value at each point x at which
ff(x,y)dy,    f(x, y) dy
are equal. At any point of that set of zero measure, at which
f (x, y) dy,  f (x, y) dy
have different values, F(x) is regarded as indeterminate; and the upper and
lower integrals are the upper and lower limits of indeterminacy. It will
now be shewn that the function F (x), so defined, is an integrable function.
Let x' be a point on the side of the rectangle, such that the component
Kx, of K, on the line x = ', has content < o. A finite number of intervals
61, e2,... e, can be determined on the straight line x= x', neither abutting
on, nor overlapping, one another, such that their sum e1 + 62 +... + el, > b - o,
where b is the length of the side of the rectangle parallel to the y-axis, and
such also as to contain, in their interiors and at their ends, no points at which
the saltus of f(x, y) is _ k. For each point of one of these intervals e there
exists a rectangle with the point at the centre, such that the fluctuation
of f(x, y) in that rectangle is < k.  The breadths of these rectangles for all
points of e must have a finite minimum, for otherwise there would exist a point
of e which would belong to K. It follows that, for the point x', an interval
(x'-a, ' +3) can be determined, such that the straight lines x  x' -a,
x = x' + 3, intersect all the rectangles corresponding to all the points of the
intervals e1, e6,... em. If X1, x2 be any two points in the interval
('-a, ' +/3),
we have             F (x1) -F(x) < bk+ a (U -L).
Now a finite number of separate intervals 8,, 8,... 8, can be determined
on the side x of the rectangle (length = a), such that 81  + 8 +... + 8, > a -,
where 7v is a prescribed positive number, and such that each point of each of
the intervals 8 is a point x', for which an interval (x'- a, x' + /) can be determined as above. By applying the Heine-Borel theorem we see that
81,  82,... 8,.
will all be covered by a finite number of these intervals (x'- a, x' + /). It
thus appears that the x-side of the rectangle can be divided into a finite
number of parts
T1, 72,... 7T




314]                   Repeated integrals                       425
and                         X1, \2 * Xq,
such that              71  2 +  2  +... + Tp > a-r,
and                     Xi + X2 +... + \q <;
and such that the fluctuation of F (x) in any one of the parts r is
< bk + o(U-L).
Let                  k = /2b, o-e=/2 (U-L);
we see then that F(x) is such that the x-side of the fundamental rectangle
can be divided into a finite number of parts, such that the sum of those parts,
in which the fluctuation of F(x) is _ e, is less than the arbitrarily chosen
number 7. It follows that F (x) is integrable along the side of the rectangle.
It has now been shewn, on the assumption of the existence of the double
integral, that the repeated integral
F(x) dx, or    dxf(x, y) dy,
taken through the fundamental rectangle, has a definite value. Moreover,
this value is equal to that of the double integral. For, let the fundamental
rectangle be divided up by means of straight lines parallel to the y-axis, through
the end-points of each interval of the two finite sets tr} and }Xj. Any one of
the rectangles so constructed, with v as base and with height b, can be
divided into parts by means of straight lines parallel to the x-axis, such that,
in each one of a number of these parts the sum of whose heights is > b - a-,
the fluctuation of f(x, y) is < Ic. The fundamental rectangle has now been
divided into a finite number of parts, such that the sum of the products of
each part multiplied by the upper limit of f(x, y) in that part exceeds the
sum of the products of each part multiplied by the lower limit of the function
in that part, by less than abk + (ao- + by) ( U - L), which is arbitrarily small.
Also           dx f (x, y) dy, and  f(x, y) (dxdy),
both lie between the two sums of products, and therefore differ from one
another by less than abl + (ao- + by) (U - L). The equality of the double
integral and the repeated integral is thus put in evidence by the mode of
sub-division of the rectangle which has been adopted.
Similar reasoning applies to the repeated integral in which the integration
is taken first with respect to x, and then with respect to y.
It has thus been established that, if the double integral through the fundamental rectangle exist, then the two repeated integrals also exist, and are each
equal to the double integral.




426


Integration


[CH. V


All the points at which x (x) vanishes are points of continuity of the
function F(x); but there may also be other points at which F (x) is continuous; because the existence of a saltus of f(x, y) at a point (x, y) is
consistent with f(x, y) being continuous with respect to x, and also with
respect to y, at the point.
The function F(x) may be replaced by r (x), the most nearly continuous
function related to it (~ 192). We thus have
f (x, y) (dx dy) =    f  (x) dx.
It has been assumed that the set of points G, for which f(x, y) is defined,
is measurable in accordance with Jordan's definition of a measurable set; and
thus, that the double integral of the limited function may be replaced by that
of the function f(x, y), defined for all points of a rectangle which contains G,
by the convention that f(x, y) shall vanish at all those points of the rectangle
which do not belong to G. If the set G be such that each straight line
parallel to the y-axis contains points of G which fill up a finite, or an
indefinitely great number of continuous intervals, or more generally, if the
set of such points for each value of x be linearly measurable, then the integral
f(x, y)dy,
taken along the whole segment of the line between the sides of the rectangle,
may be replaced by the same integral taken through the component of G on the
same segment. In particular, if the points of G on the straight line through
the point x consist of all the points in the linear interval
(fi (x), f2(x)),
we may replace                     f (x, y) dy
by                                  f(x, y) dy;
J (x)
and therefore in this case,
f I(, y)(dxay)=      dx       f(x, y) dy.
G                  to. f(x)
315. A simple proof* of the fundamental theorem of ~ 314, will be given,
which depends upon the fact that, for any limited function f(x, y), if the
operation of taking the upper integral first with respect to y, and then with
* This method of proof was first employed by Harnack; see his edition of Serret's Differential
and Integral Calculus, p. 282. Other proofs of this kind have been given by Arzela, Meem. dell' Ist.
di Bologna, ser. 5, vol. I, p. 123; by Jordan, Liouville's Journal, ser. 4, vol. VIii, p. 84, or Cours
d'Analyse, vol. I, p. 42; also by Pringsheim, Sitzungsberichte d. Miunch. Akad., vol. xxvnii,
p. 59, and vol. xxix, p. 39. See also Pierpont's paper "On multiple integrals," Trans. Amer.
Math. Soc., vol. vi, 1905, where a proof of this character for multiple integrals is given.




314, 315]


Repeated integrals


427


respect to x, be performed, the result cannot exceed the upper double integral;
and that, similarly, the result of successively taking the lower integrals with
respect to x and to y cannot be less than the lower double integral: thus
f(x, y)(cxdy)   f dxjf(x, y)dy  f dxj f(t, y)dy
f(x, y) (dxdy),
the integrals being all taken over the fundamental rectangle.
Iff(x, y) be integrable, so that
f(    y (dxd)      x, y) dy)(dxdy),
it follows that.dxf (x, y) dy = f dx f (x, y) dy
=df(xjfx, y) dy =f dxf(x, y) dy,
and thus that the repeated integral
dxjf(x, y) dy
has a definite value equal to the double integral.
To establish the theorem, let the rectangle be divided into a number of
parts 8 by means of straight lines parallel to the sides. Since the double
integral is assumed to exist, this may be done in such a manner that, e
denoting an arbitrarily chosen positive number, the conditions
fI    y) ( d dy(dy) - e  { US (8)} S {S  ()}  ff(x, y) (ddy) +
are satisfied, where the summation X is taken for all the rectangles 8, and
U(S), L (8) denote the upper and lower limits of f(x,y) in a rectangle 8.
Now, if we take the upper and lower integrals off (x, y) along a straight line
parallel to the y-axis, we have
ff(x, y) dy -S1 { {'U(8)},
ff(x, y) dy  1 {L,  L (8)},
where the summation 2E refers to all those rectangles 8 which are intersected
by the straight line along which the upper and lower integrals are taken; and
in the case when that straight line is along one or more boundaries of the
rectangles 8, E refers to all the rectangles on one side of that line: also 3'




428


Integration


[CH. V


denotes that interval along the line of integration which is in the rectangle 8.
It follows that
dx   f (x, y) dy - E {S U (8)} < Jf (x, y) (dxdy) + e
and           dx ff (x y) dy_ t {SL ()       (x y) (dJ dy) -- 
and since these inequalities hold for every value of e, we have
jdx f(x, y) dy _ Jf(x, y) (dx dy)
dxf (x, y) dy _f (X, y) (dxdy);
and thus the theorem is established.
316. The converse questions now arise whether, from the existence of one
of the repeated integrals, or from the existence and equality of both repeated
integrals, that of the double integral can be inferred. The answer to both
questions must be in the negative.   Continuity of a function f(x, y) with
respect to x and y separately does not necessarily imply continuity with
respect to (x, y); moreover, the saltus of the function at a point with respect
to x, when y has a constant value, or with respect to y when x has a constant
value, is not necessarily equal to the saltus of the function with respect to
(x, y). It may happen that the component of K on a straight line parallel
to one of the axes may consist of points some or all of which are points of
continuity of the function when considered as a function of one variable on that
straight line.  Thus K may have a plane content greater than zero; and yet
the linear content of the points on all straight lines parallel to the axes, at
which the linear saltus of the function is _ Ic, may be zero.  Hence either, or
both, of the repeated integrals may exist, whilst for values of k, the sets K are
not of zero content*; and therefore whilst f(x, y) does not admit of a double
integral. The relation of Lebesgue integrals with repeated integrals will be
considered in Chap. VI.
EXAMPLES.
1.t For the rectangle bounded by x =0, x=l, y=0, y=l, let f(x, y)=l, for all
rational values of x, and f(x, y)=2y, for all irrational values of x. We have then
i f(x, y) dy=l, whatever value x may have; and hence the repeated integral
dx      f (x,y)ddy
* An incorrect theorem relating to this point has been given by Schonflies, see his Bericht,
p. 197. In this theorem the condition that K should be closed is stated to be the condition for
the existence of the double integral. If, however, K were not closed, it could not represent the
set of points at which any function had a saltus > k. The examples given by Schonflies do not
in reality accord with his theorem.
t Thomae, Schlomilch's Zeitschrift, vol. xxIII, p. 67.




315, 316]                  Repeated integrals                             429
has the value 1: but the double integral does not exist, since (K) > 0, for any value of
k in the interval (0, 1).
2.*  Let x be represented by a finite or infinite decimal, excluding those decimals in
which every figure from and after some fixed place is 9. Let Px denote the number of
decimal places in the representation of x in the manner described. Let y be represented
in a similar manner, with a corresponding definition of py. Let the function f(x, y) be
I      1
defined in the rectangle bounded by x=O, x=1, y=O, y=l, by f(x, y)= p   + -- -
when Px and py are both finite; otherwise let f(x, y)=0.
11  1
We have       +i dg=0; for there are only a finite number of values of y in (0, 1), for
0 p —~
which py is less than an arbitrarily chosen fixed integer, or --  is greater than an
arbitrarily chosen fixed proper fraction. The function f(x, y) vanishes, except when one
at least of x and y is representable by a finite decimal; and thus the double integral
f/(, y) (dry) =o.
Now                 f(x,y)dy=p i-,      f(x,y)dy=0;
JO           Px+          f j  0   y 
and thus    f (, y) dy has no definite value for any value x of the everywhere-dense
enumerable set of points for which px is finite.
Nevertheless      f dx     f f(, y)dy=== If(, y)(dxdy).
3*. With the same notation as in the last example, let f(x, y)=0, when px, py are
both finite or both infinite; let f(x, y)=l-, when Px is finite and py infinite, and
1
f(x, y)=  ly, when py is finite and p, is infinite. In this case f(x, y) differs from 0 at
1 +]y
an unenumerable set of points; and yet the set of points at which f(x, y)>  has the
plane content zero, since all such points are on a finite number of lines parallel to the
coordinate axis, although they are everywhere-dense on those lines. The double integral,
and consequently the repeated integrals, exist in this case.
4.* An example has been given in Ex. 1, ~ 108, of a set of points K which is
everywhere-dense and unclosed, whereas the sets Kx, Ky are all finite. Let f(x, y)=c' at
every point of K, and =c at every other point. In this case, the double integral does not
exist; but
f(, y) d==c,     f(x, y)dy=c,
whatever values y and x may have in the first and in the second integral respectively.
In this case       dx f f(x, y)dy and | dy     f(x, y)dx
both exist and have the same value c.
5.*  Let a set {(x', y')} be defined as follows:-Let x' have any value for which pxz is
finite; and with such a fixed value of x', let every y' be taken for which py,  pz,. On
every line parallel to the y-axis there are only a finite number of points of the set; but
* Pringsheim, Sitzungsberichte d. Leipziger Akadenie, vol. xxvIII, p. 71.




430


Integration


[CH. V


the set is everywhere-dense on every line which is parallel to the x-axis, and which has
for its ordinate one of the y'. Let f(x, y)=c' for the set {(x', y')}, and let f(x, y)=c
for all remaining points.
We have, in this case,          f /(x, y) dy=c,
and                             dx   f (x, y)dy=c.
But f /(x, y')dx has c and c' for its upper and lower values; and the set of values
J o
of y' being everywhere-dense, | dy f f(x, y )dx does not exist.
6.* Let f(x, y) be defined at all points of the rectangle bounded by x=0, x=l,
y=0, y=1, by the condition that f(x, y) =0, except at those points (x', y') at which
= 2n+1      = 2' where f(x', y' )=  1
n, n, p and q being positive integers.
In this case the double integral exists, and therefore the repeated integrals both existt.
PROPERTIES OF THE DOUBLE INTEGRAL.
317. The following properties of the proper double integral may be
established in a simple manner:(1)  If/(x, y) have a double integral in the domain G, then If(x, y) I also
has a double integral in the same domain; i.e. f(x, y) is absolutely integrable
in the domain G.
For the fluctuation of If(x, y) I in any rectangular cell cannot exceed that
of f(x, y); hence it follows from   the condition of    311, that If(, y)I is
integrable, if f(x, y) be so.
It follows from the definition that
f     (, y)dxy)(dxdy) -  f  I (x, y) I (dxdy).
(2) If G be divided into two parts G,, G,, each of which is measurable
in accordance with Jordan's definition, then if f(x, y) have a double integral
in G, it has also double integrals in G1 and G2, which satisfy the condition
f (x, y) (dxdy)+ f(x       y)(dxdy) =     f (x, y) (dxdy).
G,                  G,                  G
For K, the set of points of G at which the saltus of f(x, y) is - c, may be
divided into its two components K1 in G1, and K2 in G2.     The only points of
* Du Bois Reymond, Crelle's Journal, vol. xciv, p. 278; also Stolz, Grundziige, vol. iiI, p. 73.
2rn ~ 1
t This is denied by Stolz, Grundziige, vol. II, p. 88, on the ground that f (x, y) for x= 2 -is not integrable with respect to y, but has - and 0 for its upper and lower values. We have,
however, shown that this is no justification for denying the existence of the repeated integral.




316, 317]


Properties of the double integral


431


G, at which the saltus of f(x, y), considered as defined for G, only, is > k,
consist of the points of the set K/, together with points forming a set K1' on
the frontier of G,. The content of this frontier being zero, K1' has zero
content; also K1 has zero content, since it is a part of K. Since K1 + K1' has,
for every value of k, the content zero, it follows that f(x, y) is integrable in
G,; similarly, it is integrable in G,. Also
f (x, y) (dxdy)
is, by definition, the limit of the finite sum,8M   (,)+ 2M(,)     + () +... +,M (
The rectangles 8 consist of (1) those which contain interior points of G, only,
(2) those which contain interior points of G, only, (3) those which contain
points on the frontiers of G and of GI and G2. The above sum may therefore
be divided into three portions containing those 8's respectively which belong
to (1), (2) and (3). The limit of the first of these is
f     (X, y) (dxdy)
that of the second is
f / (, y) (dxdy),
and that of the third is at most equal to U multiplied by the content of the
points on the frontiers of G6 and G,, where U is the upper limit of If(x, y).
Since the contents of the frontiers are zero, the limit of the third part of the
sum is zero; hence the second part of the theorem is established.
(3) If F(f,f,,... fin) be a continuous function in G of the n functions
fi, f2,... fe, each of which has a double integral in G, then F has itself a
double integral in G.
The proof of this is identical with the one applicable to the case of a
single variable, given in ~ 256.
That the sum or product of two or more integrable functions is integrable
is a particular case of this theorem.
(4) If f be integrable in G, and the function f, be defined by f =f, for
every positive value of J and byf, = 0, whenf is negative or zero; the function
f, being defined by f2= -f, for every negative value of f, and by f,= 0, when
f is zero or positive; then fi and f2 are each integrable in G. Forf, -f, being
=f, is integrable, and also in virtue of (1),/f +2f is integrable in G, hence by
(3), f, and f, are both integrable in G.




432


Integration


[CH. V


IMPROPER DOUBLE INTEGRALS.
318. As in the case of single integrals, the definition of a double
integral may be extended to the case in which the function has a set of
points of infinite discontinuity.  This set is necessarily closed, and it will be
assumed throughout that its plane content is zero. It will also be assumed
that the domain for which such a function is defined is bounded, and that the
frontier has the content zero, the domain being therefore measurable in
accordance with Jordan's definition; and consequently the function may be
replaced by another function defined for all the points in a fundamental
rectangle, the new function being taken to vanish at all points not in the
original domain, and to have the same value as the original function at all
points of that domain. These assumptions being made, a definition of the
improper double integral which is substantially the one given by Jordan*,
and adopted by Stolzt, may be stated as follows:Let Di, D2,... Dn... denote a sequence of domains contained in the fundamental rectangle, each one of which consists of a finite number of connex closed
portions each with its frontier of zero content, and in which the number of the
portions may increase indefinitely with n. Further, let us suppose that none of
these domains contain, in their interiors or on their frontiers, any point at
which f(x, y) has an infinite discontinuity, and that the sequence is such that
the measure of Dn converges to that of the fundamental rectangle; then if
the upper integrals
f (x, y) (dxdy),     f(x, y)(dx(d )... / f(x, y) (dxdy),...
D AJ D^2! De
taken over the domains D1, D,..., converge to a definite limit independent of
the particular sequence {Dn4 chosen, this limit is defined to be the improper
upper integral
Jf(x y) (dxdy)
of f(x, y) in the given domain. A similar statement applies to the case of
the improper lower integral.  When the improper upper and lower integrals
both exist, and have the same value, then the improper integral
f f(, y) (ddy)
over the given domain is said to exist, and to have this common value.
It will be observed that the domains Dn are all measurable in accordance
with Jordan's definition of a measurable set, and therefore also in accordance
with the definition of Borel and Lebesgue.
* Cours d'Analyse, vol. ii, p. 76.
t Grundziige, vol. II, p. 124.




318, 319]


Improper double integrals


433


In case the function f(x, y) be integrable in all the domains D1, D,,...,
however this sequence may be chosen, subject to the conditions stated
above, then if the sequence
f      y)(dx dy),   f(x, y) (dx dy),...
converge to a definite limit, independent of the particular sequence [D,}, that
limit defines the improper double integral
f (x, y) (dxdy).
If a function f(x, y) have an improper integral in the fundamental rectangle, then f(x, y) has a proper integral in any connex closed domain of
which the frontier has zero measure, and which is contained in the fundamental rectangle, but itself contains no points in its interior, or on its frontier,
at which f (x, y) is infinitely discontinuous. For, by the definition,
f    f(, y) (dxdy),  f(X y) (dxdy)
converge to one and the same definite limit, as D converges to the fundamental rectangle; therefore D can be so chosen that, if e be an arbitrarily
chosen positive number,
D f (, ) (dxdy) -fD f (x, y) (dxdy) < e.
Now if D' be any domain of the type defined above, in the interior of D, it is
clear that the difference between the upper and lower integrals of f (x, y)
throughout D' cannot exceed the difference of the upper and lower integrals
throughout D, and is therefore < e. Since e is arbitrarily small, it follows
that the upper and lower integrals throughout D' must be identical, and
therefore that f(x, y) is integrable in D'.  It has thus been shewn that
if f (x, y) have an improper double integral in the fundamental rectangle,
it must possess a proper integral in any connex closed domain interior to
that rectangle, such that the domain has its frontier of zero measure, and
contains no points of infinite discontinuity of the function, either in its interior
or on its frontier.
319. The necessary and sufficient condition for the existence of the improper upper double integral
f (x, y) (dxdy)
is that, corresponding to any arbitrarily chosen positive number e, another
positive number 8 can be determined, such that, if A be any connex closed
domain whatever, of which the frontier has zero measure, and which is contained
H.                                                          28




434


Integration


[CH. V


in the fundamental rectangle, but itself contains no points of infinite discontinuity of f (x, y), either in its interior or on its frontier, then, provided the
measure of A is < 8, the condition
lf (, y) (dcdy) < e,
taken over A, is satisfied.
A similar theorem applies to the improper lower double integral.
To shew that the condition stated in the theorem is sufficient, let D, D'
be two domains of the kind specified in ~ 318, such that m (D), m (D') both
differ from the area of the fundamental rectangle by less than 8; they are
both interior to the fundamental rectangle, and contain none of the points
of infinite discontinuity of the function. Let d be the set of points of D
which do not belong to D', and d' the set of points of D' which do not
belong to D; then
m(d)< A - m (D')< 8,
and                    m (d') < A -  (D) < 8,
where A is the area of the fundamental rectangle. Also, since the domains
D + d', D' + d are identical, we have
ID - ID' =  Id - Id',
where I denotes the upper double integral
f(, (y) (ddy)
taken over the domain indicated by a suffix. It follows that
I lD-ID,-  Id + I Id' < 2~;
and hence it is easily seen that any two sequences
tIDn}g {I'D'n}
both converge to one and the same definite limit.
To shew that the condition stated in the theorem is necessary, let us
suppose that it is not satisfied. We thus assume that a domain d of
arbitrarily small measure can be found, such that I Id > e.
Let D be interior to the rectangle, and such that
A- m(D)< 8.
Taking D to contain d, we then have
vn (D - d) > A - 28,
provided                     m (d) < 8.
The two domains D, D - d both converge to A, if 8 be decreased indefinitely;
and 
- D- J D-d aId;




319]                Improper double integrals                    435
thus                         I - -ID-d >e,
however small 8 may be; hence the limit does not in this case exist.
The necessary and sufficient condition that the improper upper and lower
integrals of f (x, y) in the fundamental rectangle may both exist is that the
improper upper integral of I f(x, y) I may exist.
To shew that the condition stated is sufficient, we observe that, on the
assumption of the existence of
f (, y) (dxdy),
it follows from the theorem established above that the upper integral of
I f (, Y) I
through a connex domain D, interior to A, and containing none of the
points of infinite discontinuity, tends to the limit zero as m (D) does so.
Also
(x,y) (dxdy),     jf(x, y) (dxdy)
are both                      (x f, y) (dxdy),
as is easily seen. It follows that both
S     (x, y) (dxdy),  f (x, y) (dxdy)
converge to zero as m (D) does so, and uniformly for all such domains D; and
that these are the sufficient conditions for the existence of
f f(x y) (dx dy), ff(x, y) (dxdy).
To prove that the condition stated is a necessary one, let us assume that,
for every connex domain D satisfying the specified conditions, and such that
m (D) < 8, we have;f(x, y)(dxdy) <.
Now let               f (x, y)=f+ (x, y)-f-(x, y),
where                      f  (x, y) =f(x, y)
at all points where f (x, y) is positive, and everywhere else
f+(x, y)=0;
also                      f -(, y) ---f(x, y),
at every point where f(x, y) is negative, and f- (x, y) is everywhere else
zero. The domain D may be divided into a finite number of rectangles,.
some of which may lie partly outside D; the functions being taken to be
28-2




436


Integration


[CH. V


zero in all such outlying portions. Denoting by Us the upper limit of a
function in the rectangle 8, we have
U8 {f(x, y)}= Us {f (x, y)},
in all elements 8 in which f(x, y) has positive values; and in all other elements
Us {ff(x, y)} =0.
The elements may be taken such that, if r be an arbitrarily chosen number,
the inequalities
SU8 {f,y)}-f f(x,y) (dxdy) <,
8Us {f + (x, y)} -f+(, y) (dxdy) <
are both satisfied.  These conditions are also satisfied for any domain
contained in D.
We have now
f f+(x, y)dxdy   S8Usff+ (x, y)} C  USU8{f(x, y)}
J D             D                D,
- ff(xy)(daxdy)~+n
E +,
where D, consists of the domain which is composed of those elements 8 of D
in which f (x, y) has positive values. Since qY is arbitrarily small, we have
f + (, y)(dxdy) - e.
D
Again, since
(x, y) (dxdy) = -   -   (x y)} (dxdy)
we see that               I-{f(x, y)} (dxdy)
has the limit zero when m (D) has the limit zero; and fiom this it follows as
before that
f - (x, y) (dxdy) _ e.
Since               f (x, y)= f+ (x, y)-f-(x, y),
we have                   f(x|, y) I (dxdy) >  2E;
and therefore          / f  (, y) (dxdy)
7DIfxy)J(xy




319, 320]


Improper double integrals


437


has the limit zero when m (D) converges to zero. It now appears, by employing the first theorem of this section, that
If f(, y)I (dxdy),
taken throughout the fundamental rectangle, has a definite value.
It should be observed that since
I f(x, y) I (dxdy)   f f(x, y) I (dxdy),
the lower integral
f  f (, y) (dxdy)
has the limit zero, whenever zero is the limit of
(x, y) (dxdy);
and that therefore        f(x, y) I (dxdy)
always exists when        f (x, y) I(dxdy)
does so, the integration being over the fundamental rectangle.
320. It has been seen in ~ 318 that, in case f(x, y) have an improper
integral in the fundamental rectangle, it has a proper integral in any closed
connex domain D contained in that rectangle, with frontier of zero measure,
and containing no points of infinite discontinuity of the function. It follows
by the theorem of ~ 317 (1), that If(x, y) I is also integrable in the domain D;
and we have already seen that the existence of an improper upper integral of
If (, y) I is a necessary consequence of the existence of the improper integral
of f(x, y). It thus appears that the improper upper, and lower, integrals
of If(x, y) must be identical, and therefore that, if f(x, y) be a function
which has an improper double integral, in accordance with Jordan's definition,
then If(x, y) has also an improper integral, so that every such improper
integral is absolutely convergent.
We have seen that Harnack's definition of an improper single integral is
applicable not only to the cases in which the convergence is absolute, but
also to cases in which the convergence is not absolute. Jordan's definition
of an improper double integral is however much more stringent than
Harnack's definition of an improper single integral.  In the latter case
the integral is defined as the limit of the proper integral taken through a
finite number of intervals, not chosen arbitrarily in any manner consistent
with the condition that the sum of these intervals is to converge to the
length of the interval of integration, but chosen so as to satisfy the special




438


Integration


[CH. V


condition that they are complementary to a finite set of intervals which
contain in their interiors all the points of infinite discontinuity of the
function, each interval of the finite set containing at least one such point.
If the proper integrals of which the improper integral, in Harnack's
definition, is the limit, were not subjected to the above mentioned restriction,
reasoning precisely similar to that applied above would shew that every
improper single integral must be absolutely convergent.  In order that
a definition of the improper double integral should admit of the existence of
double integrals which do not converge absolutely, it would be necessary to
subject the domains D1, D2,... D,,..., (the proper integrals through which
define, as their limit, the double integral), to some restriction which would
allow of the existence of a limit in cases in which such a limit does not exist,
independently of the particular set {D,} chosen, when no such restriction
is made.  Such a restriction as to the nature of the domains D,  would
correspond to the restriction to a special class of sets of intervals, of the
intervals through which the proper integrals in Harnack's definition of an
improper single integral are taken. The true extension of Harnack's
definition to the case of double integrals would be the following:Let the points of infinite discontinuity of f (x, y) (the set of such points
being of zero content), be enclosed in a finite set of rectangles with sides
parallel to those of the fundamental rectangle, each rectangle of the finite set
containing at least one point of infinite discontinuity, and no such point being
on the frontier of the set of rectangles; and let D, denote the remanining part
of the fundamental rectangle when the finite set of rectangles is removed, then, if
f (x, y) have a proper integral in every such domain D,,, and if this proper integral converge to a definite limit when any sequence whatever of such domains
Dn is taken, such that the measure of D, converges to that of the fundamental
rectangle, this limit shall define the improper double integral of f(x, y).
This extension of Harnack's definition would admit of the existence of
non-absolutely convergent improper double integrals, as we have seen to
be the case for improper single integrals. With this definition, the theorems
of ~ 319 would no longer be valid.
When it is asserted that non-absolutely convergent double integrals do
not exist, the assertion must be taken to mean that such integrals do not
exist in accordance with the definition of Jordan, and not that it is impossible to give definitions, such as the above extension of Harnack's, in
accordance with which double integrals exist that do not converge absolutely.
The properties of improper double integrals which are not necessarily
absolutely convergent are more restricted than those which exist in accordance with Jordan's definition, and it is consequently a matter of opinion
whether, though the former certainly exist as limits, the name integral
may be appropriately applied to such limits.




320]                    Improper double integrals                            439
EXAMPLE.
If we take as the domain of integration the rectangle bounded by x=0, x=a, y=0,
=b, then the double integral of I sin 1 is not convergent, and therefore in accordance
X    X
with Jordan's definition does not exist; although the single integral  -sin  dx is
non-absolutely convergent, and does exist in accordance with Harnack's definition.
The existence of J  - sin - dx depends upon the fact that  - sin - dx converges to
X    X                                 XrJ   X
a definite limit as e converges to zero, and this is sufficient to ensure the existence of the
single integral. Although | -sin1 (dxdy) taken over the domain bounded by x=, x=a,
/=0, y=b, converges to a definite limit, as E converges to zero, this is not sufficient to
ensure the existence of the Jordan double integral. Taking Jordan's definition, let the
domain Dn consist of the rectangular spaces bounded by the lines y=0, y=b, and the
lines parallel to the y-axis at the extremities of the intervals on the x-axis
1                I          1             1          1
\(2n  1)r' a)'   (2 + 3) r  (2 + 2)) ' ((2 + 5)r     (2 + 4) r) '
(    1       1 \
(4n+l) X' 4-;Vr'
The double integral taken through these spaces is
1
[a  [bl. 1         Pn f2lpr      /Jl. 1
b 1    Jo-sn- dxdy                    \ J  -sin -dxdy,
J    X snx      + p=+l       __         x
(2n+1)7r                      (2p+1) r
[a     fb 1    1                 p=2n     1
or              j   1       -m- sin  dxdy+b   sin z            dz;
J  1  0 X   X           0     p=n+l z+2p
(2n+1) rr
which is greater than
b                               f- sin -  zd+  sidz,
j       x    x      (4n+ 1) rJo
(2n+1) r
or than                   b          sin - dx +;
j i    x x     (4n + 1) rr'
(2n+1) rr
a 1    1     b
and this converges to          b      sin  dx + -
x X  X     27r
whereas J- sinll (dxdy) taken over the domain bounded by x=e, x=a, y=0, y=b, conal      1
verges to                         b     sin  dx.
Jo    x x
Therefore the mode of choice of the intervals D, affects the limit to which
I sin I(dxdy)
/D, X    X
converges, as Dn converges to the complete domain. Thus it is clear that the double
integral, in accordance with Jordan's definition, does not exist.




440                         Integration                        [CH. v
321. A definition of the improper double integral has been given by
de la Vallee-Poussin, precisely similar to his definition for single integrals.
This may be stated as follows:Let fn (x, y) be the function which is such that
fn (, y) =f(x, y)
at every point (x, y) at which
Mn- f(x, y) -Nn,
where M,, Nn are two positive mnmbers, and is also such that
fn (, y)= Mn,
at every point where f (x, y) > Mn; also let fn (x, y) - Nn,
at every point where       f(x, y) < - Nn.
If f (x, y) be such that the proper integral
f, ( y) (dxdy)
exists, whatever positive values Mn, Nn may have, and if also the double limit
lim.    fn(x, y) (dxdy)
Mn = c0, Nn= x
have a definite finite value, that limit is said to define the improper integral
/(, y) (dxdy).
The existence of the double limit implies that, if
M1, M2,... Mn,... and N1, N2,... N,...
be any two independent sequences of increasing numbers with no upper
limits, then the sequence of numbers
fi (x, y) (dxdy), (f2 (x, y) (dxdy)... fn (x, y) (dxdy)
converges to a definite limit, independently- of the mode in which the two
sequences {Mn}, {Nn are chosen.
It may be shewn, exactly in the same manner as in ~ 273, that the
existence, in accordance with this definition, of the improper integral
f f(, y) (dx dy),
implies the existence of   / (x, y) (dxdy);
and thus, that all improper integrals, so defined, are absolutely convergent.
As in ~ 273, it appears that, for the existence of the double integral, it is
necessary that the closed set of points of infinite discontinuity of the functior




321, 322]


Improper double integrals


441


should have zero content; and in fact that, k denoting any positive number,
the set of points at which the measure of discontinuity of the function
is i k, must have content zero. Thus all the points of discontinuity of the
function form a set of zero plane measure.
322. Since the definitions given by de la Vallee-Poussin and by Jordan
both apply only to the case of absolutely convergent integrals, it is of
importance to shew that they are completely equivalent to one another.
If, in the proof given in ~ 274, of the equivalence of the two definitions of
absolutely convergent single integrals, intervals be replaced throughout by
rectangular cells with sides parallel to the sides of the fundamental rectangle,
we have a proof that in the case of absolutely convergent double integrals,
the definition of de la Vallee-Poussin is completely equivalent to the
extended definition of Harnack, given in ~ 320.  It only remains to
prove that the latter is, in the present case, equivalent to Jordan's.  It
will be sufficient to give a proof that, for a function which is never negative,
if the integrals taken through the special domains Dn, which consist of
sets of rectangles with sides parallel to the axes of x and y, converge to a
definite limit, then the integrals taken through domains D'n, each of which
consists of a finite number of connex closed portions of any kind, also
converge to the same definite value as in the case of Dn, when m (D'n)
converges to the area A of the fundamental rectangle.
Taking D'n so that A - m (D') = en, Dn can be so chosen as to contain
D' in its interior. For, since D', does not contain either in its interior or on
its frontier any points of infinite discontinuity of the function, therefore, for each
one of the latter points, the distance from all the points of D'o has a minimum
greater than zero, D'n being closed and connex. Hence each point of infinite
discontinuity can be enclosed in a rectangle which contains no points of D'n
either in its interior or on its sides.
Since the set of points of infinite discontinuity is closed, a finite set of
the rectangles can, in accordance with the Heine-Borel theorem, be chosen
so as to enclose the whole set of these points; and the complement of this
finite set of rectangles may be taken to be Dn.
This may be done for each value of n.
If m (D'n) converge to A, it is clear that m (Dn) which is > m (D'n) also
converges to A. Also a number n' > n, can be determined, such that D'wn
encloses Dn in its interior; we have then
J,. f (x, y) (dxdy) f  d   )  f      )(x, y)(dxdy).
If                     f  (x, y) (dxdy)
nD




442                          Integration                         [CH. v
converge to a definite limit  ff (x, y)(dxdy),
n may be taken so great that
f (x, y)(dxdy) -  ff(x, y) (dxdy)
is less than the arbitrarily small number; then also
f (x, y) (dx dy) -  f (x, y) (dxdy) < 7,
and it thus follows that           (x, y) (dxdy)
D'n
also converges to the limit    f/(x, y) (dxdy).
The complete equivalence of the two definitions has now been established.
The definition of a Lebesgue integral may be extended to the case of
improper integrals, as in ~ 291. It may be shewn, precisely as in ~ 291, that
the definition of a Lebesgue improper double integral may be put into a
form which is an extension of de la Vallee-Poussin's definition to the case in
which the functions f (x, y) possess Lebesgue integrals, but not necessarily
Riemann integrals. Exactly as in ~ 291, it may be shewn that the three
definitions of an improper double integral are in accord* with one another
whenever all three are applicable.
DOUBLE INTEGRALS OVER INFINITE DOMAINS.
323. Let the function f(x, y) be defined for an unbounded domain G,
and let it be assumed that f(x, y) possesses either proper, or improper, upper
and lower integrals for every bounded domain D contained in G, such domain
being closed, and having its frontier of zero content.
Lett D1, D2,... Dn,... be a sequence of domains each consisting of a finite
number of connex closed portions, and such that Dn contains every point of G
of which the distance from the origin is < p,, where pn is a positive number
which increases indefinitely with n. If the proper or improper upper integral
of f(x, y) taken over Dn have, for the whole sequence, a definite limit independent of the particular sequence {Dn} chosen, subject to the above condition,
then this limit is said to define the improper upper integral of f (x, y) over the
unbounded domain G, and it may be denoted by  f f(x, y)(dxdy). A similar
statement applies to the improper lower integral f  f(x, y) (dx dy).
G
* Hobson, Proc. Lond. Math. Soc. ser. 2, vol. iv, p. 145.
t See Jordan's Cours d'AAalyse, vol. II, p. 81; also Stolz's Grundziige, vol. ii, p. 148.




322, 323]   Double integrals over infinite domains


443


When the improper upper and lower integrals both exist, and have the
same value, then this value defines the improper integral
f/(x, y) (dxdy)
of f(x, y) over the domain G.
The necessary and sufficient condition for the existence of
Gf f(  y)(dxdy)
is that, corresponding to each arbitrarily chosen number e, a number p can be
determined, such that for every bounded connex closed domain A contained in G,
and itself containing no points of distance from the origin < p, the inequality
if (x,y)(dxdy)  <e
be satisfied.
Let us first assume that the condition stated is fulfilled. Let D, D' be
two domains each containing as a part all those points of G of which the distance from the origin is < p, and let E be the domain common to D and D'.
Let D -E = A, D' - E= A'; then all points of A, A' are at distances
> p from the origin. We have then
f /(x, y<)(dxdy) <E, fy) <        (ddy) <e;
and since A - A' = D - D', we have
I  f (x y) (dxdy) -  f (x, y) (dxdy) < 2e;
and the condition for the existence of
f (    y) (dxdy)
is therefore satisfied.
To shew that the condition stated in the theorem is a necessary one,
let us assume that the condition is not satisfied. Then, for every pair of values
of e and p, there exists a domain A, all the points of which are of distance
p from the origin, such that
f f(x,y)(ddy) d.y
Let E be a bounded domain contained in G, and itself containing every
point of which the distance from the origin is < p; and let E contain A. Let




444                            Integration                          [CH. v
E-A= E; then E1 also contains every point of G whose distance from
the origin is <p.  Then
|    f(x, y) (dxdy) -  f (x, y) (dxdy) i  e.
It follows that, however great p may be, there exist in G two domains,
each containing all points of G of distance from the origin < p, such that for
them this last condition is satisfied; in this case the improper upper integral
through G cannot exist.    It has therefore been shewn that the condition
stated in the theorem is a necessary one.
The necessary and sufficient condition that the upper and lower integrals of
f (x, y) through the infinite domain G may both exist is that the improper
upper integral of I f(x, y) I over G may exist.
The proof of this theorem is a repetition of the proof of the corresponding
theorem in ~ 319; the only difference being that, in the present case, D is
taken to be a connex closed domain contained in G, every point of which is at
a distance from the origin _ p. The indefinite increase of p corresponds to
the indefinite diminution of m (D) in the former case.
324. It now follows, as in ~ 320, that, if f(x, y) be a function which has
an improper integral over the infinite domain G, in accordance with Jordan's
definition, then I f (x, y) ] has also such an improper integral, so that every
improper integral is absolutely convergent.
In order to obtain a definition in accordance with which non-absolutely
convergent improper integrals over an infinite domain may exist, some
restriction must be imposed upon the nature of the domains D1, D,...
which are employed in the definition in ~ 323. For example, we may
restrict D1, D2,..., as in the extended definition of Harnack, which has
been given in ~ 320.
EXAMPLES.
1*. The integral
sin (ax +by) xr'-ls-1  (dx dy),
where O<r<l, 0<s<l, taken over the positive quadrant, has no existence as an
absolutely convergent improper integral. We find that the integral taken over the rectangle bounded by x-= 0, x=h, y =O, y=k tends to the limit a-rb- r (r)r(s) sin (r+s) rr,
as h and k are increased indefinitely. If the integral be taken over the domain x >0,
y>O, ax+ by < A, then when h is indefinitely increased, the integral has no limit if
1 < r+s <; but it tends to the same limit as before, when r +s < 1. The integral may
be regarded as conditionally convergent, if we adopt a definition in accordance with which
it is sufficient that the integral taken through the rectangle x=O, x=h, y=O, y=k have a
definite double limit, as h and k are indefinitely increased.
* Hardy, Messenger of Math., vol. xxxII, p. 96.




323-325]


Transformation of double integrals


445


2*. The integrals
cos (a2 + 2A  + by2) (dxdy),  sin (a2 + 2hxy+ by2) (d dy)
where a, ab - h2 are positive, taken over the positive quadrant, do not exist as absolutely
convergent integrals. It may be seen that, if the integrals are taken over the quadrant
of a circle bounded by r =R, the value of the integral has no definite limit as R is
increased indefinitely. If the integral be taken over the rectangle bounded by x=0, x=h',
y=0, y=k', then, when h' and k' are increased indefinitely, the integrals have
1           A
0, and         cos-l1
2 ^ab-h2       fab
for limits respectively, the inverse cosine having its least positive value. These may be
regarded as the values of the integrals, subject to a suitable restriction on the domains
of which the positive quadrant is the limit.
If a=0, b=O, h=~,
sin xy (dxdy)
over the positive quadrant, has no existence, even considered as the limit of an integral
over the rectangle. But
os xy (dxdy)
exists and is equal to 2r, when the integral is defined as the limit of the integral over the
finite rectangle. It may be remarked that the single integrals
ecos ydb,     f sinxydy
are both divergent.
THE TRANSFORMATION OF DOUBLE INTEGRALS.
325. Let (x, y) be a point of a limited perfect and connex domain H, and
let x and y be expressed by means of two functions fi, f2 in terms of two
new variables A, V, which may be represented by points (I, v) in another plane.
Let us suppose that the functions
= fA (f, v),   y = f2 (, ),
and the reciprocal functions
k =    (S (X, y),  vq =  2 (x, y),
are such that the following conditions are satisfied:(1)  To each point (x, y) there corresponds one point (:, a); and, conversely, to each point (4, v) there corresponds one point (x, y); and to the
limited domain H there corresponds a limited domain H.
(2)  The functions fi (, r), f2(f, r) are continuous functions of (4, V)
throughout the domain H.
* Hardy, Messenger of Math., vol. xxxII, p. 159.




446


Integration


[CH. V


(3) The functions f(:, ), f2 (, v) have, at every point (:, 7) of H,
definite partial differential coefficients with respect to: and 7, and each one
of these is everywhere continuous with respect to (:, a), and nowhere vanishes.
(4) The Jacobian of f/ (I,?), wf (t, v) with respect to | and v does not
vanish in the domain H.  In virtue of (3) the Jacobian is everywhere continuous, and of fixed sign.
From (2) and (3) it follows that, if (I + A:, v + An), (I, v) be two points
of H, and (x + Ax, y + Ay), (x, y) the corresponding points of H, then
A    = ({  )01)A(+ ia  + 02) Az,
(Daf2 ~0
Ay = (   +(2 + 04) A,
where 01, 02, 03, 04 converge to zero as aL, AX do so, and (see ~ 237)
uniformly for all points (~, 7) in any closed domain contained in H. On
solving these equations, we find
Qf2+ o)    -(A    + 02) Ay
J+ax
with a similar expression for Aql, where J denotes the Jacobian
a (f, f2)
a(t,?) '
and ac is a function of 0,, 02, 0s, 04 which converges with them  to zero.
Since, by (4), J never vanishes, it follows from these equations that A/, Av
converge to zero with Ax, Ay, and thus that the functions  1 (x, y), b2 (x, y)
are both continuous functions.
The partial differential coefficients
a01 aif2 /    b01   a02   a02 2
ax     lJ    ay'    ax'   ay
are also continuous in H; and therefore
A=       + XI Ax+ a    + X2 Ay,
=(Yax)+ 3AX+ (ay+         4 ^Ay,
where %X, X2, X3, X4 converge to zero with Ax, Ay, and uniformly so for all
points (x, y) in H.
Corresponding to any closed set h, of zero content, contained in H, there
is a closed set h, of zero content, contained in H. It is clear, from the continuity of the functions which define the transformation, that a limiting




-325, 326]


Transformation of double integrals


447;point of- a sequence of points in H corresponds to the limiting point of the
corresponding sequence in H; and thus h is closed, since h is so.
Writing      Al = LAx + MAy,   Ax = L'Ax + M'Ay,
it follows, since
(a)2   (a)2       L'~2 +M2 + ~M'2 + 2 LM+ L'M' I
((ax)2 + (Ay)2 -
where L, L',... converge uniformly to
aS,   a     **'
ax ' ax '    '"
that, if i Ax |, I y be both restricted to be less than a fixed positive number
e, the ratio
(AA)2 + (7X)2
(AX)2 + (Ay)2
has a finite upper limit A2, for the whole domain H. Now let the points of
h be enclosed in a finite number of circles, the radii of which are all < e;
it then follows that the points of h can be -enclosed in a finite number of
circles of which the radii are all less than eA. The sum of the areas
of these circles on the (I, v) plane, which contain in their interiors all the
points of h, has to the sum of the areas of the circles on the (x, y) plane,
which enclose all the points of h, a ratio less than A2. Since the sum of the
latter circles can be taken to be arbitrarily small, it follows that the points of
can all be enclosed in a finite number of circles the sum of whose areas is
arbitrarily small. Therefore h has the content zero.
326. Let f(x, y) be a limited function, defined for all points of a closed
connex domain G contained in H, the frontier of G having content zero;
and let f(x, y) be integrable in G. If x, y be expressed in terms of I, 7 by
the relations
= fi (,,   y =f2 (f  ),
which satisfy the conditions of ~ 325, then, corresponding to f(x, y) in G, we
have a function F(:,?) in the domain G, contained in H, which corresponds
to G. The frontier of G, corresponding to the frontier of G, has also the
content zero. A point of discontinuity of f(x, y) in the (x, y) plane corresponds to a point of discontinuity of F (j,?) in the (~, q) plane, the measures
of discontinuity at the corresponding points being the same. Since those points
of (x, y) at which the saltus off (x, y) is _ kc form a closed set of zero content,
it follows that the points of (~, V) at which the saltus of F (j, r) is > k form
also a closed set of zero content; and therefore F (:, r) is integrable in G.
In order to transform  f (x, y) (dxdy), taken throughout G, into an integral
taken throughout G, it is convenient to make use of an intermediate transfor



448


Integration


[CH. V


mation* x= f (ui, u2), y = u2, followed by the transformation u =, u2 =f2 (2, V);
the function J (ui, u2) being such that
(U1~, U2)=fi(,  ).
It is easy to see that each of these transformations satisfies the conditions
of ~ 325.
Since f(x, y) is integrable in G, we may, in accordance with the result of
~ 314, replace the double integral
f(x, y)(dxdy)
by the repeated integral     y f (x, y) dx,
or by                      dy   f (x, y) dx.
Applying the transformation x =  (.u, u12) to the upper and lower integrals
(x, y) dx,      f(x, y) dx,
these may, in virtue of the theorem of ~ 306, be transformed into the single
upper and lower integrals,
(,, I 2) a du,,    (u1, U2) u du,,
where G (u1, u2) represents the function of ul, u2 which corresponds to
f(x, y). We thus have
f (x, y) (dxdy) = Jdu2 J(U, ) ax dfa,
au,
= j2du2 f (u1, u2) ax du,
r      ax
(U= j(, U2)  (du du2),
the double integral being taken through the domain in the plane (ui, u2),
which corresponds to G in (x, y).
ax
Since -  is the Jacobian of (x, y) with respect to (a1, u2), we have
am1
by a known theorem
-ax   a (ul, I2).,    am (a, ) 
* This method is employed in the general case of multiple integrals by Pierpont; see his
paper " On multiple integrals," Trans. Amer. Math. Soc., vol. vi, p. 432. It is, however, there
assumed that f (x, y) is integrable with respect to x for each value of y: but this is unnecessary.




326, 327]    Transformation of double integrals


449


hence, since J never vanishes,
ax and   a (', u,)
am,      a (I, y)
also never vanish.
Applying the same method of transformation to
a,x
1 (ui, 62z) au (dzl duz),
where                   u1=, u2 = f2 (, r),
we have        jf (x y) (dxdy)  F(       ) a  2 (dda),
au, a (ua, u  a)
where                     au2   (, 2)
ST}  a (~, v)
hence finally we obtain the formula
ff(x, y) (dxdy) =  (E,,) J(dade);
which is the formula of transformation of the integral of f (x, y) throughout
G into an integral through G.
It has been assumed that J has a fixed sign throughout the domain of
integration. If now this sign be negative, the product A:An, in JAfAn,
which corresponds to Ax Ay in the plane of (x, y), must be accounted negative,
when AxAy is positive. It is however more convenient to consider A~An
as essentially positive, otherwise the measure of a set of points in the
(I, r) plane would have to be reckoned as negative. Adopting this convention, we write IJI AAr instead of Jd:dr; and therefore the formula
of transformation will be written in the form
jf(x y)(dxdy) =  F(   I J I (ldn).
327. Let us now assume that at certain points of G, which form a set L of
zero content, either (1) f(x, y) has an infinite discontinuity, or (2) one or
more of the partial differential coefficients
afi  af, ai     af2
a8     arV  Do) Ar
does not exist, or is discontinuous, or (3) the Jacobian J vanishes. In case
J be positive over a part of G, and negative over another part, it is convenient
to divide the double integral into two portions, taken over these two parts
of G respectively, and to transform these two portions separately. It will
accordingly be assumed that J never actually changes its sign in the domain
G, although it may vanish at the points of the part L of G. We may


29


H.




450


Integration


[CH. V


denote by L the set of points on the (I, r) plane which corresponds to L: it
will be assumed that L has zero content. It will be shewn that, if one of the
two integrals
j (x, y) (ddy),
taken over G, and       jF(, v) J (d:daj),
taken over G, exists as an absolutely convergent improper integral, or as a
proper integral, then the other one exists, and the two have the same value.
Let us assume that ff(x, y) (dxdy) exists: it will then be sufficient, in
order to establish the existence of the other integral, and its equality with
the first, to shew that, for any domain G1 contained in G and itself containing no points of L either in its interior or on its frontier (which frontier
is to be taken to be of zero measure), the condition
f (x, y)(dxdy)-fF (a, ) IJ (ddn) < 
is satisfied, provided that mn (G) - m (G,) be less than some fixed finite number
dependent on a.
A domain g interior to G, and containing in its interior and on its frontier
no points of L, can be found such that
f (f, y) (dx dy) -  f (x, y) (dx dy) < e.
If h be any domain contained in g, such that m (g)- m (h) is sufficiently
small, we have
/f (x, y) (dxdy) -  f (x, y) (dxdy) < e;
and therefore      f (x, y) (dx dy) - ff(x, y) (dxdy) < 2e.
Now let k be a domain interior to G, containing in its interior and on its
frontier no points of L, and containing h, then
( / (, y) (d dy)-  f (x, y) (dx dy) < 2.
For let p denote the domain obtained by taking the two domains g and
k together, then
f j(, y) (dxdy) -  f(x, y) (dx dy) < e,
and                /(x, y) (dxdy) - f  (x, y) (dxdy) < e,
and by combining these inequalities the result follows.




327, 328]   Transformation of double integrals


451


We have now
| F         l, \J\(dd) = ( FW) )JI(dd)-f F(      D)IJ (dpdt/,
where U is the domain formed by taking all points which belong to one
or both of the domains 0G and h; and V consists of those points which
belong to h but not to G,.
Now U corresponds to a domain in the (x, y) plane which contains h, and
which domain may be taken to be identical with k; therefore
|f F(B  ) I J (dad) differs from  f(x, y) (dxdy)
by a number numerically less than 2e. Again
|  F(,) J (dada) < L {m (G)- m       } (G)}
where /u is the upper limit of F(, q) J I in the domain V obtained by
removing from h those points which belong to G1.
We thus have
|f /(x, y) (dxdy) -f F(  11) I J (dd) |< 2e + At {m (G) - m (G,)j.
Now let e be fixed so that it is < ~r, then h is fixed, so that /u cannot
exceed a fixed finite number 1. If then 0G be so chosen that
in (G)- i (G1) <,
the inequality
fG (x, y) (dxdy) -  F(, j) \J (dada) -  < 
will be satisfied. Therefore it follows that
I F(, v) JI (dads)
exists, and is equal to
f (x, y) (dxdy).
328. This method of transformation may be extended to the case in
which one of the domains G, U is infinite, or to the case in which both are
infinite. It can be shewn that, if either of the integrals
f (, y) (dxdy),     |F (, ) I J I (dads)
exist, the definition of ~ 323 being applied when G or G is infinite, then the
29-2




452


Integration


[cH. V


other integral also exists, and the two integrals are equal. The proof can be
given by slightly modifying the procedure of ~ 327.
There may be a set of points of zero content, in the domain of G, such
that the corresponding values of ~, sv are infinite, or such that one of them
is infinite. This set now takes the place of the set L. Whether G be finite
or infinite, the finite, or infinite, domain h contained in G may be so fixed as
to exclude all points which correspond to infinite values of 4 or q. The
domain k including h, and containing no points which correspond to infinite
values of k and qr, may then be fixed as before, and will satisfy the condition
f /(x y) (dx dy) -  f(,) y  ) (dx dy) < 2,
it being assumed that the integral of f (x, y) over G exists.
The finite domain h contains all points of G of which the distance from
the origin is less than some number R depending on the domain G - h, which
contains in its interior all points (x, y) that correspond to infinite values
of 4 or r or of both.  The same statement holds for k, which contains h..
When the finite domain G1 is such that the condition
jF(t, ) I JI (dd)< 
is satisfied (and, in order that this may be the case, G1 must certainly contain
all points of G whose distance from the origin is less than some fixed
number R, _ R), we have as before
f    f y ( ddx,)  -(ddy)- f  F (, ) l Jl (dd) <;
and as 7 is an arbitrarily fixed number, we thus see that
F(~ /) IIJI (d dn)
exists and is equal to
ff (, y) (dxdy).




CHAPTER VI.
FUNCTIONS DEFINED BY SEQUENCES.
329. LET us suppose that u,, u 3, u3,... un,... is an unending sequence of
numbers, so that u, has for each value of n a definite numerical value,
assigned by means of a prescribed rule or set of rules. Let the sums u,,
u1 + u2, u, + u2 + u3,... U +  2 + *.. + Un,,... be denoted by s 2, s2, s,... s,,...,
and let us consider the aggregate (s,, s2,..., sn... ). If this aggregate be a
convergent one, in the sense described in ~ 28, it has a limit s, or s, which
is said to be the limiting sum of the infinite series u1 + u2 +... + un +...
in which case the series is said to be convergent.
The condition that the sequence (si, 82, 83,... Sn,...) be convergent is that,
corresponding to each arbitrarily chosen positive number e, a value of n can
be found such that Sn+m - n I < e, for m = 1, 2, 3,.... This is then the condition that the infinite series u. + u2 ++... + u+... may be convergent.
The difference s+m - sn  Un+i + Un+ +... + un+m is called a partial
remainder of the infinite series, and may be denoted by Rn,m. Thus the condition of convergence of the infinite series may be stated in the form, that,
corresponding to each arbitrarily chosen positive number e, a value of n can
be found such that all the partial remainders Rn,i, Rn2,... Rn,m,... are
numerically less than e.
Since Rni_, 1- u, it is seen to be a necessary, but not a sufficient, condition for the convergence of the series, that I tn be arbitrarily small when n
is sufficiently great; this condition may be written in the form lin un = 0.
I = 00
If the series uq + u2 +... + Un +... be convergent, then, for any value of n,
the series un+, + Un+2 +... is also convergent, and has, in the sense defined
above, a limiting sum which may be denoted by Rn. This limit is called the
remainder after n terms of the original convergent series; thus s = Sn + RnIt is clear that, the given series being convergent, the sequence
R,, R2,... Rn,... is also convergent, and that its limit is zero. That this may
be the case has sometimes been given as the necessary and sufficient condition
for the convergence of the given series; such a statement of the condition is,
however, circular, because the existence of the numbers Rn cannot be assumed
unless the given series is already known to be convergent.




454


Functions defined by sequences


[CH. VI


It is important to observe that the number s has not been defined as the
sum of the infinite series uzl + u2 +... + un +...; for that would have implied
the completion of an indefinite series of operations of addition: but, conversely,
the limiting sum, or simply the sum, of the infinite series has been defined to
be that number s which was itself defined, as in ~ 28, by means of a convergent sequence.
330. If all the terms of the series u1 + u2 +... + UL +... be positive, the
numbers of the sequenrce s&, 2,... n,... continually increase; and hence it is a
sufficient condition for the convergence of the series that a number K exist
such that Sn <K for every value of n; for the numbers 1, s2,... Sn,... then
have an upper limit s.
If the terms of, the series u, + 'u2 + t3 +... + un +... be of alternate signs,
or if this be the case from and after some fixed term, then the necessary and
sufficient conditions for the convergence of the series are that un, should
continually diminish as n is increased, and that lim un = 0. For, in this case
n= oo
Sn+m- Sn \ < [un+ 1, for every value of m, from and after some fixed value of
n, and  Un+ I is arbitrarily small, if a sufficiently great value of n be chosen;
hence the sequence sl, s,... s,... is convergent.
If the series u U + u2 -...+ Un +... be convergent, and a,, a2,... an,... be a
sequence of numbers which, from and after some particular value of n, are all
positive, and do not increase as n is increased, then the series
a1ul - C+a2U2 +... + antun +..
is also convergent.
In proving this theorem, it is clear that, without loss of generality, we
may suppose all the numbers a, a2,... to be positive and not increasing,
since we need only remove a definite number of terms from each series to
reduce the general case to this one. We have
an+1,n+1   + an+2 n+22 +.. + an+muqn+n
= an+, R  + (a,+2 - an+i) R+, +... + (an+- - an+m-1) Rn+m-i -- a2+mnt Rn t,.
Since the series Zu is convergent, we can, corresponding to an arbitrarily
chosen e, find n such that Rn, Rn+1, Rn+2,... are all numerically less than e;
also a,+-2- an+i, an+ 3-an+2,... are all of the same sign, therefore
an+lun+l + a2 +2 u+2 +... + an+nu+m2n+   I < a,+1e +I an+m - an+1 6 + an+me
< 2an+le < 2aie;
and thus, from and after a large enough value of n, all the partial remainders
of the series lau are arbitrarily small, and therefore the series is convergent.
It is clear from the preceding proof that the theorem also holds if, from
and after some fixed index n, the numbers a,, a2, as,... do not diminish, but
be such that all of them are less than some fixed number.




329-331]


Arithmetic series


455


NON-CONVERGENT ARITHMETIC SERIES.
331. The partial sums s, s2,... n,... of a series U, +2 +... + u+...
may be represented in the usual manner by an enumerable set of points G, on
a straight line. The following cases may arise:(1) The set G may all lie between two fixed points A, B, and the derivative G' may consist of a single point s; in this case the series is convergent,
and s is the limiting sum.
(2) There may not exist any two points A, B between which all the
points of G lie, and G' also may not exist; in this case I s,n has no upper
limit, and the series is said to be divergent.
A divergent series may be such that, from and after some fixed term, all
the partial sums are of the same sign and increase without limit. An example
of a divergent series of this kind is the series
1         1  1  1
i +2+     1++...+-+....
A divergent series may be such that, although I s I increases without
limit, there are an unlimited number of positive elements s,, and also an
unlimited number of negative elements sn. An example of this class of
divergent series is the series 1-2 + 3 - 4+... +(2n- 1) - 2n +..., for
which s2n- = n, s2n= - n.
(3) The set G may consist of points all lying between two fixed points
A and B, and the derivative G' may consist of more than one point; in this
case the series is said to be an oscillating series. The set G' may contain
a finite, or an infinite, number of points, but it must be a closed set; it consequently has an upper boundary U and a lower boundary L; and these
boundaries U, L are called* the limits of indeterminacy of the series. It
is always possible to find a sequence (s,, sn, sn,,... ) of partial sums, where
nI < n2 < n,..., which converges to the point U, and another such sequence
which converges to L, or to any point of G' which may be chosen. It thus
appears that, by introducing a suitable system of bracketing, according to
some norm, the terms of an oscillating series, the series may be converted into
a convergent one of which the limiting sum is any chosen point of G', including either limit of indeterminacy. The set G' may be non-dense in the
interval (L, U), or it may consist of all the points of that interval, or it may
consist of a closed set of the most general type, as described in ~ 86.
The series 1-1 + 1-1 + 1... has 1 and 0, for the upper and lower limits
of indeterminacy; and G' may be regarded as consisting of these two points.


* This term is due to Du Bois Reymond: see his Antrittsprogramm, p. 3.




-  &  -   -  -


I

456              Functions defined by sequences              LCH. VI
1      1       1      2      1      2      1
Let   S-=     2 -, s3 -, 4,, S -,     7 -.
2,=   Ss~=, s4       s      s~=$,  s7=,
and generally
1                 2              m+l
Sm(n+i)+i 2m + 2' Sm(, +l)+2= 2m. S (m 1)2   + 2
1                 2                 Mn+l
s(m+l)2+l =m +3 '  s(+l)2+2 2m  + 2 ' S ( + +1)2  ) (m+2)  n + 3.
It follows that the series
1    1    1      5    7
2 3   -...
2   2.3   3.4   3.4  3.5
has 1 and 0 for the upper and lower limits of indeterminacy. The set G
consists of all the rational numbers between 0 and 1; so that G' consists of
the whole interval (0, 1). By introducing a properly chosen system of
brackets, the series may be converted into one converging to a limiting sum
which is any prescribed number of the interval (0, 1).
(4) The derivative G' may exist, but one or both of the points A, B may
be absent; in this case also the series is said to be an oscillating series. If the
points s, have no upper boundary, then the upper limit of indeterminacy
is said to be + oo; and if they have no lower boundary, the lower limit of
indeterminacy is said to be - e. In this case the series may be made to
diverge, by introducing a properly chosen system of brackets, or, on the other
hand, it may be made to converge to any point of G'.
It should be observed that oscillating series are frequently included in the
term divergent series.
For example, a series* may be constructed which oscillates between
infinite limits of indeterminacy, but which, by introducing a suitable system
of brackets in accordance with a norm, may be made to converge to any
prescribed number whatever.
2x - 1
If x'=, where the positive sign is ascribed to the radical, the
V/x(1 -x)
points x of the interval (0, 1) have a (1, 1) correspondence with the points x'
of the unlimited straight line (- c, o ). It is clear that a set of points {x}
in the interval (0, 1) corresponds to a set {x'} in (- c,,o), the relation of
order being conserved in the correspondence.  Further, a limiting point
of the one set corresponds to a limiting point of the other set. The rational
points of the interval (0, 1) of x correspond to a set of points x' everywheredense in (- oo, o ). This method of transformation may be applied to the
series obtained in (3), which oscillates between the limits of indeterminacy
0, 1, and which can be made, by introducing suitable brackets, to converge to
any prescribed number in the interval (0, 1).
* See Hobson, Proc. Lond. Math. Soc., ser. 2, vol. III, p. 50.




331, 332]                Arithmetic series                       457
We find that
Si/1             2         1         3
Ss=O, s7=-i,         8-6       s      s5  ^""v^
and generally
42m         1           2   -3
+i ) +1 —5 /(2m6' + 1'  m(n+]) +2  V{2 (2m- )},      m,           2m +1,m                     2m - 1
Therefore the series
I     I   ~2. +   2_+ 1I'3       I
V/2    /2   3)    \3 +\/ 2    2    \2 +2   V 5
has the required character: it may be made to converge to any assigned
number whatever, by suitably bracketing the terms together in accordance
with a norm, and amalgamating the terms in each bracket.
ABSOLUTELY CONVERGENT, AND CONDITIONALLY CONVERGENT, SERIES.
332. Let us suppose the terms of a convergent arithmetical series to be
all positive. The order of the terms in the series is defined by the norm
which defines the series.
If now a new series be defined by another norm, and be such that any
assigned term in either series is identical with a definite term in the other
series, then the new series is said to be obtained by rearranging the terms
of the original series; and the two series are conventionally regarded as
identical with one another. It can be shewn that the new series converges to
the sum of the original series.
Let (si,, 2... Sn,...), (8s '/, s,... S'w,...) be the aggregates of partial sums
of the two series, and let s be the sum of the given series defined by the first
of these aggregates. If e be an arbitrarily chosen positive number, n can be
determined so that s - s, < e; and then, since all the terms of the series are
positive, it follows that s - sn+n < e, for m = 1, 2, 3,....
A number n' can be determined such that the first n terms of the first
series all occur in the first n' terms of the second series; and therefore
Sn< S'n'  Again a number n" can be determined so that the first n' terms
of the second series all occur in the first n" terms of the first series; then we
have Sn < s'n' < S,. Since s - s, s - Sn,, are both < e, it follows that s - s', < e;




458


Functions defined by sequences


[CH. VI


and this clearly holds for all values of n' greater than the one employed.
Since e is arbitrary, it has thus been shewn that the second aggregate of
partial sums converges to s; and therefore the sum of a convergent series
whose terms have all the same sign is unaltered, if the order of the terms be
altered in accordance with some nornm, such that each term of the original
series has a definite place in the new series.
333. Next let us suppose the arithmetical series u, + u2 + u3 +... to have
both positive and negative terms, each indefinitely great in number. Let the
positive terms, in the order in which they occur, be al, a2, a3,... and the
negative terms, in the order in which they occur, be - bl, - b2, - b,...; and
consider the two series
a +   a2+   a  +...,.............................(1)
b, +   b2 +   b3 +..................................(2)
Denoting by on, Cao' the sums of mn and m' terms respectively of these series,
we see that, if in the first n terms of the given series there are m positive and
m' negative terms, then
S     rn = fm - 0' n?' 
If both the series (1), (2) be convergent, then ar,, Ca',l have finite limits,
and the given series is itself convergent, its sum being independent of the
arrangement of the positive and negative terms, provided only that each term
of the original series occurs in the series obtained by the rearrangement of
the order of the terms.
The series U1 +- u2 + 3 +... is said to be absolutely convergent, provided
| 1I + I u2 + u31 +... be convergent, which is the case when both the series
(1), (2) formed respectively by the positive and by the negative terms, are
convergent.  The sum of an absolutely convergent series is thus unaltered by
a rearrangement of the terms, in accordance with some norm.
If one of the two series (1), (2) be convergent, and the other be not convergent, the given series is not convergent.
If both the series (1) and (2) be divergent, it may happen that the given
series itself is convergent; in this case oM - aM' has a definite limit, although
Crm, C r' have no limits.
If the series u1 + u2 +... be convergent, whilst the series l 1 + u1 +... is
not convergent, then the given series is said to be conditionally* convergent.
It will be seen that the order of the terms in a conditionally convergent
series cannot in general be altered without affecting the sum of the series, or
of possibly rendering it no longer convergent.


* By Stokes the term accidentally convergent is used; by many writers such series are spoken
of as semi-convergent, but this term is also used in quite another connection.




332-334]


Arithmetic series


459


334. It will now be shewn that the terms of a conditionally convergent
series can always be so rearranged in accordance with a norm, that (1) the new
series is convergent, and has as sum an arbitrarily given number; or (2) so
that the new series is divergent; or (3) that the new series oscillates between
arbitrarily given limits; moreover each of these rearrangements may be made
in an indefinitely great number of ways.
Let kI, k2, Ick,... Ik,... be a sequence of positive numbers, assigned in accordance with any prescribed law, and which are such that no one is less than the
preceding one. Take p, terms of the series (1), so that a,+a2 +.. +ap,
or rP1, is greater than kc, whilst op_- < kl; next take qx terms of the series (2)
such that a + a, +...+a, - b - b -... - bq1, or o, - oql', is less than k,,
whilst a-, - o-'i_,  k.  Next take p, more terms of the series (1) so that
p PI- ig + (ap,+l + ap,+2. +. + ap+p,) is greater than k2, whilst
Pi- - I   + (a)i+1 + a+2 +. + al+, + )  k,:
then take q2 more terms of the series (2), such that
Y'pi -  ql + ('pi+p~ - 'p1) - (aql+,- aq, ) < k,
whilst           - -q, + (op+p2 - 'p) - (a q+q2- - q)  k2*
By proceeding in this manner, we define an arrangement of a number of
terms of the given series, which is such that
(a, + a2 +... + ap) - (b, +.2 +... + bq)
+ (aP,+ +... + aP,+P2) - (bq,+ +... + bql+q2)
+ (aPi+P2+.- +Pn-+ + * * + a. ++ -... +) --  (bqf+...  +l +  + bql+.. +qn) (3)
is less than kIn, whilst, if we leave out the last bracket, the expression is
greater than Ic. It will be observed that none of the numbers p1, p2,.. Pn,
q1, q2,... q,, can be zero. The expression we have obtained differs from k,
by less than bql+,+.. +q.; and if in the expression we leave out the last
bracket, the resulting expression differs from kn by less than ap,+P2+ +,,;
also, since the given series is convergent, both the numbers ap,+p2+...+^,
bq+q2+..+qn are as small as we please, if n be taken sufficiently large. Now
suppose (kI,, k... kI,...) to be a convergent sequence which defines the
number k, arbitrarily given, then, taking n large enough, k,, kn+,... all
differ from k by less than an arbitrarily small number; hence a number n can
be found such that, for it and for all succeeding integers, the series we have
found differs by an arbitrarily small number from kl, even when we suppress
the last bracket. We have thus assigned, by means of a norm, an arrangement of the terms of the given series so that the new series, so defined,
converges to the arbitrarily given number k; and it is clear that this may
be done in an indefinitely great number of ways, since there are an indefinitely great number of convergent sequences which define the same
number k.




460


Functions defined by sequences


[OH. VI


Next suppose the sequence k,,... kn,... increases without limit; then
we have defined an arrangement of the terms of the given series which makes
the new series divergent. It is clear that the same method would have been
applicable if we had taken the numbers k1, k,,... all negative and numerically
increasing.
It remains to be shewn that the terms of the series can be arranged so
that the sum of the new series oscillates between two arbitrarily given numbers
k, ' (k > k'). We may suppose without loss of generality that k, k' are both
positive.
Let (kl, k2,... kn...) be an aggregate of increasing numbers which defines
the number k, and (k,', /... I'...) another such aggregate which defines I';
where we may suppose that, for all values of n, kn > kn'.
Choose p, so that a-p, >ki whilst ap_,ki 1; next choose q, so that
p, - aq,' < ki, whilst op, -    kC-'q -i Ic'; then choose P2 so that
UOP- aq' + (P1+P2- ~P1) > c2,
whilst               Op, - %  + (op,+p2 - -pl) < k2;
then choose q2 so that
OP — o'q1 + (ol+P2 - oI)- (aqi+g- 2 - oq) < I2',
whilst       a-p, - a-qq + (-p+p,2 - op) - (o-aq+qi - (rq/l) > k2.
Proceeding in this manner, we define a series of the form (3) whose sum
is < kn', but differs from kIc by less than bq,+q2...+qn; and moreover, if the last
bracket be suppressed, the sum is then > kn, but differs from kn by less than
aPl+p2+.+pn. It is now clear that an arrangement of the terms of the given
series has been assigned, such that the resulting series oscillates between k and
k'; and this can, as before, be done in an indefinitely great number of ways.
A special case of this general theorem as to the nature of the new
series obtained by rearranging the terms of a conditionally convergent series
in accordance with some norm, is that in which a,= b, a = b2,... a = bn...;
so that the original series is a, - a, + a2 - a2 + a3 - a3 +..., which, provided
lim an = 0, has zero for its sum.  It thus appears that, from the terms of a
n=00
divergent series
a, +    a2 +... + a +...,
which is such that an is arbitrarily small when n is taken large enough, we
can, in an indefinitely great number of ways, construct series which are convergent and have a given sum, are divergent, or oscillate between given
limits, by taking the various series of the form
a, + a2 +...+ ap, -  a, - a2   -a aq~ + a,,+l + apl+2 +... + aP+p2
- aq+l - aq1+2 -  -... - + aq,,+2 + 
the numbers P1, p2,...ql, q2,... being assigned in the manner explained
above.




334, 335]               Arithmetic series                      461
EXAMPLES.
1. The series
I       1  I 1     1
1 ---+ ---+    +. —+ 
12 +3 4      2n-1  2n'
is conditionally convergent, its limiting sum being loge 2. The series
1 1 11 1
1+ — - + - -...,
3-2 5 7 4
which is obtained by systematically rearranging the terms of the first series, converges to
loge 2.
For we find that
t=h / (1     1     1     1\
S4 =:  4m-3 4m -2  4m-1 41 m
and that, for the second series
lit=t  /          1 \
S'3n =    =        1 An)
n==I" V4m- -3 +4m - 1 2m 
m=n/ 1     1 \ 
therefore          S'3n-4 - =l 4n- 2 4,) = 82n
Since s4,, sS2 both converge to loge 2, as n is indefinitely increased, it follows that s'3,,
3
converges to 2loge 2.
2. By * rearranging the terms of the convergent series
1        1 1 1  I  I
+-+-       +1 -1+. +- -+...
2 2   3 3      n n
we obtain the series
1     11111           1      1        1 1 1 1 1
21 + 3          + 4 2 5  3  n-1 2n- I +      +2n n+
l+-+ ---1+-+-+ ----+...-       +     +      +
2 3     4 5      2     n     3n -  3n- 1  3n n
1      1         1          11
-- + (- + (- +-)  +  "  + -      +....
n-l1(n-1)2+1 (n-1)2+2     f -+2 -The first of these new series converges to loge 2, the second to log 3, and the third
diverges.
SERIES OF TRANSFINITE TYPE.
335.  If       S1, S, s,... SS,,... SY, S+1,... Sy,...
be a set of numbers each one of which is definite, and in which every index
that is less than, or equal to, some number 8/ of the second class, occurs as a
suffix, and if the series
ul+   +...I + Un... +. +  U,+1 +...  +...
* Dini's Grundlagen, p. 133.




462


Functions defined by sequences


[CH. VI


be formed, where
U1 = S-, 812 = 82-S-,.-.. -, = Sn - S-_1... Uo= So+1 - S.,
t(o+l = So+2- Swol ~...y = S+i - y,...
in which the indices of u, include every number less than /, then the series is
said to be a convergent series* of type /. If 3 be a limiting number, the
series has no last term, but if / be a non-limiting number, the last term of the
series is
UP_ = So - sp-1.
An ordinary infinite series
%1 + 42 +... -+ 3 +   +...
is of type o. A series
U +2 +            -.  + VIq  +.....+ v V  + +.+ v, +...
is of type to2; and a double series.Z Crs,
'=1 s=1
a,, - al2 + a13.+.. + a,,,...
+ a21    + a     +    a22  +... + a2n +  *. 
3+      a32 + a 2 +................... o.........
+ anm - ac2 + an +-...........,...........
*,,.,....,........
is of type w2, if it be taken in columns successively, or in rows successively;
but it is of type to if the terms are taken diagonally in the form
all + (a12+ a)+ (a2)   +  3 22 + a31) +.
Conversely, a series of any type / is convergent if all the sums
S1, S2,... Sl,... So,... Sf
be definite numbers.
It is clear that, /3 being any given number of the second class, any series
of the ordinary type w can have its terms so arranged that the series becomes
of type /. For a correspondence can be defined between all the ordinal
numbers up to, and including /3, and the ordinal numbers of the first class.
Let us now suppose that all the terms of a convergent series
U1l+  2+   + U2  +  (+     + +1  + U.  +...
of type /3, are positive, and thus that
S, < 82 < s,... < S,,... < S3.
If we represent the numbers
S1, 8 9,... S,,... So
* Such series have been investigated in a different manner by Hardy, Proc. Lond. Math. Soc.,
ser. 2, vol. i.




Arithmetic series of transfinite type


463


in the usual manner, by points on a straight line, the terms of the series
are represented by a set of intervals
(0,, (s1  2),... (s   s+1)...
on the straight line; each interval abuts on the next; and all the points sa
where a is a limiting number of the second class, are semi-external points of
the set of intervals. The end-points and the semi-external points of the set
of intervals form an enumerable closed set which has consequently zero
content; and it follows, from the theory of the measures of sets of points, that
the set of intervals has a measure equal to that of the whole interval (0, sp),
which is therefore sp. Since the measure of an infinite sequence of intervals
is equal to the sum of the measures of the intervals, it follows that, if the
intervals be arranged in a sequence of type co, their sum is s3. The following
theorem has thus been established:If a series of positive numbers be convergent, and of type /, it will also be
convergent when arranged in type o; also the sums will be the same; and
conversely.
We may pass to the consideration of series of type /, of which the terms
are not necessarily all positive, but of which the convergence is absolute.
An absolutely convergent series of type / is a series which is convergent when
each term is replaced by its modulus.
Let us suppose the intervals constructed as before, which represent the
terms of the series
U z + I 2 + I i.t. +  +t 1I +. + +y I... I
If we choose out from this set of intervals those which correspond to
positive terms of the series
uitu + 2 + * + u(0 +... + ut +...,
we have a set of intervals which has a definite finite measure; and the same
is true of the set of those intervals which correspond to negative terms of the
given series. The given series converges to a sum which is the difference of
the measures of these two sets of intervals, and this sum is unaffected by the
order in which the intervals are taken in either the positive or the negative
component. It has thus been shewn that:If a series be absolutely convergent, and of type /3, then the series is
convergent, and its sum is independent of the type.
An important particular case of this theorem is Cauchy's theorem that an
absolutely convergent double series has the same sum whether the sum be taken
by rows or by columns.




464


Functions defined by sequences


[CH. VI


DOUBLE SEQUENCES AND DOUBLE SERIES.
336. A set of numbers {Smn}, where each of the indices m, n may be any
positive integral number, and the single number sm, is defined, in accordance
with some norm, for each pair of values of m and n, is said to form a double
sequence*.
The numbers of the sequence may be regarded as arranged in rows and
columns, in accordance with the scheme
113,  812,  813,..  81in, *
821,  822  8S23 ).  823)......,,,,,,,.......................................
Sn1, S? )2  5  s 3,,  ** Sn,)...............o................,........................
in which no column and no row has a last constituent.
If, for a given double sequence, a number s exist, such that, corresponding
to each arbitrarily fixed positive number e, the condition I s-smn l<  be
satisfied, for every value of m and n such that nm i p, n _ p, where p is some
fixed integer dependent on e, then the double sequence is said to be convergent,
and the number s is said to be the limit of the double sequence.
This is denoted by
s=   lim   snL,.
m = oo, n= 0o
The theory of double sequences may be correlated with that given in
~ 231, 232, of the double and the repeated limits of a function of two
variables. For, if we assume x= 1/m, y = 1/n, then the number Smn may be
taken to define the value of a function f(x, y) at the point x = l/m, y = 1/n.
That the function f(x, y) is not defined for all positive values of x and y in
the neighbourhood of the point x = 0, y = 0 makes no difference as regards
the validity of the results obtained for a function of two variables. These
results may now be interpreted as properties of the sequence {Sm,,j.
The double limit    lim  f(x, y), when it exists, is identical with
x=O, y=O
lim   s,,,,,, and the existence of the one double limit implies that of the
^1= Wo, It = 
other one.
Corresponding to
lim f(x, y), lif     y, lim  (f(x, y ),
y=0         y=0         y=0
* The theory of double sequences has been treated by Pringsheim, Munch. Sitzungsberichte,
vol. xxviIi, 1898; also in Math. Annalen, vol. LIII. See also a paper by London in Math. Annalen,
vol. LIII, 1900.




336]


Double sequences and double series


465


the notation          li S,      i Smn, lim s.,
o,=00     n,=00     = 00
may be employed to denote the upper limit, the lower limit, or the limit
of the mth row of the sequence; the limit existing when the upper and
lower limits are identical. The notation lim smn may be used to denote the
upper, and the lower, limits, when either is to be taken indifferently. The
corresponding notation
lim Smn,  lim srn  lim smn,   lim sm,
m=oo      g=oo      n=oo      m==oo
is applicable to the nth column of the sequence.
The repeated limits lin lim smn, lim lim smn correspond precisely to
Mn=-c n'=oo,n=0D m=0co
the repeated limits lim lim f(x, y), lim lim f(x, y) respectively.
x=O y=0         y=0 x=O
The following results are obtained from those in ~ 232.
The existence of s=  lim  sm,n implies the existence of the repeated limits
m==o, n=o0
lim lim smn, lim lim Smn, which have both the value s.
m=i oo '=   n= oo m= o
The existence of s is however not a necessary consequence of the existence,
and the equality, of the two repeated limits.
The existence of the repeated limit lim lim smn does not necessarily involve
m=oo -n=-o
that of lim smn, as a definite number; but it implies that
n=oo
lim lim snn, lim limn Sn
m=00 oo ==o m=o nco
have one and the same value.
In case the sequence be such that s,n'n _ Smn, for every value of m', n',
such that m' > m, n' - n, and for every value of m and n, then the sequence
is said to be monotone. It is also said to be monotone in case the relation
sam, < sm is always satisfied.
The following theorem may be easily established:If the sequence {[sn be monotone, then the existence of any one of the three
limits  limn  sn, limn lin smn, lim lint s;U implies the existence of the other
m=00, n=oo   n=oo =o     n=OC o A=C-o
two; and all three are equal.
Let us now assume that all the rows of the sequence converge, i.e. that
lim sm exists for each value of w.
n=00
If lim s,, be, for every value of m, numerically less than some fixed
positive number, and if further, corresponding to any arbitrarily chosen
positiveH.  numberandiffr30
H.




466


Functions defined by sequences


[CH. VI


positive number e, an integer ne, dependent on e, but independent of m, can
be determined, such that sn- lim s m< e, provided n > n_, and for every
value of m, then the convergence of the rows is said to be uniform. A similar
definition applies to the uniform convergence of the columns.
The rows of a double sequence may be uniformly convergent, and yet the
columns need not converge.
From the theory of the limits of a function of two variables, the following
theorem is easily deduced:In order that the double sequence {sn}, of which the rows are known to
converge, may converge, it is necessary and sufficient that the convergence of the
rows be uniform, and that lim lim sa shall exist.
Mn=oo w=00o
The double sequence may however be convergent, without the rows being
convergent.
337. The theorems of ~ 233 and ~ 234 may be employed to obtain the
necessary and sufficient conditions that the two limits
lim  im       i   li s, li  lim
m=co j2'=0    n=oo m=oo
may both exist, and have the same finite value.
We thus obtain the following theorems:In order that the repeated limits lim lim s,,., lim lim sn, may both exist
m==xo =oo    -=oo -12=oo
and have the same finite value, it is necessary and sufficient, (1) that
lim s, -lim su   should have the limit zero, for m = oo, and also that
== 00     =oo
lim s   m- lim sm, should have the limit zero, for n = co; and (2) that, corm= o i   m=cO
responding to each fixed positive number e, arbitrarily chosen, a positive
integer N  can be determined, such that for each value of n that is > N,
a positive integer M,, in general dependent on n, can be determined, such that,
for this value of n, s0 lies between lima sn + e and lim, s  - e, for all values
X' —=00 o-=00
of m that are > Mn.
If My, when found for n, is also applicable for all greater values of n,
then the conditions, thus rendered more stringent, ensure that  lim  sn also
m=o0, =00o
exists, and is equal to the repeated limits.
If* the rows and the columns of the sequence {[sn} be both convergent,
* This theorem was given by Bromwich, Proc. Lond. Math. Soc., ser. 2, vol. I, p. 185; except
that in the statement there given a redundant condition is contained, viz. that lim lim su must
l=oo, qm=oo
exist; this condition is however contained in the one stated in the theorem above. This arose
from the fact that Bromwich deduced the theorem from a theorem corresponding to the alternative
theorem given below, in which this condition is required.




336-338]       Double sequences and double series                  467
i.e. if lirn s.m, limn s, both exist as definite numbers, then the necessary and
n=oo    m=osufficient conditions that the two repeated limits may both exist and have the
same finite value is that, corresponding to each fixed positive number e,
arbitrarily chosen, a positive integer N can be determined, such that for each
value of n that is > N, a positive integer M,, in general dependent on n, can be
determined, such that, for this value of n, S  - lim sm  < e, for all values of m
n=ao
that are > M.
The following alternative set of conditions for the existence and equality
of the repeated limits is obtained from the theorem in ~ 235:The necessary and sufficient conditions that lim lim, sm =lim lim smn
m=0o 2n=0 n^=    m=oW
their value being finite, are (1) that lim smn converge to a definite value
m=-     
lim lim sn, when n is indefinitely increased, and that liim sn- lim sm
n=  c'O M=0                                            no       n=__
converge to zero, as m is indefinitely increased; and (2) that, corresponding to
each arbitrarily chosen positive number e, and to an arbitrarily chosen integer
1V, a value N, (> N) of n can be determined, and also an integer M, such that
the condition that s.mN lies between limt s,, + e and lim sn - e is satisfied for
n=oo           n==oo
every value of m that is > M.
In case linm s, exists for every value of m except for a finite number
of such values, the condition (2) is that smN- lim sm  <, for every value of m
n=co
that is > M.
338. The preceding results may be applied to questions concerning the
convergence of a double series
a11 + a12  a, +... + an +...
+ a21 + a22+ a23 +... + an +...
+ am, - al2 + am3 +... * + am +..........................................
which has been defined in ~ 335, as a series of type )2. For this purpose we
denote by sm the sum of the finite series
a,, + a,2 + a,3 +... + ain
+ a  -  22a  + am3 +... + a2n
+ am, + am2 + a,3 + * * *+ am,30-2
30-2




468


Functions defined by sequences


[CH. VI


The condition of convergence of the double series is then equivalent to that
of the double sequence {s,}. If    lim   smn be + so or- 0o, the double
n= C, &= co
series is divergent; if  lim  sm does not exist, then the double series is
m=00, n=0o
said to oscillate.
In case the double series be convergent, it is not necessary that the single
rows, or the single columns, should separately converge; but only a finite
number of rows, or of columns, can diverge, or have infinite limits of indeterminacy; and the difference between the limits of indeterminacy of a row,
or of a column, must be arbitrarily small, from and after some fixed row, or
column.
If the double series be convergent, and if also every row be convergent,
then the series of the sums of the rows must converge to the limits. A similar
statement applies to the columns.
When all the rows, and all the columns, converge, and when the sum
of these sums in each case converges, the double series may oscillate, and
this is necessarily the case if the two limiting sums are different.
The series
a,1  + aa2 + a2l + a13+ a22+ a  +... + - an (n) + a2(n-+) +...  a.,
which has the type co, is said to be the diagonal series corresponding to the
double series. If this series converge, its sum is said to be the diagonal sum
of the given series.
In accordance with a theorem in ~ 335, if all the terms of the double
series be positive, the existence of the sum s ensures also that of the diagonal
sum; and the converse is also the case. The convergence of the double series
also ensures, in this case, that all the rows converge, and that the sum of
their sums converges also to s. A similar statement holds for the columns;
the sum of the series being independent of the type.
A convergent double series is said to be absolutely convergent, if the double
series of which the terms are | am, be convergent*.
The theorem of ~ 333, that the absolute convergence of a series of any
term implies the convergence of the series, and of all the series obtained by
rearranging the terms in another type, shews that, if the given series be
absolutely convergent, then the diagonal series converges to the sum of the
double series; and also the sum taken by rows, or by columns, converges to
the sum of the double series.
* It has been asserted by Jordan that there exist only absolutely convergent double series; see
his Cours d'Analyse, vol. i, p. 302. This statement rests upon a defective definition of convergence,




338, 339]     Double sequences and double series


469


Thus, for an absolutely convergent series, each one of the four equations
00          00  00          CO  00
an=s, =          a,=s,     S X   am=s,
mn, n       n=1 M=l        wm=1 n=1
00
E (ak  + a2(n-)... +   a m) = s
n=1
implies the other three.
A convergent double series which is not absolutely convergent can be
replaced by a new series which diverges, and is such that each term amn occurs
in a definite place in the new series, and that no terms occur in the new series
which do not belong to the original one.
EXAMPLES.
1. Let s,,n=( —l)m+n (m-P+n-q), where p > 0, q > 0. In this case lim sn and
'n=oo
lim Sm, do not exist as definite numbers, but the three limits  lim  s,,,,, lim lim Smn,
-m=oo                                               =0o, m n=oo  m= O oo =coo
lim lim smn all exist, and are zero.
n=oo m=oo
2. Let Sm,=  1 (m     In this case lim  smn does not exist, but the two repeated
t 1 + (nm - n)2      m=- oo, n=oo
limits both exist, and are zero. The same remarks apply to the case Smn-,,=22.
f_-mm2+n1  1  \
3. Let snn=2 (= 1  - +-), where a > 1. In this case the double limit exists and
2 (a+l~ \1j ) CK/
is zero, but neither the rows nor the columns of the corresponding series are convergent,
but are oscillating series; consequently the double series does not converge absolutely.
4. Let a=,,= - 1 )m+n(  -11 + m-n). In this case the single rows and the single
columns converge, and the double series converges, but the sum of the diagonal series
oscillates between log 2 + 1 and log 2-1.
FUNCTIONS REPRESENTED BY SERIES.
339. Let             u1 (x), u2 (x),... Un (x)...
be an unending sequence of functions, defined for a given domain of the
variable x, which domain is most usually a continuous interval (a, b), but
may be any given set of points G. The infinite series
Ue (X) $+ U2 (,X) +... + Un (x) +...
is taken to define a function s(x), for the domain of the variable, in the
following manner:-At any point x = a, for which the series




470


Functions defined by sequences


[CH. VI


converges, the limiting sum of the series is taken to be the value s(a) of the
function; if, at the point a, the series
VI (a) +  2 (a)+...
diverges, the function s (x) is undefined, but it is frequently convenient to
say that the value of the function at that point is one of the improper
numbers
+oo, or - oo, or ~ o,
according to the mode of divergence of the series. If, at x = a, the series is
an oscillating one, the function s(x) may be regarded as multiple-valued,
and as having all the values to which a sequence
Snl (a), Sn' (),...
of the partial-sums may converge, s (a) having thus the same limits of indeterminacy as the series itself.
If                 u1 (X) + u2 (X). + *  + n (X)
be denoted by Sn(x), the function s(x) is definable as the limit of the
sequence of functions
S, (x),... S(x) (. )....
It will be observed that the term "limit" is here used in an extended
sense, which covers the cases when, at a point a, the sequence of functional
values is divergent or is oscillating. Stated in this form, the theory may be
regarded as a theory of functions defined as the limits of sequences of given
functions, the serial form being, in fact, only a particular mode of presentation. Thus
S, (x),  S (x),... S   ($),...
may be a sequence of functions represented in any manner, for example by
continued fractions, or by determinants; but, in whatever manner the sn (x) be
represented, the limiting function s (x) can always, of course, be exhibited in
the form of the series
S, (so) + [S2 () - S (x)] + [S3 () - S, ()] +....
The function s (x) may be termed the sum-function of the series..
UNIFORM CONVERGENCE OF SERIES.
340. If the series
u, (X) + U2 () +... + U(X) +...
converge, for the point x = a, in the domain for which the functions u (x) are
defined, then, corresponding to each arbitrarily assigned positive number e, an
integer n can be found such that
I Rn, i  (), I Rn,(a),...  Rn, s (a) I




339-341]


Uniform convergence of series


471


are all numerically less than e, this being the condition of convergence of the
series at the point a.
A similar statement holds -as regards each point at which the series
converges. The least value of n for which the condition stated is satisfied will
in general depend upon the arbitrarily chosen number e, and also upon the
value of a; but it is important to consider the case in which n can be chosen,
for each fixed e, so as to be independent of a. Let it now be assumed that the
series is everywhere convergent in a given domain.
If a value of n can be found, corresponding to each arbitrarily assigned
positive number e, such that, for all values of x which belong to a given domain,
(R    n, () |, [ R,2(),... I Rn,s(...
be all less than e, then, if this value of n be independent of x, the series
Ui (x) + U2 (X))  +...
is said to converge uniformly in the given domain of x.
If we denote by ~ (e, x) the least value which n must have, for a fixed
value of x belonging to the given domain, in order that
I Rn,l(X)|, |IRn,2(X)|,...
may all be less than 1e, the series is, in accordance with the above definition,
uniformly convergent in the domain of x, provided that, corresponding to
each fixed value of e, the values of b (e, x) for all values of x in the domain
be all less than some fixed integer n,; and this integer n, is such that all the
numbers I Rn,s (x) 1, for every value of x, are less than e, for all values of n
that are _- n. The definition which has been given of uniform convergence
in a given domain includes the condition that the series converges at each
point of the domain. If it be assumed that this is already known to be the
case, the definition of uniform convergence may be stated as follows:If the series       u( (x) + u2 (x) +... + u (x) +...
converge for each value of x in a given domain to the value s (x), then the series
is said to converge uniformly in the domain, provided that, corresponding to each
arbitrarily chosen positive e, a number n, independent of x, can be found, such
that all the remainders
I S (x)-Sn (x), S (X) - s+() I,... I (X)-Sm () I...,
for every value of x, are less than e.
341. A mode of convergence of a series in a given domain, less stringent
in character than that of uniform convergence, has been considered by Dini
and by other writers. This mode of convergence has been termed by Dini
"simple-uniform convergence," and is defined by him* as follows:

* See Grundlagen, by Liiroth and Schepp, p. 137.




472


Functions defined by sequences


[CH. VI


The series       u1 (x) + u2 (x) +... + u (x) +...
which converges at each point of a given domain to the value s (x), is said to be
simply-uniformly convergent in the domain, if, corresponding to each arbitrarily
chosen positive number e, and to each integer m', only one or several integers m,
not less than m', exist, which are such that, for all values of x in the domain,
the I Rm (x) I are < e.
The condition of simple-uniform convergence is less stringent than that
of uniform convergence, in that, in the latter case, all the remainders after
a certain one are numerically less than e, whereas in the former case one or
several, but not all the remainders, need be numerically less than e.
As regards the above definition, it may be remarked that, if there be, for
each e, one value of m which satisfies the prescribed condition, there must
be an infinite number of such values; because we have only to ascribe to m'
a series of values which increase indefinitely, and for each of these exists a
corresponding value of m. Any one of an infinite set of values of m may thus
be taken to correspond to one value of m'. Moreover, the definition can be
reduced to a simpler form, thus:-Let us first suppose that there exists no
value of n such that Rn (x) = 0 for every value of x in the domain; it will
then be shewn that, if, corresponding to each e, one value of n can be found
such that i R, (x) I < e, independently of x, then there must be an indefinitely
great number of such values of n. Let us denote by Rs, the upper limit of
I Rs (x) I for the whole domain of x; Rs may have a definite value, or it may
be indefinitely great. If I Rn (x) I <e for every value of x, we have R  - e;
let us therefore take a positive number e1 less than Rn, and also less than the
least of the numbers
R1, R2,... Rn-i_
then, by hypothesis, a number n, can be found such that, for all values of x in
the domain, I Rn (x) < < e. This number n? cannot be one of the numbers
1, 2, 3,... n; for it is always possible to find a value of x for which I R1 (x) ] is
arbitrarily near its upper limit RS, and is thus > e1; hence a number nj, > n,
has been shewn to exist, such that, for all the values of x, I Rn (x) < e.
Similarly it may be shewn that a number n2, > n1, exists which has the same
property; and thus there is an indefinitely great set of values of n such that
I R,,n () <. If there be an indefinitely great number of values of n such
that Rn (x) = 0, for every value of x in the domain, Dini's definition of uniform
convergence is satisfied. In the case in which there are a finite number of
such values of n, it will be sufficient, in order to ensure simple-uniform convergence, that, for each e one value of n shall exist, such that Rn (x) I < e, and
also such that Rn (x) is not zero for every value of x; in this case the above
reasoning is applicable, provided e1 be taken less than R,, and also less than
all those of the numbers R,, R2,... Rn-,, which do not vanish. The
definition of simple-uniform convergence may now be stated as follows:



341]


Uniform   convergence of series


473


A series which converges for every value of x in a given domain is said to
converge simply-uniformly either, (1) if there be at most a finite number of
values of n such that Rn (x) vanishes for every value of x, and if, corresponding
to each arbitrarily chosen positive number e, a number n can be found such that,
independently of x, i Rn (x) I < e, whilst Rn (x) does not vanish for every value of
x, or (2) if there be an indefinitely great number of values of n for which R( (x)
vanishes for every value of x.
A series which is uniformly convergent is also simply-uniformly convergent;
but the converse does not hold.
If the series be simply-uniformly convergent, but be not uniformly convergent, there must, corresponding to each sufficiently small e, be an
indefinitely great number of values of n for which the condition
I Rn() l<,
for all values of x, is not satisfied; for if there were only a finite number of
such values, n could be taken greater than the greatest of these, and thus the
condition for uniform convergence would be satisfied, which would be contrary
to hypothesis.
If all the terms u (x) of a series be positive for every value of x in the
domain of the variable, then, if the series E u(x) be simply-uniformly convergent, it is also necessarily uniformly convergent. For the condition of
simple-uniform convergence ensures that, corresponding to an arbitrarily
chosen e, n can be found such that tie sequence
(Rn,, R, 2) Rn,3...) *)
converges for every value of x to a value which is less than e; and, since the
terms of the series are all positive, each element of this sequence is less than,
or equal to, the next one; and therefore
R   n,2, R,,, R2n,3... are all < e.
It follows that Rn+m,s, which equals Rn,s+m - Rn,m, is also < e, for all values
of m and s; and thus that Rn+m < e; hence the series converges uniformly.
It may easily be shewn that, if the two series E u (x),  i u (x), be both
simply-uniformly convergent, then X u (x) is uniformly convergent.
Let e1, 62,... be a sequence of diminishing positive numbers which
converges to zero. If the series Su(x) be simply-uniformly convergent,
a number nz can be found such that Rn, (x)l< e1, for all values of x; a
number n2, > n, can then be found such that I R.2 (x) | < 62; then n3, such that
I Rn (x) < e3; and so on. If now the first n, terms be amalgamated into one,
then those after the first n, up to and including u,, (x), and so on, the series
may be written in the form
Sl (,) + [n2 () - s  )] + [3 (c) - Sn (x)] +...;




474


Functions defined by sequences


[CH. VI


and in this form the series is uniformly convergent. It thus appears that a
simply-uniformly convergent series can be changed into one which is uniformly
convergent, by bracketing the terms suitably, in accordance with a norm, and
taking each bracket to constitute a term in the new series.
Conversely, a uniformly convergent series may be replaced by one which is
only simply-uniformly convergent.
If each term Un(x) of a uniformly convergent series be replaced, in
accordance with some norm, by the sum of rn functions, such that
Un () = Un, l (x)+ Ui,2 (x)+ +... + lUn,., (X),
then the new series
U1,1 (X), ()+ U,2(x)+   U,, U, (X) + U2,1 (x) + 
is not necessarily convergent, but may oscillate. If, however, the functions U
be so chosen that the series converges in the whole domain of x, then the series
converges at least simply-uniformly. It thus appears* that the distinction
between uniform convergence and simple-uniform convergence is an unessential one.
NON-UNIFORM CONVERGENCE.
342. If we denote by p (e, x) the least value which n can have, such that
Rn (), I i Rn+(), I Rn+2 (x) I...
may all be < e, where the series  u (x) converges, for every value of x in
a given domain, to the value s (x), then the condition of convergence ensures
that for any fixed value of x, - (e, x) has a definite finite value, for each value
of e, which however may increase indefinitely as e is indefinitely diminished.
Taking a fixed value of e, sufficiently small, it may happen that j (e, x) has
no finite upper limit for all values of x in the domain; and this will happen
in case the convergence of the series be non-uniform. The function t[ (e, x)}-1
has, in this case, zero for its lower limit; and therefore in accordance with
the theorem of ~ 171, there must be at least one point x, such that, in an
arbitrarily small neighbourhood of it, zero is the lower limit of ti (e, x)}-1.
There may be a finite, or an infinite, set of such points; and, in an arbitrarily
small neighbourhood of any one point of this set, J (e, x) has no upper limit,
and thus has values greater than any arbitrarily chosen number A. Nevertheless - (e, x) has a definite finite functional value at each point of the set,
provided that such a point belong to the domain of the variable; for otherwise
the series would not converge at such a point.


* See Arzela, Bologna Rendiconti, 1899; also Hobson, Proc. Lond. Math. Soc., ser. 2,
vol. I, p. 376.




341-343]


Non-uniform convergence


475


A point, in the arbitrarily small neighbourhood of which Fr (e, x) has no
upper limit, provided e be sufficiently small, is said to be such that the series is
non-uniformly convergent in its neighbourhood.
Frequently, for shortness, such a point is said to be a point at which
the series is non-uniformly convergent, or to be a point of non-uniform
convergence.
It has been shewn that such points exist whenever the series is nonuniformly convergent in the domain of the variable; and, in the case in which
the domain is a closed set, which case is alone of importance, these points
themselves all belong to the domain of the variable.
When there are only a finite number of points of non-uniform convergence,
the series is frequently said to be in general uniformly convergent. It
becomes in this case, uniformly convergent, if arbitrarily small intervals containing these points be removed from the domain of the variable.
If x = a, be a point of non-uniform convergence, a sequence
al, a2,... an) *..
of values of x in the closed domain can be found which converges to the
value a, and is such that the numbers
(e, al),  t(e, ca),... 2   (e,  )...
form a sequence with no upper limit, where e has a fixed value chosen
sufficiently small. Thus one of the limits
(e a+ o),(e a-0)
is infinite, or both are so, although  t (e, a) must be itself finite. Therefore a
is a point of infinite discontinuity of the function r (e, x).
If one, but not both, of the limits
* (6, o + 0),  (e, a - 0)
be infinite, the point is said to be one of non-uniform continuity on the right
or on the left, as the case may be.
THE CONTINUITY OF THE SUM-FUNCTION.
343. Let us suppose that the domain of x is either the interval (a, b), or
else a perfect set of points in that interval, and further that the functions
U1 (x), u2 (X), 3 (x),...
are continuous throughout the domain. It will then be shewn that:If the series $u (x) converge at least simply-uniformly in the domain of
the variable, the sum-function s (x) is everywhere continuous.




476


Functions defined by sequences


[CH. VI


Let a be any point in the domain of x, and a + 8 another such point on
the right of a; then
s (a)= S(a) + Rn (a),
s (a+  )=  (a+)+ R(a+ );
thus
s (a + ) -  (a)= [Sn (a + ) - n (a)] + [Rn (a + 8)- Rn (a)].
Since the series converges simply-uniformly, a value of n can be found,
corresponding to any arbitrarily small e, such that
Rn (a), IRn (a+) are each <   e,
for all positive values of 8 such that a + 8 belongs to the domain of x. Suppose
n to have this value; then, since sn () is continuous, a value &i of 8 can be
determined such that
Isn(a+8)-sn(a)|< Ce, if 0<8-81.
It follows that  s (a + 8) - s (a) < e, provided 0 < 8 _   81;
and, as e has been arbitrarily chosen, the condition of continuity of s (x) at a,
on the right, is satisfied. In a similar manner it may be shewn that s (x) is
continuous at a on the left.
A fortiori, the condition that the series converge uniformly is sufficient
to secure that the sum-function may be continuous.
The above proof also suffices to establish the following more general
theorem:If the functions un (x) be all continuous at the point x = a, but not necessarily
elsewhere, the condition of simple-uniform convergence of the series in an
interval containing the point a in its interior is sufficient to ensure that s (x) is
continuous at the point a.
344. If the function s (x) be discontinuous at a, say on the right, then 83
cannot be chosen so that, for
0< 8-8, [s(a+8)-s(a)| < c,
provided e be chosen sufficiently small; hence, in this case, it is impossible to
choose n such that
R, (a + )- Rn (a) < Ie,
for all the values of 8 concerned, sn being a continuous function for the domain;
and it follows that it is impossible to choose n such that
I Rn (a+ 8)| < 4e,
for all values of 8 such that  0 < 8- 81.
Therefore, in this case, the series converges neither uniformly nor simplyuniformly, and the point a is a point of non-uniform convergence.
It has long been known that the sum of a series of which all the terms




343-346]


Continuity of the sum-function


477


are continuous is not necessarily itself continuous. The important discovery
that such a discontinuity is due to the non-uniform convergence of the series
was made independently by Stokes* and by Seidelt.      It was not until a
later time that, under the influence of Weierstrass, the great importance of
the notion of uniform convergence in the Theory of Functions was fully
recognized.
The question whether non-uniform convergence necessarily implies
discontinuity in the sum-function remained for many years an open one.
It was decided in the negative sense when Darboux and Du Bois Reymond
constructed examples of cases in which the series are non-uniformly convergent, and yet nevertheless have continuous sum-functions.
TESTS OF UNIFORM CONVERGENCE.
345. In certain cases it can be easily established that a series is uniformly
convergent. This can frequently be done by applying the following theorem:un( (x) denoting a series of functions such that I un (x) I has, for each value
of n, an upper limit u, for the whole domain, iJ the series
Ul + U2 + * + un +...
be convergent, then the given series is itself uniformly convergent, and is
absolutely convergent for each value of x.
The remainder              n+1 + un+2 +...
of the series Zu, is greater than, or equal to, the remainder
n+l (x)l +  Un2 (X) | +...
of the series  | u,, (x). If n be so chosen that the former remainder be < e,
the latter remainder is also < e, for every value of x; and the convergency
condition of Zun states that, corresponding to each e, a number n, exists, such
that all the remainders, of index > n,, are < e; therefore the same holds as
regards the series  I Un, (x)) I. Hence this latter series is uniformly convergent;
and since no remainder of 2un (x) can exceed numerically the corresponding
one of; I u (x) 1, it follows that >2u (x) is uniformly convergent, and converges
absolutely for each value of x in the given domain.
346. If all the terms of the series    (x) + u (x) +... + u (x)+... be
positive or zero, for all values of x in (a, b), and the series converge uniformly
in that interval, then the series obtained by rearranging the order of the terms,
in accordance with some norm, is also uniformly convergent in the interval.
That the new series obtained by rearranging the order of the terms
* "On the critical values of the sums of periodic series," Math. and Physical Papers, vol. I,
p. 236.
t "Note fiber eine Eigenschaft der Reihen," Abhd. d. Miinch. Akad. vol. vII. On the history
of this discovery, see Reiff's Geschichte der unendlichen Reihen, p. 207.




478


Functions defined by sequences


[CH. VI


converges to the sum of the original series, everywhere in (a, b), has
been proved in ~ 332. An integer n' exists, such that the first n terms
of the given series all occur amongst the first n' terms of the new series;
it follows that RA (x) < R',g (x), where Rn (x), R'~, (x) denote the remainders
after n and n' terms respectively, in the original series and in the new series.
If n be so chosen that Rn (x)< e, for all values of x in (a, b), we have also
R', (x)< e for all values of x; therefore the new series is also uniformly
convergent.
If the series  u (x) I + I, (x) +. +..  u, (x) +... converge uniformly in
(a, b), then the series u (x) + u (x) +... + u, (x) +... converges uniformly in
(a, b); also any series obtained by rearranging the order of the terms of the
latter series, in accordance with any norm, is uniformly convergent.
The second series is necessarily convergent everywhere in (a, b); also its
remainder after n terms cannot exceed, in absolute value, the remainder
after n terms of the first series.  It follows that, if n be so chosen that the
remainder of the first series after n terms is less than e, for every value of x,
the absolute value of the remainder of the second series satisfies the same
condition. Therefore the second series is uniformly convergent. Since, from
the last theorem, a rearrangement of the order of terms of the first series
does not affect its uniform convergence, it follows that a corresponding
rearrangement of the terms of the second series does not affect its uniform
convergence.
The converse of this theorem also holds, and may be stated as follows:If* the series u, (x) + u, (x) +... be uniformly convergent in (a, b), and if
all the series obtained by systematic rearrangement of the terms of the series be
also uniformly convergent, then the series u, (x) + u2, (x) I +... is uniformly
convergent in the same interval.
347. The following theoremt is sometimes useful:If the terms of the series u, (x) + u, (x) +... be continuous in (a, b), and
never negative, and if the series converge to a continuous sum-function s(x),
then the series converges uniformly in (a, b).
To prove this theorem, let x, be any point in (a, b), then
s (x) - s (x,) = {s, (x) - s (x)}j + tR (x) - R, (x,)}.
For the fixed point x,, and corresponding to any fixed positive number e,
an integer n can be so chosen that R, (x,) < ~e. This value of n being fixed,
an interval (x, -, x~ + 8) can be so determined, that, if x be in this interval,
both Is (x) - s (xi) i and  s,, (x) - s, (x,) are <  e; this follows from the con* This theorem has been proved by G. D. Birkhoff, Annals of Math., ser. 2, vol. vi, 1905,
p. 90.
+ See Dini's Grundlagen, p. 148.








346-348]           Tests of uniform    convergence                  479
tinuity of s (x) and s (x) at x,. We now see that, throughout the interval
(x - 8, x, + 8), the condition Rn (x) < e is satisfied; and since the terms of the
given series are never negative, it follows that Rt (x) < e, for every value of n'
that is > n, and every value of x in (x, -, x, + 8). It has therefore been
proved that x, is a point of uniform continuity of the series; and since x,
is any point whatever in (a, b), the convergence of the series is uniform
in (a, b).
If a sequence s, (x), s2 (x),... S, (x),... be such that, for every value of x in
an interval (a, b), one of the sets of conditions Sn (x) > sn+, (x) for every value
of n, or Sn (x) - Sn,+ (x) be satisfied, then the sequence is said to be monotone
in (a, b).
The above theorem may be stated in the following form:A sequence of continuous functions {s, (x)} which are monotone in a given
interval, and which converges to a continuous function s (x), converges uniformly
to s (x).
348. If* u, (x), u2 (x),... un (x),... be defined for the interval (a, b), and
be limited in that interval, and positive for all the values of x, and if further
un (x)  Un+l (x), for every value of n and x; then, if S a, be any convergent
series, the series S anun (x) converges uniformly in the interval (a, b). Moreover, if X an do not converge, but oscillate between finite limits of indeterminacy,
then, provided the additional conditions that the functions un (x) be all
continuous, and that lim un (x) = 0 for each value of x, be satisfied, the series
an u,, (x) is uniformly convergent in the interval (a, b), and its sum  is
consequently continuous.
In case the series 2 a, be convergent, the partial remainder R,,, of the
series X au, (x) being
(a+, - + an+2 +... + an+m) Un+m+l (x)
r=m
+ X (an+ +      + an+... + a ) {Un+r (X) - Un,+r+ (X)j,
r=l
we see that, by choosing n so great that all the partial remainders of the
series X a, after the nth term are numerically less than the arbitrarily chosen
number e, the condition
Rn,m < En+l(x)
is satisfied; and therefore, for every value of x in (a, b), we have R,,., < e. U,
where U denotes the upper limit of u, (x) in (a, b). Since eU is arbitrarily
small, it has thus been shewn that the condition of uniform convergence of
a au, (x) is satisfied.
It is easily seen that this part of the above theorem also holds when the
terms of the series X ac are functions of x, provided E a, converges uniformly
* See Hardy, Proc. Lond. Math. Soc., ser. 2, vol. Iv, pp. 250, 251.




480


Functions defined by sequences


[CH. VI


in (a, b).  When the series 2 a, oscillates between finite limits, K  can be
determined such that can+l    a,1+2 +... + an+r I < K, for all values of n and r.
Also, since the sequence u (x), u2 (x), u (x),... u (x),... is by hypothesis
monotone, and converges to the continuous limit zero in the interval (a, b),
it follows from the theorem of ~ 347, that the sequence converges uniformly
to the limit zero. We can consequently choose n so that u,+,. (x) < e, for every
value of r and x; therefore I R, (x) I < 3eK, and since 3eK is arbitrarily
small, it follows that the series is uniformly convergent.
EXAMPLES.
x                           -x
1. Let      u2,n-_(x)=x_(_     ), (x)= —+l+l)2 -1nX2+(1 _ nx)2         (n + 1) x2 + {1 - (n + 1) X12 '
In this case, the series converges for all values of x, and
()  X2 (] _)2;    R2n-1 (X) = 0,  R2n - 2 (X)-= 2n-l (X)
In an interval (a, 2), which contains the point x=0, the series converges simply-uniformly,
but it does not converge uniformly, since R2,,2  ) =1, however great n may be.
n2X
2. Lett sn(x)=-ln 3, s(x)=0, for 0   ~xl. This series converges non-uniformly
Let  n3X2' 
0                       1      x
FIG. 1.
in the neighbourhood of the point x=0. The approximation curves y=s5(x) have peaks
of height tjn, which increase indefinitely in height as n is increased. At the same time, the
point -i, at which the ordinate is a maximum, continually approaches the point 0; and
thus, for any value of x which is >0, n may be taken so great that s,(x) is arbitrarily
small. At the point x=0, we have s,,(x)=O, for every n.
* Tannery, Theorie des fonctions, p. 134.
+ Osgood, Aner. Journal of Math., vol. xix, p. 156; also G. Cantor, Math. Annalen, vol. xvl,
p. 269.




348]


Tests of uniform  convergence


481


3. Let s,, (x)= 1   22, s ()=O, O<   1. The curves y=s (x) have peaks all of the
1                                   1
same height a at the points x =-. As in the last example the point x=-, below the peak,
n7                                  n
continually approaches the origin as n is increased. The convergence is non-uniform in
the neighbourhood of x=0.
y
O                                   i
FIG. 2.
n sin2krTx                   1         1              1
Let k (X)   +n2; s  (x)= 1 i, (X)+ 21 2! (X)+.  3! ()+...    L!(X).
The series which defines s,(x) converges uniformly, and thus s, (x) is a continuous function
of x.  In the neighbourhood of any rational point x=p/q, the curve y=s,(x) has peaks
arising from the term, ( k!(x), where k is the smallest integer such that k! is divisible
by q. The series converges to the limit s (x) =0, non-uniformly in any interval whatever
(a, b), taken in the interval (0, 1).
4. Let*   u2n - I ()-=X + 1 u2n (X)= -n+ {1-   1) }, where 0<!x<l, and
1
zb((l) = H.
The series 2u(x) is simply-uniformly convergent in (0, 1), but it is not uniformly convergent.
5. Lett u,(x)==x (1-x), 0<xl. In this case s(x)=x, for 0_x<1; but s(x)=O,
for x= 1; and the series converges non-uniformly in the neighbourhood of the point x= 1.
6. Lett Un(X)=xn(l -xn"). If I x <1, we find s (x)=-1; alsos(l)=0; whereaslims(x)
is indefinitely great. The series converges non-uniformly in the neighbourhood of the
point 1, and its sum-function has an infinite discontinuity at that point.
7. Let X u (x)= - 2 (n- 1)2 Xe-(-1)22 +2n2Sxe-2$x2. Here s (x) =0 for every value of x;
R,(x)= - 2n2xe-22x; and at x=-, R  -)= - -. The series converges non-uniformly in
n     (n \n/  e
* Volterra, Gior. di Miat., vol. xix, p. 79.
t Arzela, Memorie di Bologna, ser. 5, vol. vIII, p. 139.
+ Darboux, Ann. de l'cole normale superieure, vol. v, "Sur les fonctions discontinues 


hI,


31




482                  Functions defined by sequences                       [CH. VI
the neighbourhood of x=O, since arbitrarily large values of the Rn(x) I exist in such neighbourhood; but the sum-function is continuous at x=0.
1               1
8. Let*         s (c)= xn(x)+ 2  n(2! x)+... +..  kn(k! x)+...
where q5/ (x)= /2e.n sin2 7rx. e-n2si14 rx. The series which defines sn(x) converges uniformly,
since ] On ( k! x) < 1; and thus sn (x) is a continuous function of x. The sum-function s (x)
is also a continuous function of x; but the convergence of the functions sn (x) to s (x) is
non-uniform in every sub-interval of the interval (0, 1).
9. Considert the series
1+5x        x(x2)n2+x (4- x)n+ 1 - x
2 (1 +x)     n (n+ 1) t(n- 1)x+ l} (nx+ 1)
Here u,(x)=    +(n- 1)+l -         +   n    ]; thus s(x) =3, unless x=0, when
s (0)= 1; and the sum-function is therefore discontinuous at the point 0.
1      2
Since R,B(x)=  I +    2    we find on equating this to E, and solving for n,
it-= {x + 2 -            2(x  ( 1) [ + 2 - E(x + )   4x (3-x)]}2ex;
thus, for a fixed E, the value of n increases indefinitely as x approaches the value 0.
10. The series:
2      2      22         2        2
X2 - X2 +           +    x    -( —      +(l+x2)'
If +2  1 g2  1 (1 +x2)2  (1 +'2)2  (l+,2)3
is uniformly and absolutely convergent in any interval (-A, B).      For 2n (x)=O,
2                        1
S2 +1 (X) - (1+ X2)  and hence s2,,+ l()<; therefore the series converges uniformly to
the sum zero. The series
2 2     X2       x2        x2       X2       X2
+ 1     (1+ X)2 (1 + X2)2 + (1 + 2)3  (1 +2)4    +2)2
obtained by rearranging the terms of the given series, is however non-uniformly convergent in (-A, B). Fors 3-($)= (1+ X2)n-1 1
vergent (in  A, B). For     1= (l+x2)2 —2;and forx:= +(2n-1)
S3n- 1 (X)=W
The given series does not satisfy the condition stated in the theorem of ~ 346, that the
series whose terms are the absolute values of those of the given series should be uniformly
convergent. For the series
X2     X2       32
X2+x2+      2 + +  2 + ( +    +'
has its sum discontinuous at the point x=O, and therefore does not converge uniformly in
an interval (-A, B).
* Osgood, "A geometrical method for the treatment of uniform convergence," Bulletin of the
American Math. Soc., 1896.
- Stokes, Math. and Phys. Papers, vol. I.
+ B6cher, Annals of Math. ser. 2, vol. iv, 1904, p. 159.




in A r  ''A f —X


o1 *' /:4~, 64Wj     _on-unJiorm            convergence                  483
RELATION OF THE THEORY WITH THAT OF FUNCTIONS OF TWO
VARIABLES.
349. If the functions s,(x), s2(x),... Sn(x),... converge for all values
of x in a given domain to the value s(x), the functions Sn(x), and
_n (x) = s (X) - s (x),
may be regarded as functions of two variables x, y, where n =1/y, and may
be written s(x, y), R (x, y). These functions have been defined only for
values of y which are the reciprocals of positive integers; it is however
frequently convenient to assume that, for a value of y between two values
y, yn+i, which correspond to consecutive integral values m, mn + 1 of n, the
functions are defined by
s8  yM               (x,  n+= y)+ ym+ -y s (xS, Y),
Ym+ - m       y       m+i - Ym
R (x, y) =     Yy -  y R (, YI) +  m+l - Y R (, y)
Ym+i~ - yIM          ym+i - ym
so that s (x, y), R (x, y) are continuous linear functions of y in the interval
(yim, ym+l)
If we further assume
R (x, ) = 0,, s  ) = s (x),
the functions s(x, y), R (x, y) are defined for all values of x in the domain
of x, and for all values of y in the interval (0, 1), the ends included. These
functions are everywhere continuous with respect to the variable y. That this
is the case for y = 0, follows from the condition of convergence
lim s (x, y)= s (x)= s (x, 0),
y=0
limR(x,y)=    0  =R(x 0,O).?/=o
The functions s(x, y), R(x, y) may be termed* the transformed sumfunction, and the transformed remainder-function respectively. The study of
the properties of series or sequences of functions of a variable may be thus
reduced to the study of the properties of functions of two variables, and this
is frequently a very convenient procedure.
The function R (x, y) is continuous with respect to x in the domain of x,
upon the line y = 0, since it is everywhere zero; but it is not necessarily
continuous with respect to (x, y).
* Hobson, "On non-uniform convergence and the integration of series," Proc. Lond. Math.
Soc., vol. xxxiv, p. 247.
31-2




484


Functions defined by sequences


[CH. VI


Let P be a limiting point of the domain of
a semi-circle qpq' of radius p, with P as
centre. The upper limit of I R(x, y)[ in
this semi-circle will have a value /3(p)
which is a function of p, and which has
a limiting value i3p, when p is indefinitely
diminished. It may happen that fp is
indefinitely great. If /p be zero, P is a
point of continuity of R (x, y) with respect
to (x, y); but if,3p be not zero, P is a point
of discontinuity.


x, on the x-axis; describe


q P P P'q'
FIG. 3.


x


It is easily seen that a point P of discontinuity of R (x, y) is
non-uniform convergence of the sequence s,, (x)}.
At a point of uniform convergence, corresponding to an
assigned number e, in a sufficiently small
neighbourhood NM, a value ye of y can be
found, such  that for y _ ye, we have
R (x, y) < e; and this will be the case for
all points of the domain of x in the rectangle
MNSR. Within this rectangle semi-circles
with centre P can be described in which 
R(x, y) I < c, for all points within the semi-  N q'  P (I
circle, and thus the value of 3p is zero; and
therefore P is a point of continuity of the
transformed remainder-function.


a point of
arbitrarily


R
ye


I M


If at P, the number /p be not zero, the convergence is non-uniform in the
neighbourhood of the point P; and the number ip, which is the saltus at P
of IR (x, y), may be called the* measure of non-uniform convergence at P.
The number tip may be regarded as existent at every point which is a
limiting point of the domain of x, although there may be points at which it
has the improper value + o; and at a point of uniform convergence it has the
value zero. If 8f is regarded as a function of the point P, it may be termed
the convergence-function*.
If the semi-circle used in defining 13p be divided into two quadrants by
means of the radius Pp, the upper limits of R (x, y) I in the quadrants Ppq,
Ppq' may be considered separately.  When p has the limit zero, these upper
* The term "Grad der ungleichmassigen Convergenz" is employed by Schoenflies, see Bericht,
p. 226, who uses the definition given by Osgood, American Journal of Math., vol. xix, p. 166.
The term " Convergence Function" is also that employed by Schoenflies.




349, 350]


Non-uniform convergence


485


limits have for their limiting values two* numbers /3p, /-, which may be
called the measures of non-uniform convergence at P on the right, and on
the left, respectively. If ti3+ = 0, /p > 0, the point P may be said to be one
of uniform convergence on the right; a corresponding definition holds for the
left. The measure /p is the greater of the two numbers /,, /3,; and at a
point of uniform convergence R/3 = /3 = 0.
THE DISTRIBUTION OF POINTS OF NON-UNIFORM CONVERGENCE.
350. Let u, (x) + u2 (x) +... + u, (x) +... denote a series of continuous
functions which converges everywhere in the interval (a, b) to the sum s (x).
The most general possible distribution of the points of non-uniform convergence of the series will be here investigated.
In the first place, it can be shewn that the points, at which the measure
of non-uniform  convergence /3 exceeds any fixed positive number A, form  a
closed set.
For, if P be a limiting point of this set, in any semi-circle with P as
centre there are points on the x-axis at which 3 > a, and therefore there are
points within the semi-circle at which I R(x, y) I > a; and since this is the
case however small the radius of the semi-circle may be, it follows that P
is itself a point at which 3 > -.
Next, it will be shewn that the closed set, for which /3 > a-, is non-dense in
the interval (a, b).
At any point P, (x, y), let a straight line of length 2p be drawn parallel to
the y-axis, with P as its middle point, and let o (p) be the fluctuation of the
function R (x, y) in the line 2p. The function w (p) is a continuous function of
p, since R(x, y) is continuous with respect to y. If P be in the boundary
y = 0, it will be sufficient to take the straight line of length p within the
rectangle. Let a, (x, y) be the upper limit of the values of p which are such
that co (p) - ao. The function a, (x, y) is defined for every point in the rectangle,
and is essentially either positive or zero. Since R (x, y)= s (x)- s (x, y), and
since s (x) is independent of y, it follows that the function a, (x, y) is the same
as the corresponding function defined for s (x, y) instead of for R (x, y).
The function s (x, y) being everywhere continuous with respect to y, and
being also continuous with respect to x, for every value of y except zero,
it follows from the theorem of ~ 243, that a, (x, y) is an upper semi-continuous
function.
* These numbers are equivalent to Osgood's indices of a point (b+, b-) of which he gives a
different definition. The definition in the text is given in the paper in the Proc. Lond. Math.
Soc. already quoted.




486


Functions defined by sequences


[CH. VI


Let P be a point of the boundary y = 0, at which the minimum of
a, (x, 0) is not zero; it can be shewn that the
saltus of l R(x, y) at P is _ 2o-. To prove
this, we observe that a neighbourhood pp' of
P can be found such that a, is at every
point of pp', greater than a fixed number 7
which is less than the minimum of ar at P.             X 
Let X, Y be any two points in the rectangle          ___
of base pp' and height V, and let Xm, Ym'                m        m' P'
be perpendicular to the x-axis.   We have               FIG. 5.
then
IR(X)-R(Y)        R (X) -R     l(m) + [R(Y)- R(n')
2o7,
and thus the required neighbourhood has been found.
It follows that, if the saltus of R(x, y) I at P be > a, the minimum  of
al,(P) with reference to the x-axis, must be zero. It has been shewn in
~ 184, that in every sub-interval of the x-axis, there are points at which this
minimum   of a, (P) is positive.  It follows that the closed set, for which
3 > a-, cannot be dense in any interval; and thus*:If a series of continuous functions converges to the sum s (x) at every point of
a given interval, then the points, at which the measure of non-uniform convergence
exceeds a given positive number a, form a non-dense closed set.
If we take a sequence of values of a- which converges to zero, we see that
the set of all the points of non-uniform convergence of the series is the limit
of the sequence of the closed non-dense sets which correspond to the values
of a. It follows that:The points of non-uniform convergence of a series of continuous functions
which converge in a given interval to the sum s(x), form in general a set of
points of the first category; and the points of uniform convergence form  a set
of the second category, which is consequently everywhere-dense, and of the power
of the continuum.
It is clear that the set of points at which the convergence function is
indefinitely great, when it exists, forms a closed non-dense set.
If the functions ul(x), u (x), u3(x),... of which the sum in the interval
(a, b) is the function s(x), be discontinuous functions, there may be points of
non-uniform convergence dependent upon the discontinuities of the given
functions; and then it is no longer necessarily true that the points of
uniform convergence are everywhere-dense.
* This theorem was given by Osgood for the case in which s (X) is continuous, see American
Journal of Math., vol. xix, 1897. The proof in the text was given by Hobson, Proc. Lond. Math.
Soc., vol. xxxiv, p. 245, also Acta Mathematica, vol. xxvII, p. 212.




350, 351] The limits of a sum-function at a point


487:


EXAMPLE.
Let* ul(x)=0, at all points of the interval (0, 1) except at the point x=~, where
ul(x)=l. Let u2(x)= -1, at x=~, and u2(x)=l, at x=-,,and 2 (x)=0, everywhere
else; let u3(x)= -1 at x=4,, and U3(x)=1, at x=-, 3,, 7, and u3(x)=0, at all other
points; and so on. Then sl(x) is zero except at x=, where sl(-)=l; s2(x) is zero
except that S2 ()=s2 () =1; s3 (X) is zero except that s3 (1-) =3 ()=s3 ()=s3 ()= 1, and so
on. The function s (x) is everywhere zero, and therefore continuous in (0, 1); but the series
everywhere converges and is non-uniformly convergent at every point of the interval (0, 1),
since, in the neighbourhood of every assigned point, there are discontinuities of measure.
equal to 1, of R. (x).
THE LIMITS OF A SUM-FUNCTION AT A POINT.
351. Let a be a limiting point of the domain of the variable x, for which
the convergent series Zu (x) is defined; a may, or may not, itself be a point of
the domain. Let us suppose further that the limits u (a +-0), 2 (a + 0),...
un(a + 0),... at a on the right, all have definite values; it follows that
s8(a + 0), 2 (a + 0),.8.. s (a + 0),... also exist and have definite values.
We propose to examine the circumstances under which the limit s (a + 0),
of s (x) on the right at a, exists, and the series Zu (a + 0) is convergent, with
s (a + 0) for its sum.
Let us assume that an interval (a, a + 8) exists, such that, for that part of
the domain of x which falls within
it, the given series is at least simplyuniformly convergent; and let us consider the transformed sum-function
s (x, y). If e be an arbitrarily chosen
positive number, a value y' of y can
be- found such that, for every value of        P' Q'
x interior to (a, a + 8) which belongs
to the domain, Is (x, 0) - s(, y') < e.
With this value of y', an interval             t, Q x +            X
(a, a -+ D), where v  a 8, can be found,           FIG. 6.
such that the fluctuation of s(x, y')
in the interval is < e, since the limit s (a + O y') exists. If P, Q be any two
points within the interval (a, a + r) on the x-axis, and P', Q' are the points
on the line y = y', with the same values of x, we have
s(P) - s (Q)= [s(P')- s(Q')] + s (P) - s (P')] + [s (Q')- s (Q)],
it follows that Is(P)-s(Q) < 3e; and since e is arbitrarily chosen, we see.
that s(a + 0) has a definite value. The following theorem has therefore beei
established:

" See W. H. Young, Proc. Lond. Math. Soc., ser. 2, vol. II, p. 94.




488


Functions defined by sequences


[CH. VI


If a be a limiting point of the domain of x for which the convergent series
Eu (x) of which the sum-function is s (x) is defined, and if, in a certain neighbourhood of a, on the right, the series converge at least simply-uniformly, then
the function s (x) has a definite limit s (a + 0) at a, on the right. It is here
assumed that s (a+ 0) has a definite value for each value of n.
This theorem specifies a sufficient condition for the existence of s (a + 0);
but the fulfilment of the condition does not ensure that the series I u1, (a + 0)
-=1
is convergent.
352. Necessary and sufficient conditions will now be determined, that
s (a + 0) may exist, and that the series 2e (a + 0) may converge to the value
s (a + 0). If s (x, y) denotes the transformed sum-function, then the required
conditions are those that the two limits lim lim s (x, y), lim lim s (x, y) should
x=a y=0      y2=0 x=a
both exist, and should have the same value. The required conditions may
consequently be obtained by applying the theorem of ~ 234, relating to
repeated- limits. We thus obtain the following theorem:The necessary and sufficient condition that the sum s (x) of the convergent
series xu (x) may have a definite limit s (a + 0) at the limiting point a of the
domain of x, and that also the series  tu (a + 0), of which the terms are
assumed to have definite values, may converge to the limit s (a + 0), is that, corresponding to each arbitrarily chosen positive number e, and to each integer n
which is greater than some fixed number n1 dependent on e, a number 0 can be
found, such that, for every value of x belonging to the given domain and interior
to the interval (a, a + 0), the condition I Rn (x) I < e is satisfied, the number 0
being in general dependent upon n.
In the particular case in which 0 is, for each value of e, independent of n,
the point a is a point of uniform convergence on the right, of the series $u (x),
the point a itself being supposed to be excluded from the domain of x for
which the series is defined; therefore uniform convergence at a on the right is
a sufficient condition that s (a + 0) may have a definite value, and that the series
Yu (a + 0) may converge to s (a + 0); the point a being itself excluded, for the
purpose, from the domain of x.
By employing the alternative set of conditions for the existence and
equality of repeated limits, given in ~ 235, we obtain the following
theorem:The necessary and sufficient conditions that the sum s (x) of the convergent
series au (x) may have a definite limit s (a + 0) to which the series,u (a + 0)
may converge, the terms of this series being assumed to have definite values, are
(1) that s,(a+ 0) should converge to a definite limit as n is indefinitely
increased, and (2) that, corresponding to each arbitrarily chosen positive




351-353]


The continuity of the sum-function


489


number e, and to each integer n, there should exist a value of n > n,, and also a
number 0, such that I Rn (x) < e for every value of x within the interval (a, a+0).
This theorem contains the completion and generalization of that of ~351.
It is clear that the condition stated in the latter theorem is insufficient,
without postulating that the condition (1) of the above theorem is satisfied,
to ensure the existence and equality of the two repeated limits.. When the
conditions stated in the theorem are not satisfied, either or both of the limits
s (a + 0), Su (a + 0) may exist; but they cannot both exist, and at the same
time have one and the same value.
THE NECESSARY AND SUFFICIENT CONDITIONS FOR THE CONTINUITY
OF THE SUM-FUNCTION.
353. If the functions ul (x), u2 (x),... u, (x),... be all continuous
throughout the domain of x, which will be taken to be the continuous
interval (a, b), it has been shewn that a sufficient condition for the continuity of the sum-function s (x) at a point xc is that a neighbourhood of xc
can be found within which the series converges simply-uniformly; it has
however been shewn, by means of examples, that this condition is not
necessary for continuity of s(x) at x,.
The theorem of ~ 352 may be applied to obtain the necessary and
sufficient condition for the continuity of s (x) at xi. That theorem shews
that, in order that s (x) may be continuous at xc on the right, it is necessary
and sufficient that, corresponding to any arbitrarily chosen e, an integer n,
should exist, such that, for each n which is > n,, a neighbourhood (x1, cx + 0)
can be found, 0 depending on n, such that I R, (x) < < e, for every point x within
this neighbourhood; where the integer n, may be so chosen that I Rn (xi) < e.
A corresponding condition is necessary and sufficient to ensure that s (x) is
continuous at xc on the left. We can therefore state the necessary and
sufficient condition of continuity at xl as follows:In order that s (x) may be continuous at the point x1, it is necessary and
sufficient that, corresponding to each arbitrarily chosen positive e, a number n,
can be found such that for each value of n >n,, a neighbourhood (x1-81,
i + 82) of x1 can be found, at every point of which I Rn (x) < e, the numbers
81, 82 being dependent in general upon n.
For a prescribed e there is a certain range of values of y from zero
upwards, for which I R (x, y) I < e; and the upper limit of these values of y may
be denoted by p, (x): but there may be other greater values of y not continuous
with the interval (0, O. (x)), for which the condition I R(x, y) I < e, is also
satisfied. At a point xc of non-uniform convergence of the series, the lower




490                Functions defined by sequences               [CH. VI
limit of d, (x), for the values of x in any neighbourhood of x1, is zero, provided
e be chosen sufficiently small; whereas, for a point x, of uniform convergence,
a neighbourhood of x1 can be found for which the lower limit of e (x) is
greater than zero.
The second theorem of ~ 352, shews that, in the statement of the theorem,
when e and n, have been arbitrarily chosen, it is necessary and sufficient that
a single integer n > n1 should exist, and also a neighbourhood (x - 81, x1 + 2),
at every point of which IRn(x) <e, the numbers 1, 82 being dependent
upon n. This is an alternative form of the theorem just stated.
The distinction between the three classes of points in the interval (a, b),
viz. (1) those at which the series is uniformly convergent, (2) those at which
the series is non-uniformly convergent, but at which the sum-function is
continuous, and (3) those points at which the function is discontinuous, may
be illustrated by means of figures* which indicate the regions of (x, y) in
the neighbourhood of (xi, 0), at which IR (x, y) is less than an arbitrarily
chosen e.
my
0                   P                    X
FIG. 7.
Fig. 7 represents the neighbourhood of a point P at which the convergence of the series is uniform. The blackened lines represent those
portions of the lines whose ordinates are l/n, 1/(n + 1), 1/(n + 2),... at which
I R, (x) 1, R,+i (x) I... are < e. These portions consist of all those parts of
the lines which are bounded by the curve y= f (x), there being also
possibly such pieces outside the curve. An area, for example semi-circular,
can be drawn, bounded by a portion of the x-axis containing P, and such
that for every point within it I R(x, y) <e; and that this should be
possible for every value of e is the condition that R(x, y) be continuous
at the point P with regard to the two-dimensional continuum (x, y).
* See Hobson, "On modes of convergence of an infinite series of functions of a real variable,"
Proc. Lond. Math. Soc., ser. 2, vol. i.




353]            The continuity of the szum-function             491
Fig. 8 represents the neighbourhood of a point P at which the function
s(x) is continuous, but at which the series is non-uniformly convergent. In
this case the function p, (x) is for all values of e < e0, discontinuous at P. The
y
1/n
1/(n+ 1)
e.          1~!/(n + 2)
0t                 P
FIG. 8.
value of (e (x) at P is itself finite; but the functional limits p, (xi + 0), b> (x,-O)
at P are both zero. The breadth of the blackened portions of the straight
lines parallel to the x-axis, which represent the portions of those lines at which
R,, (x) -< e, diminishes indefinitely as y approaches the value zero at P. In
this case no semi-circle can be drawn with P as centre, for all internal points
df which |.R (x, y) I < e; and thus the point P is one of non-uniform continuity,
the measure of non-uniform convergence being e0. In the figure, the convergence is non-uniform on both sides of P; it is clear however in what
manner the figure must be modified for the case in which the convergence is non-uniform on one side only of P. In case the measure of nonuniform  convergence be indefinitely great the figure will be essentially
similar to the above figure, whatever value of e be chosen; otherwise the
figure applies to an e which is less than the measure e0 of non-uniform convergence, viz. the saltus at P of R (x, y) J in the two-dimensional continuum.
Fig. 9 represents the neighbourhood of a point P  at which s(x) is
y
--- - _,-   --------— 1/n l
~ — ~~~~~~..x~/(n +l)


I zr3 i.
IT   \.   a


0


P
FIG. 9.




492


Functions defined by sequences


[CI. VI


discontinuous, the value of e being less than the measure of non-uniform
convergence of the series at P. In this case, as before, <,e(x) is finite at P,
and  b, (xi + 0), b, (x - 0) are zero; but, on the parallels to Ox intersecting
the ordinate at P, there are no intervals near P intersecting the ordinate, at
which ] R (x, y) I < e, but only points on the ordinate through P itself.
EXAMPLE.
As an example we may take the case in ~ 348, Ex. 3, R (x)-=  n22 and thus
R (X, y) =  X+
and we may suppose the domain of x to be the interval (0, 1).  In this case, the point
x=0 is a point of discontinuity of R (x, y), and we find that if e<~, the condition
IR (x, y)<e,
is satisfied for the space bounded by the x-axis, and by the straight line
Y  I) (X) [=      (2  ) ]2
The same condition is also satisfied for the space between the y-axis and the straight line
y=x [iE+ 4    -2
and thus the point x.=0 is a point of continuity of the function s (x), although the convergence is non-uniform at that point. If E>, then I R (x, y) <e, for the whole space
between the axes; and thus the measure of non-uniform convergence at the point x=0 is 2.
354. The necessary and sufficient conditions will now be determined
that s (x) may be continuous in the whole interval (a, b). First let us
assume that s(x) is everywhere continuous. Choose an arbitrarily small
positive number e; then, if n be sufficiently large, there are points in (a, b) at
which Rn ((x) < e. If P be such a point, a neighbourhood of P can be found
within which the condition   Rn (x) < e is everywhere satisfied; the size of
this neighbourhood may be extended in both directions until points p, q are
reached, at which I Rn (p) = I Rn (q) = e; for this follows from the fact that
Rn (x) is a continuous function of x. Every such point P in (a, b), at which
I Rn (x) < e, may, in a similar manner, be enclosed in an interval of finite
length; and in all internal points of such intervals the condition I Rn (x) < e,
is satisfied. Thus, for any fixed value of n which is sufficiently large, there
exists a finite, or infinite, set D,, of intervals in (a, b) which do not overlap,
such that, at every point which is interior to one of the intervals of the set
Dn, the condition I Rn (x) I < e, is satisfied; moreover, the intervals contain in
their interiors all points except a, b, at which the condition is satisfied. Two
intervals of Dn may abut on one another at a point in which IRn (x) is
equal to e; otherwise the intervals will be separated from one another.




353, 354]


The continuity of the sum-function


493


Let us consider the systems of intervals D,, Dn,+, D+2,,...; every value of
n being taken from a fixed value onwards. The whole set thus formed is such
that every point in (a, b), except the end-points, is interior to an infinite
number of intervals of the compound set; this follows from the fact that, for
any point x, a value of n, say n, can be found such that
i Rn, (x) I < e,  i Rnl+ (x) [ < 6... 
Moreover, intervals can be found with a, b as end-points, which for a
sufficiently large value of m belong to D,+m. In accordance with the HeineBorel theorem, established in ~ 68, a finite set of intervals can be selected
from the set which consists of Dn, Dn+1, Dn+2,... which contains every point
of (a, b) as an internal point of one of the intervals at least, and such that a, b
are end-points of two of the intervals of the finite set. It thus appears that,
on the supposition that s (x) is continuous in the whole interval (a, b), if n be
any integer chosen arbitrarily, a finite set of numbers n + tj, n + L2,... n + t,.
all greater than or equal to n, can be found, such that, for every value of x,
I Rm (x) I < e, where mi has one of the values n + L1, n + t2,... + t+,.. The particular value of m varies with x, but the same value of m is applicable to the
whole of one of a finite number of continuous intervals; also the set of values
of m is dependent on the chosen e. The intervals of the set will overlap; but
an overlapping portion may be considered to belong to one of the intervals
only, so that (a, b) may be divided into a finite number of parts, in each of
which, for some value of m, constant for that part, the condition I Rm (x) < e is
satisfied.
It should be remarked that the set Dn, for a fixed n, is not necessarily a
finite set; for, besides those intervals for which I Rn (x) > e, and the intervals
Dn themselves, there may be points of (a, b) which are not in either set of
intervals but are limiting points of end-points of the intervals Dn; and at
such points  Rn (x) l= e. For example, if | Rn (x) l= -i +  3Csin (  )
where a < c < b, and if e = 1/n2, the point c is a point of continuity of Rn (x),
and is a limiting point of end-points of those intervals for which I Rn (x) < e.
Conversely, if for every value of e a finite set of intervals exists, which
has the property described above, the function s(x) is continuous in (a, b).
For let us consider a point P of non-uniform convergence of the series. Then
for a given e, P is inside an interval for every point of which, for a fixed value
of m, I RA (x) I <, and it has been shewn that this is the condition that s (x)
may be continuous at P: hence every point of non-uniform convergence of
the series is a point of continuity of s(x). The condition has now been
obtained in the following form:The necessary and sufficient condition for the continuity in (a, b) of the
sum-function of a series u, (x) + u2 (x) +..., each term of which is a continuous




494                  Functions defined by sequences                    [CH. VI
function of x throughout (a, b), and which converges at every point of this
domain to a definite value s (x), is that, corresponding to any arbitrarily chosen
number e, and to an arbitrarily chosen integer n, the condition  R, (x) < e is
-satisfied for every value of x in (a, b), where m has one of a finite number
of values all greater than or equal to n, the value of m depending in general on
x, but being constant for all points x which lie in one of a number of finite
portions of the interval (a, b).
This theorem, which was first established by Arzela*, states that a certain
mode of convergence in the interval is the necessary and sufficient condition
for the continuity of the sum-function; and this mode has been termed by
Arzela*, convergenza uniforme a tratti (uniform    convergence by segments).
The term is perhaps not altogether appropriate, because the intervals are
dependent in number and length upon the arbitrarily chosen e. Uniform
convergence, and simple-uniform convergence, are special cases of this mode of
convergence; for in these cases the finite set of intervals which corresponds
to a given e, reduces to one interval, viz. the whole interval (a, b).
EXAMPLES.
32 X3       Xn
1. The series            1 ++!+ 3       c++t+
is convergent in any finite interval (a, b) whatever. It is shewn in elementary treatises
that the series converges to ex, for all rational values of x. In order to extend the proof
of the exponential theorem to the case of an irrational value of x, we observe that the above
series converges uniformly in the interval (a, b), since  I <-! ' where k is a fixed number
greater than lal, and \bl; and hence, in accordance with the theorem of ~ 345, since
S   is convergent, the given series converges uniformly in (a, b). It follows that the
sum-function s(x) of the series is continuous in (a, b). Further, the function ex has
been defined for an irrational value of x, by extension (see ~ 191) of the function as
defined for rational values of x; and it was shewn in ~ 37, that the function ex, so
defined for the whole domain, is single-valued at the irrational points, and therefore it is
continuous. The two functions ex, s (x) are both continuous in (a, b), and have identical
values at the rational points; therefore, in accordance with the theorem of ~ 173, they
are identical everywhere in (a, b). Therefore ex is the sum-function of the series in any
finite interval (a, b).
2. It is proved in elementary treatises that, for a value of x which is numerically less
than unity, the binomial series
nx+  (n - 1) 2  n (n - 1)... (n - r+ 1) X+
1 +nx+   2!      +... + 
* See the memoir "Sulle serie di funzioni," Mem. delta R. Accad. degli Sci. di Bologna,
ser. 5, vol. vIII, 1900. The proof given above was published by Hobson in the Proc, Lond.
Math. Soc., ser. 2, vol. I, p. 380..




354, 355]           The convergence of power-series                      495
converges to a suitable value of (1 +)., when n is a rational number. To extend the
theorem to the case in which n may have an irrational value, consider an interval (ni, n2)
of n, where n1 and n2 are rational numbers.
We have         n(n-1)...(n-    ) +l)' < (  1)... (+- 1)
r!                    r!
where N is the greater of the numbers I n i and I n2[. The number x remaining fixed, we
thus see that, for all values of n in the interval (nl, n2), each term of the series is numerically less than the corresponding term of the convergent series
N(N+ 1)
I+NIx+            x12+...;
therefore the series converges uniformly for all values of n in the interval (n1, n2). Hence
the sum-function of the series, for a fixed value of x, is a continuous function of n in the
interval (ai, n2). The function (l+x)"' of n, was defined in ~ 37, for irrational values
of n, by extension of the function considered as defined only for rational values of n;
and it was shewn that the function so obtained by extension is single-valued, and it is
therefore continuous. As in example (1), it now follows, that, for the fixed value of x,
numerically <1, the sum of the series is for all values of n in (nl, n2) represented by the
suitable value of (1 +x). The interval (ni, n2) is arbitrary.
THE CONVERGENCE OF POWER-SERIES.
355.  A series of which the (n + 1)th term is of the form anxn is called a
power-series. It will be assumed that the domain of x is a continuous one.
If the power-series ao + alx + a2x2 +... + anx  +... be such that, for a
positive value x = X, every term   ax1L is numerically less than some fixed
positive number A, the series converges absolutely for every value of x which is
numerically less than X.
The partial remainder
Rn,m (x) = axn + a,+Ixl+l +... + a.+m —1X^-+mis such that
Rn,m (x) I a1" I     a+ Al nXL I +...+ I a~+m- l+-l 1
<           -X) I  {1-A -X + IX2+...+XM
-Xy W       X      X1 X|-1 -< AXl             ) 
hence, for any fixed value of x such that I x < X, n may be so chosen that
all the remainders Rn,m(x) are numerically less than an arbitrarily chosen
number; and therefore the series converges at the point x.    It is also clear
that the convergence is absolute.
If the series diverge for the value x of X, it diverges also for every value
of x which is numerically greater than X.




496


Functions defined by sequences


[CH. VI


For if the series converged for a value x, of x numerically greater than X,
the condition of the preceding theorem would be satisfied by I x,, and hence
the series would converge absolutely for the value X; which is contrary to
the hypothesis.
The power-series may converge (1) for no value of x except zero, (2) for
every value of x, or (3) for a value X of x different from zero, but not for every
value of x.
In case (3), there exists a definite interval (- R, R) such that the series
converges for every value of x in the interior of the interval, and diverges for
every value of x exterior to the interval. The series may, or may not, converge
at either end-point of the interval.
The interval (- R, R) is called the interval of convergence of the series. To
establish the existence of this interval, we observe that, if the series converge
for any value x, of x, it converges for every value numerically less than I xa 1,
because, then, every term anxcn is numerically less than some fixed number A.
It has been shewn that, if the series diverge for any particular value of x, it
diverges for all numerically greater values. Hence those numbers I x which
are such that the series converges for x 1, must have an upper limit R, which
must also be the lower limit of those values of Ix for which the series
diverges; and this limit R determines the interval (- R, R) of convergence.
356. If* the power-series converge for a value X of x, greater than zero,
it converges uniformly in the interval (- Xo, X), where 0 < Xo < X.
We have
Rn (w) = Rn, (X) X) +r {Rn,p (X) - Rnp-1 (X)) (lX
-,Rnp(X)(R -X)(X     12 -=5RX (x    ) 1 -X
p=l              \X '  A
hence                                     x1
X  'IO x n'^p-i      X   X 1 ^,
Rnp d  n() <e 1that   (X) -  <  fe            of X; w   h 
X
provided n be so chosen that I Rnp(X) [ < e, for every value of p; which is
possible by reason of the convergence of the series for the value X of x. For
such value of n, and for all greater values, Rn (x) < e. -X X, for every point
in the interval (- X, X); hence the series converges uniformly in this
interval, including the end-points.
It follows from this theorem that the sum-function s (x) is continuous in
the interval (-X0, X).


* See Abel's (Euvres, vol. I, p. 223.




355-357]


The convergence of power-series


497


In case the series be convergent at the point R at the extremity of the
interval of convergence, we see from the theorem that the convergence is
uniform in the interval (- Ro, R), where R, < R; and that consequently the
sum-function is continuous in this interval, and is continuous at the points
R, and - R0.
We have thus established the theorem due to Abel*, that:If the series a, + ax  + ax2 +..., which converges within an interval of
convergence of which R is one of the ends, be such that the series converges
for x = R, then the sum of the series for x = R is continuous with the sumfunction for the interior of the interval of convergence.
This theorem may also be deducedt from the theorem in ~ 348. For we
have (-) -- (, for 0  x  R, and for all values of n; hence, since the
series  a,,Rn is, by hypothesis, convergent, it follows that the series E a,,Xn
n=O                                                        n=O
is uniformly convergent in the interval (0, -). Therefore the sum-function
is continuous in that interval, including the end-point R.
It should be observed that this theorem has been established only for a series
in which the powers of the variable are ascending, and that it is not necessarily
true in any other case. For example, the series x -  2x + x3 -... is convergent
within the interval (- 1, 1); and as the series is, for such values of x, absolutely
convergent, the series x +  x3 x- X2 + 1 X5 +  7 -  x4 +.has the same sumfunction within the interval, that function being log, (1 + x). At x= 1, the
series 1 — +  -   +... is convergent, and in acn cordance with the theorem its
sum is log, 2; but the series 1 + ~- +-+ + i- I +..., although convergent,
has the sum - loge 2 (see Ex. 1, ~ 334), which is not continuous with the sum
of the series x + ~x3 - iX2 +.
357. If the series a, + ax + a2x2 +... converge within an interval (- R, R),
and be such that, in every interval (- 8, 8), where 8 is an arbitrarily chosen
number < R, s (x) vanishes for some value of x which is not zero, the coefficients
ao, a1, a2,... must all be zero.
If 8 have any value < R, the function s (x) is continuous in the interval
(- 8, 8); hence, if e be an arbitrarily chosen positive number, 8, may be chosen
so small that I s (x) - a0 < e, if x be in the interval (- 8, 81); and by hypothesis
there is in the interval one value of x such that s (x) = 0; therefore I a0 < e,
and since e is arbitrary, we have a, = 0. The series a, + ax + acx +... converges in the interval (- 8, 8), and its sum vanishes for some value of x which
is not zero, hence the same argument as before establishes that a, = 0; proceeding in this manner, it can be shewn that a2, a3,... vanish.
* Abel's (Euvres, Crelle's Journal, vol. I, p. 223; also Dirichlet, Liouville's Journal, ser. 2,
vol. vii, p. 253.
+ Hardy, Proc. Lond. Math. Soc., ser. 2, vol. iv, p. 252.


I,


32




498


Functions defined by sequences


[CH. VI


It follows, as a corollary from this theorem, that there cannot be two
distinct power-series, each of which converges within some interval, and such
that in every sub-interval (-8, 8) there is a point distinct from zero, at which
the sum-functions have identical values.
358. Let us suppose that all the series
al + al2x + a1sX2 +... + al.xr-  +...
a21 + a22x + a23x2 +... + a2rxr-l +...,...,..........,......***........................................................
am + ab2x + an3X2 + $... + anrx1-...,.............................................
are, for every value of n, absolutely convergent at the point x = R; each series
is then both absolutely and uniformly convergent in the interval (- R, R). Let
us denote by u, (x) the sum of the series an + anx +... + anrx- +... at a
point x in the interval, and let Un denote the sum of the series
an. I+I n2,R+ +... +l,, anR"Rr j....
If the series            U + U +... + Un+...
be convergent, the series
t1 (x) + t2 (X) +... + U, (x) +...
is, in accordance with ~ 345, uniformly convergent in the interval (- R, R), the
end-points included; it follows that s (x), the sum of this series, is continuous
in the interval (- R, R).
The series u (x) + U2 (x) +... + u, (x) +... can be arranged as a series of
type Co2, by substituting for u1 (x), u2 (x),... the various series in powers of x.
Moreover this series is absolutely convergent; for the terms of the series
la,, al2x a  +    a ]+...    + a,.  +ar...+ a2I +  X...
are each less than the corresponding terms of the series obtained by writing
R for x; and the latter series is U1 + U2 +..., which is convergent.
Since the series u1 (x) + 2 (x) +... is absolutely convergent when the
power-series are substituted for u1 (x), u2 (),..., it remains (see ~ 335) absolutely convergent when it is arranged in the form
b, + bx + bx2... + b,x'-l+...,
where                    bl = a11 + a2 + a3 +...,
b2 = a2 + a a2 32 +...,


and its sum is unaltered. It has thus been shown that the continuous




357-359]


The convergence of power-series


499


function s(x) can be represented in the closed interval (- R, R) by the
power-series
b, + b2x + b3x2 +... + b2xr-l +....
The following theorem has therefore been established:If ui (x), u2(x),... be functions which can be represented by power-series
that are all absolutely convergent at the point R, and therefore in the interval
(- R, R), and if the series u1 (x) + u2 (x) +... be absolutely convergent at x = R,
then the series u1 (x) + u2 (x) +... converges in the interval to a continuous sumfunction s (x), which is the sum of the power-series obtained by substituting the
different power-series for the terms  1 (x), u. (x),..., and rearranging the
resulting series.
359. Let s(x) denote the sum of the power-series a1 + a2x + a3x +...
which converges at all points interior to the interval (- R, R); then, provided
x, x + h be both interior to the interval, the series
a, + a2 (x + h) + as (x + h)2 +... + a,, (x + h)'-4 +...
converges absolutely to the sum s(x + h); and if we expand each term
(x + h)n-l in powers of h, we have for each n a finite series of powers of h
which may be regarded as absolutely convergent. In accordance with the
theorem of ~ 358, if we arrange the series in powers of h,
(a, + ax + a32 +...)+ (a2  +2a3x +... + n - 1 a,,n-2 +...) +...,
this series will represent a continuous function of h, provided x, x + h lie in an
interval (- r, r), where r < R; and the sum-function is s (x + h). We have then
s (x + h) - s (x)
h)      = {Ia2 + 2a3x... + (n - l) ax-2 +. + hv. () + h2V2 (x) +.,
where v1 (x), v2 () are continuous functions of x. As h converges to zero, the
convergent series hv, (x) + h2v2 () +..., of which the sum is a continuous function of h, converges to zero; therefore (x +h) —  () has as its limit, when h is
h
indefinitely diminished, the sum of the convergent series
a2 + 2ax +... + (n- 1) anxt-2+...;
and thus the function s (x) possesses a differential coefficient, which is the sum of
the convergent series obtained by differentiating the terms of the power-series
which represents s (x). We have thus obtained the following theorem:If s (x) be the sum  of a power-series which converges within a given
interval, the function s(x) has a differential coefficient s' (x) at each point x
within the interval of convergence; and the series obtained by /means of a term
by term differentiation of the given series converges at such a point to the sum
s (x).
32-2




500


Functions defined by sequences


[CH. VI


THE CONVERGENCE OF THE PRODUCT OF TWO SERIES.
360. Let the two series
al + a,2 +- a3 +-... + cn +...,
b6+ b4 + b3  +..b  + b6 +...
be both convergent, and let the series
C + C + C3 +...+ Cn +...
be formed, where
cl = alb1, c2 = ab2 + a2b1,... cn = abn + abn-1 +... + ab,...;
then the last series may be termed the product series of the other two. If the
sums of the first two series be s, s', a sufficient condition for the convergence
of the product series is given by the following theorem:If* one at least of the two convergent series  a, b be absolutely convergent,
the product series  c is convergent, and its sum is ss', the product of the sums of
the two given series.
Let Sn, s,, denote the sums of the first n terms of the series Sa, 2b, and
Sn the sum of the first n terms of the series 2c; let us assume that the series
2b converges absolutely, and that 2' is the sum of the rnoduli of the first n
terms.
We have
Sn = (a + a2+... + an) (b + b2  +.* + bn) - b2an - b3 (an + an-)
- b4 (an + a,- + an-2)-... - bn (an + an- +.. + a2)
= SnSn2 - b2 (Sn - Sn-l) - b3 (Sn - Sn-2) -..- b (Sn - Si),
therefore
I Sn -- ss' l I SnsS'-ss' + I b2 I ] sn - sn-i I + I b3 Sn - Sn-2 |..+ bn I I Sn -s i1 j
Let m be an integer < n, and let m be so great that
|n ~- Sm \1n S - S,-+l |,  |S -n Sn-i
are all less than a fixed number t. We have then
Sn - SSI < snsn'- SS' + 1 I{ b, I + I b3 I +... +- bn-m+i I I
+ Sn - Sm-1i bn-n+2 + 5ln - Sm-21 Ibn-m+s 4+.  + | sn - S I bn.
Now there exists a fixed number A, such that, for all values of m and n, the
numbers
Sn - S'm —1i,n | SM —2  _ * * * - S[ n 1 i
are all < A; hence
n - SS' I < I Sn  - 'SS' +r (,'n-m+i - I) + A (/n'- 'n-m+)* This theorem was first given by Mertens, Crelle's Journal, vol. LXXIX (1875). The theorem
had been given, for the case in which both series converge absqoltely, by Caighy in the Cours
d'Analyse.




360-362]             The product of two series                    501
The numbers m, n can be chosen so great that SnS' - ss' I <0, where 0 is
arbitrarily fixed, and also such that I,- I'n-m+1 I is less than 0', where 0'
is arbitrarily chosen; and if this be done we have
I Sn- ss' ] < 0 + v (s' -:]') + A0 '.
But 0, V, 0' are each arbitrarily small; hence n can be so chosen that Sn - ss' I
is arbitrarily small; and thus the series Zc converges to the sum ss'.
361. In case both the series Sa, Eb be only conditionally convergent, we
are unable to assert that the series Z c converges; but in case it do converge,
its sum is given by the following theorem due to Abel*:In case the product series Z c of two convergent series  a, 2 b be itself also
convergent, its sum is the product ss' of the sums of the two given series.
Since the series
ca + a2 +... + an +..., bl + b2 +... +  bn +...
are both convergent, the series
a1 + a2x + ax2 -... + anxn-l...,
b, + bx -- b32 +... + bn'-l +...
converge absolutely for all values of x, such that 0 < x < 1; this follows from
the theorem established in ~ 355. The product series formed from these two
series is
C1 + C2X+ C+ X2 +... + C Xn-1 +.;
and this, in accordance with Cauchy's theorem as to the product of two
absolutely convergent series, is convergent for 0  x < 1, and converges to
s (x) s' (x). Since the series is by hypothesis convergent when x = 1, its sum
for this value is (see ~ 356) continuous with its sum, s (x) s'(x), for all values
of x which are numerically < 1; also lims (x) = s, lims' (x) = s', and hence the
x=l        x=l
series Zc converges to the value ss'.
It will be observed that the above theorem  of Abel does not give any
criterion which decides whether the product of two conditionally convergent
series is, or is not, convergent. Criteria applicable to special classes of cases
have been given by Pringsheimt.
TAYLOR S SERIES.
362. If a function f(x) be such that, at every point within the interval
(-R, R), it is the sum of the convergent series


* Crelle's Journal, vol. i, also (Euvres, vol. i, p. 226.
t Math. Annalen, vol. xxi, " Ueber die Multiplication bedingt convergenter Reihen."




502


Functions defined by sequences


[CH. VI


it has been shewn in ~ 359 that f'(x) exists at every point within the
interval, and that it is the sum of the convergent series
a2+ 2a3x +... + (n - 1) a,-Xn-2 +....
A second application of the same theorem shews that f" (x) exists, and is the
sum of the convergent series
1. 23 + 2. 3c4X+... + (n - 1) (n - 2) a,,-3 +....
Proceeding in this manner, it can be shewn that f(r) (x) exists, for every value
of r, at every point x interior to the interval (- R, R), and that the series
1.2.....+ + 2.  3...  (r + 1) a,.+2x +...
converges to the value f(3) (x).
It has further been shewn that, if x+h also lies within the interval
(-, R), the series obtained by arranging the series
a, + a2 (x + t) + a(x + h)2 +...
as a series in powers of h converges to the value f(x + h). The coefficient of
h1 in this series is
+(r+ 1)a,. (r + 2) (r + 1) 
a,.+i + (r + I) 2C,.+2X +     - a+3x2 +...,
1
which is - f(") (x). It has thus been shewn that the series
h,2           hr
f() + /f (.) + - f/ (x) +...   f  ()...
converges to the value f(x + h), provided x, x + h be both interior to the
interval (-R, R) of convergence of that power-series of which f(x) is
the sum.
This theorem is a particular case of Taylor's theorem for the expansion of
a function f(x + h) in powers of h, and has here been established for the particular case of a function f(x) which represents the sum of a convergent
power-series. It has moreover been proved that such a function possesses
differential coefficients of all orders within the interval of convergence of the
power-series.
We proceed to investigate the necessary and sufficient conditions that a
corresponding theorem may hold for functions which are not defined by
means of a power-series.
363. Let f(x) be a function defined for the interval (a, a  X), where X
is a positive number, and satisfying the conditions (1) that, at the point a, the
first n - 1 derivatives of f(x) on the right all exist; (2) that at every point x
such that a < x < a + X, the first n - 1 differential coefficients off (x) all exist,
and are continuous, being also continuous with the derivatives on the right




362, 363]


Taylor's series


503


at a; (3) that f(n) (x) exists at each point in the interior of the interval
(a, a+ X), having values which are finite, or infinite, with fixed sign.
Let the number K be defined by the equation
f (a + ) -f (a) - f' (a) -  f (a)-. -       f (1-) (a) = h- K
where h is such that 0 < h < X, and v is a fixed positive or-negative integer, or
zero, but such that n - v is positive; the derivatives f' (a), f" (a),... are those
of f(x) on the right at a.
Next, let F(x) denote the function
f(a.A)-f()-(  +h-)f'()_ (a + h - X)2
(a + ht) -f (x) - (a +   h- x)f' (x)  (n2 +  (-a + h 
(CL +- h - x)~ —h
(,_ -  1)! f(~-~ (x)  - (_  + h - x)-vK,
(n -i)
where K has the value defined above, and x is in the interval (a, a + A). The
function F(x) is continuous in the interval, and F'(x) exists everywhere
in the interior of the interval. Moreover, since F (x) vanishes for x = a, and
for x=a+ h, it follows from the theorem of ~ 203 that F'(x) vanishes for
some value of x within the interval (a, a+ h); let this value be a + Oh,
where 0 is a number such that 0 < 0 < 1.
Since
(a + h - X)fn- )
F' W  =    -  n      '-   (x) + ( - v) (a + h - x)n-v-1,
we see that          K= (    ( -(  1) I!f() (a + Oh);
therefore, from the definition of K, we have
A2 /hn-1
f(a + h) =f(a) + f' () + ( a)  + f"... +  - f (1-)  ) (a)
2!            (n (- 1)!
+      (a —)   /) ( a) (a + Oh).
- -v) (n-!       f n)
It is clear that a corresponding result holds for an interval on the left of
the point a, provided corresponding conditions hold as to the existence of the
differential coefficients; the derivatives at a being in this case those on
the left.
In case f(x) be defined for an interval (a - X', a + X), and the first n - 1
differential coefficients of (x) exist at every point x in the interior of the
interval, andf ) (x) everywhere exist in the interval of which the end-points are
a, a + h, where h is any number such that - X' <h < X, the theorem holds for
every such value of h, positive or negative; f'(a), f" (a),... denoting in this
case the differential coefficients at a.




504


Functions defined by sequences


[CH. VI


This theorem is frequently spoken of as Taylor's theorem, although that
name was originally, and is still usually, applied to the case in which it is
possible to suppose n to be indefinitely increased, so that the series becomes
an infinite convergent one.
The expression Rn =     ( - O    f(n) (a + Oh), where v is a positive or
(n - v) (n - 1)!
negative integer such that n - v > 0, is spoken of as ' the remainder in Taylor's
series." In this general form it was obtained by Schlomilch* and by Rochet.
The particular case in which v = 0, Rn, = W!(n) (a+ Oh), is known as Lagrange's
form+ of the remainder in Taylor's series; another particular case, due
originally to Cauchy~, of the general form given by Schlomilch, is that in
hn (1 _)n-i
which v = n -1, or Rn    (1=  -    (n) (a + Oh).
364. If f(x) possess differential coefficients of all orders within a
prescribed interval (a - X', a + X), then, provided Rn have the limit zero,
when n is indefinitely increased, for each value of h, the series
f (a)+ li/f '(a) + 2 /" (a) +...+   f (a)...,
where - X' < h < X, is convergent, and has f(a + h) for its limiting sum. This
is Taylor's theorem in the original sense of the term.
It will be observed that the existence of differential coefficients at the
extreme points - X', X has not been presupposed, but only their existence for all
points for which -  ' < h < X. If the condition lim R,, = 0 be satisfied for each
n=oo
value of h within the interval (- X', X), and if the series converge also for h = X,
then, since it is a power-series, it follows from the theorem of ~ 356, that at
h = X the series converges to the value f(a + X).
The value of 0, in any of the forms of the expression for R,, is in general
dependent upon n; and consequently it is not a sufficient condition of convergence of the series that Rn have the limit zero as n is indefinitely increased,
whilst 0 remains fixed, even though this be the case for each fixed value of 0 in
the interval (0, 1). In connection with the theory of non-uniform convergence
of series we have already seen in ~ 349, that a function Rn (x) may have the limit
zero, as n is increased indefinitely, for each fixed value of x in a given interval,
and yet that lim R,, (x) may not necessarily be zero when x varies with n.
* Handbuch der Differential- und Integralrechnung, 1847.
t Mem. de lAcad. de Montpellier, 1858. See also Liouville's Journal, ser. 2, vol. II, pp. 271
and 384.
+ Theorie des Fonctions, vol. i, p. 40.
~ Calcul Diff., p. 77.




363-365]


Taylor's series


505


nOh
For example, if Rn = (1 +  h), then RB has the limit zero for every fixed value
of 0; but if 0 = 1/n, Rn has the limit he-7.
A sufficient condition for the convergence of the series is that Rn, for each
fixed value of h within the given interval, as n is indefinitely increased, should
converge to zero uniformly for all values of 0 in the interval (0, 1). Thus, for
each value of h, and each value of an arbitrarily chosen positive number e, a
value n, of n, would exist such that
hn (I - 0)V
(n-v!) (n - 1)!f         <
provided n i - n, for every value of 0 in the interval (0, 1).
This condition, though sufficient for the convergence of the series, has not
been shewn to be necessary. An investigation, due to Pringsheim*, will now be
given of the necessary and sufficient conditions for the convergence of Taylor's
series.
co
365. If the series  c, hn converge for every positive value of h which is
0
< B, and if f(x) denote the sum of the series 2c,, (x - a)n, where a is a fixed
number, and 0   x - a < R, then (1) f (x) possesses for every value of x, such
that a  x < a + R, a definite finite value,; and (2) for every x, such that
a < x < a + R, f(x) possesses finite differential coefficients of every order,
and at a, derivatives on the right of every order; also (3) the condition is
satisfied that --   f(n) (a + h). lCn+p converges uniformly for all values of
(nsatisfied that  + p)! 
h, k such that 0 < h < h + k - r, to zero, as n is indefinitely increased, where
r < R, and p is any arbitrarily chosen integer, which may be zero.
This theorem contains necessary conditions for the convergence of Taylor's
series for an interval on the right of a point a. A similar statement holds as
regards an interval on the left of a; and it is clear that the theorem can be
stated so as to apply to the more general case of a neighbourhood which
contains a in its interior.
It has been shewn in ~ 359, that the function f(x) is differentiable within
the interval (a, a + R) of x, and is obtained by means of a term by term differentiation of the series. The same theorem may be applied to the function
f'(x), and to the series which represents it, and thus successively to the
higher derivatives off(x). We have therefore
(P) () =       -        - (n- p + ) C. (-p  - a)n-;
n=p


* See Math. Annalen, vol. XLIV, "Ueber die nothwendigen und hinreichenden Bedingungen
des Taylor'schen Lehrsatzes fur Functionen einer reelen Variabeln."




506              Functions defined by sequences               [CH. VI
hence f(a) = c, fP) (a) =p! p, where f(m) (a) is the derivative at a on the
right; and thus the conditions (1) and (2) are satisfied.
We have now
f(P) (a + h) =  -    f(n) (a). 91
fX, (C + h)=  E (-_p)!
where 0 _- h < R.
From the theorem of ~ 355, that a power-series converges absolutely at all
points within its interval of convergence, we see that the function +(x),
defined, for the interval a ' x < a + R, by
0
has properties similar to those of f(x); and thus that
o        oon, (a + h) =   I n I hn        (_), ) (a)). -it,
0        0
for 0 ~ h < R. The functions  ( (a + A), +(P) (a + A) are continuous functions
of h in the interval 0 _ h < R; and for each value of p,
If)> (a + h) |- (P)( (a + h).
In order to prove that the condition (3) is satisfied, it will be sufficient to
shew that the corresponding condition is satisfied for the function b(n) (a + A).
If 0 - h _   h + k < R, we have
X       (n) 1           ON)
(a      + hk) =! '  (a) (h + k) =     (a + A);
o  nI!              o  it;
and it will now be shewn that the series Z 1 t (n) (a + A) kn converges uniformly
for all values of h and k which are such that 0 -O  h ~ h + kc r, where r < R.
Let / (a + h + k) = Sn (h, k) + Tn (h, k), where Si, denotes the sum of the
first n terms of the series., ( (a + h) kn, and Tn denotes the remainder
after those terms. Let A, k have arbitrarily assigned values which satisfy the
condition above stated, and let 8 be a positive number less than h and I, and
such that h+ +  + 28 < R; also let ', } be two numbers in the interval (- 1, 1),
arbitrarily chosen. In case h = 0, k > 0, the positive number 8 is to be so
chosen that k + 28 < R, and that 8 < k; also ' is to be in the interval (0, 1).
Similarly if h > O, k = 0, the number 8 is to satisfy the conditions h + 28 < R,
8 < A; and?q is to be in the interval (0, 1). If h = 0, k = 0, then 8 is to be




Taylor's series


507


so chosen that 28 < R, and both 4 and q are to be in the interval (0, 1). We
have then
C (a + h + k + + T8) = Sn (h + t8, A + 78) + Tn (h + 8, 1 + v8),
and hence
T (h + ~8, k + 16) -  (a + h + F+  +  8) - (a + +    kh ) 
+1 Sn (h + g, k + 8) - S, (h, k) I + TX, (A, k).
If e be a prescribed positive number, n may be chosen such that, for
the prescribed values of h and k, T, (h, k) <  e. Since ( (a + h + k) is a continuous function of h + k, and S, (h, k) is a continuous function of the two
variables h, k; it follows that 8 can be so fixed that
I ((a + h + k + +  +  8) -  / (a + h + ) < <,
and also              | Sn (h +  8, k +  8s) -  f (h, A) | < 3e.
The numbers n, 8 have now been so chosen that, whatever values ' and Vq may
have, subject to the restrictions already stated,
Tn (h +, k - +  r8)< e;
moreover, since the series, -  (n) (a + h) kI contains only positive terms, we
have
T, (h + ~8, lc + r8) < e,
for v _ n.
It has thus been shewn that the series representing the function  (a+h+ k),
considered as a function of the two variables (h, k), converges uniformly for
a certain neighbourhood of each single point (h, k). Since there are no points
in the neighbourhood of which   b (a + h + k) converges non-uniformly, it
follows from the theorem of ~ 342, that the function converges uniformly
for all values of h, k such that 0   h- h + k= r, where r < R.   Thus
T, (h, k) < e, for v _ n, and for all values of h, k which are not negative, and
are such that h + k - r.
The analogous result holds a fortiori for the series
OZ   1
o     (n + p)! '( ) (a + h). kn+P,
o (n +p) v
and also for the series
~o   1
p_     )   (n) (a +  lb). kn-p,
E (n -p)!   ) (a + h).
which is obtained from Z -I 4 (n) (a- +h) kn by differentiating p times with
respect to k.
respect to k.




508


Functions defined by sequences


[CH. VI


Since all these series contain positive terms only, it follows that n can
be so chosen that
1 -  (V) (a + h). Jcv+P < 
for v > n, and 0  h h  + k =< r; and thus the condition (3) in the statement
of the theorem is satisfied, since I /(v) (a + h) l _ (v) (a + h).
366. If f(n (x) be defined for every integral value of n, where x is such that
a - x < a + R, and if, for some one value of p which is a positive or negative
integer, or zero, it satisfy the condition that (  )!f   (a + h) Ic+p converges to
zero when n is indefinitely increased, uniformly for all values of h, k, such that
O _ h: h + kc - r, where r < R, then the same condition holds for every value of
p which is integral or zero.
If we denote (n  )   (2) (a + h) k'+P by Fp,, (h, k), we have
Fp+l,n (h, k) = -  +  Fp,n (h, k);
and hence, since k < r, I Fpn (h, k) < Fpn,, (h, k) l, if in 4p + 1 > r. It follows
that lim Fp+,,n(h, k), and generally lim Fp+q,, (h, k), for q > 0, converges uniformly
91= O                     n=oo
if Fp,n (h, k) does so.
Again      F,_, (h, kc) = -(n) ( + p) 1
(n +p - 1)!
= Fp, n(h, k +  ) (k  +P   + P'
if r < R be fixed, 8 can be so chosen as to be positive, and such that r + 8 < R.
Hence, if 0 < h -  r,
r y\n+p-ln + pL )
Fp-,n (h, k) < I Fp (h, k + 8) (+) 
if n, be so chosen that
r+'S/ )  +-   P <  1, and Fv, (h, k + 8)<,
for 0 < h _ h + k -  r, we have, then, for such values of n, I Fp-i,n (h, k) < e.
It is now seen, also, that the corresponding result holds for Fp_-n(h, k)I,
where q > 0.
367. The necessary and sufficient conditions that Taylor's theorem should
hold for the function f(a + h), where 0 < h < R, can be most simply expressed
if Cauchy's form of the remainder be used, and may be obtained as follows:The condition as to the existence of differential coefficients of all orders
being assumed to be satisfied, it has now been shewn in ~ 365, to be a necessary




365-367]


Taylor's series


509


condition for the validity of Taylor's series, that (-1  )!(* (a + h) k~"(n - 1f if
should converge to the limit zero, as n is indefinitely increased, uniformly for
all values of h and k such that O h h    + k <r, where r < R. If we write
Oh for h, and (1 - ) h for k, then, if the condition be satisfied, the expression
(n- )!(X (a + Oh) (1 - e)'-lhn(n - 1)!
converges to zero, when n is indefinitely increased, uniformly for all values of
O and h such that 0 < 0 < 1, and 0 < h < r; and it follows that, for each value
of h, (  - 1) f  (a + Oh) (1 - O)n-~lh converges to zero uniformly for all values
(n - 1i)!
of 0. It has been shewn in ~ 364, that this last condition is sufficient to
secure the convergence of the series to the sum f(a+h). We have now
established the following theorem:That the function f (a + h), defined for all values of h such that 0 < h < R,
1
may be represented for all the values of h by the series fS () (a) ha, it is
o n
necessary and sufficient, (1) that f () have differential coeficients of all orders
for a < x < a + R, and derivatives at the point a on the right, of all orders;
and (2) that Cauchy's remainder  -      1) f) (a + Oh) (1 - O)n-lhn, for each
value of h such that 0 - h < R, converge to zero, when n is indefinitely increased,
uniformly for all values of 0 in the interval (0, 1).
In case Lagrange's form of the remainder in Taylor's theorem be employed
instead of that due to Cauchy, the necessary and sufficient conditions cannot be
stated in so simple a form. The following theorem has reference to this form
of the remainder:
In order that the function f (a + h), defined for all values of h such that
0 - h < R, may be represented, for all the values of h, by the series  P.! f (l) (a),
it is necessary, besides the condition of unrestricted differentiability previously
stated, that - f  ') (a + Oh) hn should converge, for each value of h such that
o  h<   R, to the limit zero, when h is indefinitely increased, uniformly for all
values of 0 in the interval (0, 1). It is not necessary, but it is suficient, that the
expression should converge to zero for each value of h such that 0 < h < R,
uniformly for all values of 0 in (0, 1).
In accordance with the theorem  proved in ~ 365, it is necessary that
kf(n) (a + h) c should converge to the limit zero, when n is indefinitely
increased, uniformly for all values (h, k) which are such that 0 - h < h + k  r,




510


Functions defined by sequences


[CH. VI


where r < R. If we write h for Al, and Oh for h, we see that this condition
includes the condition that  f (n) (a + Oh) hn should converge to zero, for each
value of h such that 0 _ h < 1 R, uniformly for  all values of 0 in the interval
(0, 1). This condition is therefore a necessary one.
Let us next assume that -f (1) (a + Oh) hA converges to zero, for each fixed
value of h such that 0 < h < R, uniformly for all values of 0 in the interval
(0, 1). Let p, r be two positive numbers such that p  r< R, and in the
0on  -!
1
It follows that the series E,.f (n (rt + p)?  converges absolutely for every
positive value of k which is < r, and therefore for every positive value of k
which is < R, since r may be taken arbitrarily near to R. In the series.f (n) (a + p) kn, the functions f/() (a + p) may be replaced by the absolutely convergent series in powers of p; and by the theorem of ~ 358, the
terms of the series E.f(n) (a + p) k, thus expressed, may be rearranged
without affecting the convergence or the sum of the series. The series then
becomes S -!f(n) (a) (p + k)1, and this series converges for p < R, k < R, i.e.
for p + k < 2R. It follows that the series E  f- ( (a) h converges if h < 2R.
Now it is clearly not necessary for the validity of Taylor's theorem within the
range 0 < h < R, that the series should converge for all values of h which are
< 2R; for R might be the upper limit of the values of A for which the powerseries converges, and then the series could not converge for values of h which
are > R. It has thus been shewn that the condition for all values of h in the
interval 0 < h < R, is not a necessary one. It is clearly however a sufficient
condition.
hitn
It was remarked by Cauchy* that the series E  f (?!) (a) may be convergent
in an interval, and yet that its sum need not be f(a + h). This happens
whenever the remainder R, in Taylor's theorem, which is defined as the
difference between f(a +h) and the sum of the first n terms of the series,
converges, for each value of x, to a limit which is different from zero, as n
is indefinitely increased.
1
Let the function f(x) be defined by f(x) = e ~2, for x2 > 0, and f(0) = 0;
it can easily be shewn that this function and all its differential coefficients
* Calcul Diff. p. 103; see also P. Du Bois Reymond, iMath. Annalen, vol. xxi, p. 114.




367]


Taylor's series


511


exist and are zero at the point x = 0; and that for x2 > 0, the remainder in
1                          1
the Taylor's series has for its limit e x2. If now   ) (x) = ex + e Ad, (x2 > 0),
h2l
b (0) = 1, and the series   -T ~(n) (0). in the neighbourhood of the point x= 0
be formed, then the series converges, not to the value 4 (h), but to the value eh.
EXAMPLES.
1. Let f(x)=(l +x)p; then, in a neighbourhood of the point x=0, we have
f (x)1=+px +  P P   X2.... +  (      p-    2) (     Rn!       "         ((n-i)!
where R, can be expressed in Lagrange's form by
p(p - 1)...(p-n+l1)   xtt
n          (1 + Ox)'n-P'
or in Cauchy's form by
( - 1)...(p-n+l 1) (1 - )tx1 -(n- )!       ( + Ox)n -P
Using Cauchy's form, we see that
1Rn < I P  -1)...(p-7 nl+) x:.-l —(-i)!                '
provided n>p. If xl <1, the expression
p (p - 1)...(p-n+l)
(n-l)!
continually diminishes as n is increased: for, denoting it by z,,, we find
un      n
where E is a fixed positive number < 1- I x, provided n be sufficiently great, and it follows
that the limit of u, is zero; and thus lim Rt=0.  The series therefore converges for all
values of x such that Ix < 1.
To find the limit of p(p  ) —(p- n+) I when n is indefinitely increased, suppose
n I
first that p + 1 is negative, say - k. We may write the expression in the form
(1+k)(/) k      l    ),
and this is >l+Ik (+ 2+...+); thus the limit is indefinitely great. Next suppose
that p +l is positive. Then the expression may be written in the form
P (p -- 1)... (p-X+2) (1  + )(1- X + 1)1p+       f *   -S
— (X-        -        x A      x+'       x- -2] -\ X  '"
where X is the integer next greater than p + 1; this is less than
p (-l )...(p- X+2)            1
(-)!    (1 +_X) (1 +I+   1)..(l +P+
X   X-~i         n



Functions defined by sequences


[CH. VI


or than              -l.)                        1
hence the limit, when n is indefinitely increased, is zero. If p= - 1, the limit is unity.
If x= 1, Lagrange's form of the remainder shews that the series converges if p> - 1.
The series diverges if p< - 1, because the general term of the series increases indefinitely
with n. The series oscillates if p= - 1.
If x= - 1, Cauchy's form of the remainder shews that if p - 1> -1, or p>O, the series
is convergent. It is divergent if p<O, for the sum of n terms of the series is
(p - 1) (p -- 2)...(p-  + 1)
(n - i)!.) (       -1)r  a"2. Let*f(x)= E      r   a- 2r+x ' where a>l.  For this function
r==0  ' o -2
f(o)=e -,  f(2-) (0) = 0  f (2k)(O)=(- )(2k)! e-a~ I +;
thus the series for f(x) is
U ( - )k e ai + 1 X2k
o
which is everywhere convergent.
The sum of the series, for x=0, is f(0), but in every neighbourhood of x=0, the sum of
the series and the value of f(x) are different except at most at a finite number of points..x 1   a3. Let f (x) =   -    a —     where a> 1. For this function, the Maclaurin's series
r=O T ' a-2r'"x'L'
is 2 (-l)kea2k+1x2k, which diverges for every value of x except  =0.
4. (- 1)"  1
4. Lett /f( )=- =       1 a X, where a> 1.   This function is continuous on the
o  n! 1+ax
right of the point x=0, and has derivatives on the right of all orders at that point;
the Maclaurin's series 2 (- 1)" (-. x thus obtained, converges for all positive values
of x, but does not represent the function f(x).
00   1  1
5. Let f(x) =, I  I    where a> l. For this function the Maclaurin's series does
0 n!l +anx
not converge in any neighbourhood of the point x=0.
MAXIMA AND MINIMA OF A FUNCTION OF ONE VARIABLE.
368. It has been shewn in ~ 207, to be a necessary condition that a
function f (x) may have an extreme at the point x= (, that the differential
coefficient at that point should be zero, provided the function be such that a
differential coefficient at x = 0 exists. Let us assume the function to be
such that the first n differential coefficients f' (x),f" (x),... fn) (x) all exist
and are continuous, at every point x such that - 8 < x < 8. Let us further
assume that f' (0), f" (0),... f(-l/) (0) are all zero, and thus that f(n) (0) is
that differential coefficient of lowest order which does not vanish at x = 0.
* Pringsheim, Minchener Berichte, 1892, p. 222.
t Pringsheim, Math. Annalen, vol. XLII, p. 161, and vol. XLIV, p. 54.




367, 368]


Maxima and minima


513


We have then      (x)-f(0)=      f(n) (Ox); where 0<0<1, and x is
such that - 8 < x < 8. Since f(n) (x) is continuous at x = 0, a neighbourhood
(- 8', 8') of that point, interior to (- 8, 8), can be so determined, that f( n (Ox)
has the same sign as f/() (0), provided - 8' ' x _ 8'. If n be odd, the sign of
f(x) -f(0), in the interval (- 8', 8'), depends upon that of x; and therefore
f(x) has neither a maximum     nor a minimum    at the point x=0.    If n
be even, the sign of f(x)-f (O) is the same as that of f' (0), in the whole
interval (- 8', 8'), and therefore f(x) has a maximum or a minimum at x = 0,
according as f(n) (0) is negative or positive. The following theorem for
determining whether a maximum, or a minimum exists at a point at which
the differential coefficient of a function f(x) vanishes has therefore been
established:If the first n differential coefficients of a function f(x) all exist, and are
continuous, at all interior points of the interval (- 8, 8); and if f ( (x) be the
differential coefficient of lowest order which does not vanish at the point x = 0,
then (1) if n be odd, there is neither a maximum nor a minimum of the
function f(x) at the point x= 0; and (2) if n be even, the point x = 0 is a
maximum or a minimum of f(x), according as f'(n (0) is negative or positive.
It is unnecessary for the application of the criterion given in this
theorem that f(x) be capable of representation in a neighbourhood of the
point x = 0 by a convergent power-series. The theorem cannot be applied
to the case of a function with differential coefficients of all orders, when
they all vanish at the point x = 0.
EXAMPLES.
1
1.* Let f (x)=X2- e X2, and f(O)=O. In this casef'(0) = O" (O)=2; andf' (x), f" (x)
are continuous in any neighbourhood of x=0. The theorem establishes that f(x) has
a minimum at x=O, although f(x) cannot be represented by a power-series in any
neighbourhood of the point.
1
2.* The function defined by f(x)==e x2, f(0)=0, has a minimum at x=0; and
yet the theorem is not applicable, because the differential coefficients of all orders vanish
at x=0.
1
3.* The function defined by f(x) =xe x2, f(0)=0, has neither a maximum nor a
minimum at x=0. As in (2), the above theorem is in this case inapplicable.
* These examples are given by Scheeffer, Math. Annalen, vol. xxxv, p. 542.


H.


33




514


Functions defined by sequences


[CH. VI


TAYLOR'S THEOREM FOR FUNCTIONS OF TWO VARIABLES.
369. Let us assume a function f(x, y) to be defined for all values
of x, y in the domain defined by a -       x   a + 8, b - 8' y  b+3'.
Under proper conditions as to the existence and continuity of the partial
differential coefficients off(x, y), of a finite number n of orders, it is possible
to obtain an expression for f(a+ h, b +k)-f(a, b) consisting of terms
involving the first n powers of h and k, together with a remainder analogous
to.the remainder in Taylor's theorem, such expression being valid for values
of h, k, such that I h < 8, 1 k < 3'. It is however, for the present purpose,
unnecessary to consider the least stringent set of conditions relating to the
partial differential coefficients of the various orders, which are sufficient to
allow the extension of Taylor's theorem to the case of a function of two
variables. It will here be assumed that, for all values of x and y such that
a - 8< x < a + 3, b - 8' <y < b + 8', the partial differential coefficients of
f(x, y) of the first n orders all exist, and are finite; and further, that they
are all continuous, for this range of values of x and y, with respect to (x, y).
In accordance with the theorem of ~ 240, the order of differentiation, in each
of the mixed partial differential coefficients, is in this case immaterial.
Taking values of h and k which are numerically less than, 8'
respectively, let f(a + th, b + tAc) be denoted by F(t), the variable t having
the domain (- 1, + 1). The conditions contained in the last theorem  of
~ 236 being in this case satisfied, the differential coefficient F'(t) of F(t)
exists, and is equal to (h - + k) f(x, y), where x =a +th, y=b + tk.
Similarly, it is seen that all the differential coefficients
F" (t), F"' (t),... F) (t)
exist and are continuous; and that
F( (t) = (ha  + k a- f (, ).
In accordance with the theorem of ~ 363, we have
t2             " tnl-1         tn
F(t)= F(O) (0) +       F"(0 ( O)...+  +.(. — 1 ) (0) +  F(n) (0t),
(n -!             n!
where 0 is a number such that 0< 0< 1.
Since this holds for t= 1, we see that
f(a+h, b +k)=f(a, b)+ (h     +k   )f(a, b) +  (    + k    f(a, b)+...
(, b) +                                +    )   a + Oh b + Ok).




369, 370]


iaxima and minima


515


This is an extension of Taylor's theorem to the case of a function of two
variables. It has been established for all values of h, k such that h I < 8,
k ] <', on the hypothesis that f(x, y) and all its partial differential
coefficients exist for all values of x and y such that a - 8x <   a<  + 3,
b - 3' < y < b + 8', and that they are all continuous with respect to the twodimensional continuum (x, y).
MAXIMA AND MINIMA OF FUNCTIONS OF TWO VARIABLES,
370. Necessary and sufficient conditions have been stated, in the
theorem of ~ 242, that the point (0, 0) may be a point at which a function
f(x, y) has a maximum, or a minimum. The general theory of maxima and
minima of functions of two variables has been discussed by Scheeffer*,
Dantscher+, and Stolz+, the last of whom has extended Scheeffer's method
to the case of functions of any number of variables. The account which will
here be given of the general theory is based upon the investigations of
Scheeffer, as modified by Stolz.
Let the function f(x, y) be such that either f(x, y)-f(O, 0) is representable in a neighbourhood of the point (0, 0) by a convergent series
consisting of powers of x and y, or else that it is such that the theorem
of ~ 369 is applicable, so that
f (, y) -f (, 0) = Gn (x, ) + Rn+1 (x, y),
where Gn (x, y) consists of terms of dimensions not higher than n, in x and y;
and Rn+~ (x, y) is either a convergent series of which the terms of lowest
dimension are of the order n + 1, or has the form of the remainder given in
~ 369, consisting of terms of dimension n + 1 in Ox, Oy, where 0 < 0 < 1; and
in the latter case it will be assumed that the differential coefficients in that
remainder are limited in the whole domain. It will be shewn that, under a
certain condition, the problem of determining whether the point (0, 0) is
a point at which f(x, y) has a maximum or a minimum is reducible to the
solution of the corresponding problem relating to the rational integral
function G (x, y). The following general theorem will be established:The function f (x, y) having in the neighbourhood of (0, 0) the character
above described, if an index n and two positive numbers c, 8 can be so
determined that (1) for all values of x such that 0 < Ix I < 8, the upper and
lower limits of G, (x, y), Jor a constant value of x, and for all values of y
in the interval (- x, x), are in absolute value not less than c  ax n; and
(2) that, if 0 <[ y | <, the upper and lower limits of G, (x, y), for a constant
* Math. Annalen, vol. xxxv.
+ Math. Annalen, vol. XLII.
+ Beri~hte of the Vienna Academy, vols. xcix, c; also Grundziige, vol. i, p. 211.
33-2




516


Functions defined by sequences


[cH. VI


value of y, and for all values of x in the interval (- y, y), are in absolute value
not less than c ly n; then the two functions f (x, y), G, (x, y) have both either a
proper maximum, or both a proper minimum, or both neither a maximum nor
a minimum, at the point (0, 0).
To prove this theorem, we first observe that Rn+i (x, y) can be regarded
as a homogeneous function of x and y of degree n+ 1, in which the
coefficients depend upon x and y. By giving each of the coefficients its
greatest possible value, for I x < 8, l y <, we see that
| Rn+i (X, y) I < Ao x n+1 + Al x n I y I +... + A,+1 I y I n+1;
where A0, A,... An+, are positive numbers.
If now I y - x, we have
Rn+ (, y) < (Ao + Al +...   An+ ) |11 ]x I;
hence we see that a number 8' <   can be so chosen that
-Rn+i (X, y) | < e X,
where e is an arbitrarily chosen positive number, provided I x < 8', y i 5 x.
In a similar manner we can shew that 8' can be so chosen that
I Rn+ (X, y) l< eyln,
provided Ixl  ly, and Iy< 8'.
Let now the upper and the lower limits of GO (x, y), for a constant value
of x, and for all values of y such that y I < x 1, be denoted by Gn (x, 4 (x)),
G, (x, 4 (x)) respectively. Also let the upper and the lower limits of
GO (x, y), for a constant value of y, and for all values of x such that x1  - l Y,
be denoted by G, ( (y), y), Gn (J (y), y) respectively. We have then,
provided Ix I<', and!y -< x1,
Gn (x, 0 (X)) - e ] x I 2 < /(, y) -/f(O, ) < Gn (x,  (x)) + e | I;
also, provided  y <   x', Ix  I y, we have
G, (, (y), y) -E I y I < f(x, y) - f(O, O) < Gn( (y), y) + e y 
First, let us assume that O,} (0, 0) is a proper minimum of GO (c, y), and
that the conditions of the theorem are satisfied. By the theorem of ~ 242,
Gn (x, 4) (x)), Gn (r (y), y) are both positive, for sufficiently small values of
x and y; we may suppose 8' to be so small that these conditions are satisfied,
provided Ix < 8', | y < <'.
We have then    Gn(x, O(x))clIxI n, if O< x <8', l       yll:x; and
Gn((y), y) _ c Iyln, if 0< y < 8', 1Ix  - y.
It now follows that
(c-e) I x  <f(x,y)-f(0,0), for 0< x <8I<     y}   x,1




370]                  Maxima and minima                        517
and that
(c-e) yl <f(x,y)-f(O, ), for 0<I y<8', x         y.
Since e can be chosen so as to be less than c, we see that f(x, y) -f(O, O)
is positive for all values of x and y such that 0 < I x < ', 0 < y < 8', and
therefore f(0, 0) is a proper minimum of f(x, y).
Next, let us assume that G,(0, 0) is a proper maximum of Gn (x, y); then
Gn (x, ( (x)), Gn (I (y), y) are both negative, for sufficiently small values of
x and y. We therefore assume that
Gn (x, p())  - c lxn, for O < x <', and lyl [ x;
and that
Gn((y),y),  -c yI, for 0< yI <3', 1xl    IyI
We have then f(x, y) -f(0, 0)<-(c -e)[ x, for 0< x <8', and
I y\ Cx i; and alsof(x, y)-f(O, 0)<-(c-e)lyln, for < Iy}<8', \x\- iy!.
Since e may be taken to be < c, it follows that f(0, 0) is a proper maximum
of f(, y).
Lastly, let us assume that Gn (0, 0) is neither a maximum nor a minimum
of Gn(x, y). In this case we may, for example, assume that
Gn (X, q) (X)) _ Ccn, 0n (x, ( (x)) _ - cn, for 0 < X < 8.
We have then,   f(x,  (x)) - f (0, 0) > (c - e) x,
and                f(x,   ) (x) - f(0, 0) <- (c - ) X1,
provided 0 <x< 8'. Since e may be taken to be less than c, these two
differences are of opposite signs; therefore f(O, 0) is neither a maximum nor
a minimum of f(x, y).
It should be observed that this theorem does not always suffice to decide
whether the point (0, 0) is a point at which f(x, y) has an extreme value,
or not. For it may happen that, for a given function f(x, y) of the assumed
type, no value of n can be determined, for which the conditions stated in the
theorem hold, and therefore the theorem is inapplicable however great n
may be taken.
If f(a, y) = [u (x, y)]2, where u (x, y) vanishes at points of a locus which
passes through the point (0, 0), then the function f(x, y) is one for which the
theorem is inapplicable; the point (0, 0) is in this case a point at which
f(x, y) has an improper minimum.
In general the theorem is inapplicable in the case of any function which
attains the value zero, at points other than (0, 0), in every neighbourhood of
that point, but which has one and the same sign at all points at which it does
not vanish.




518               Functions defined by sequences              [CH. VI
371. The simplest case in which the theorem of ~ 370 can be applied is
that in which the function G,, (x, y) is a homogeneous function of degree n.
For such a function Gn (x, y), three cases arise.
(1) If Gn(x, y) be a definite form, i.e. if Gn (x, y) has one and the same
sign for all values of (x, y) except (0, 0), then Gn (0, 0) is a proper minimum,
or a proper maximum, according as that sign is positive or negative.
(2) If Gn (x, y) be an indefinite form, i.e. if there are points in
every neighbourhood of (0, 0) at which GO (x, y) is positive, and others at
which it is negative, there are other points besides (0, 0) at which the
function vanishes, and there is no extreme of the function Gn (x, y) at the
point (0, 0).
(3) If Gn (x, y) be semi-definite, i.e. if GO (x, y) vanishes at points other
than (0, 0), but has a fixed sign at all points at which it does not vanish,
then Gn (0, 0) is an improper extreme of Gn (x, y).
It should be observed that, if n be odd, GO (x, y) is necessarily an indefinite form.
It will be shewn that, when Gn(x, y) is definite or indefinite, it satisfies
the conditions stated in the theorem of ~ 370; accordingly f(x, y) has a
proper maximum or else a proper minimum, when Gn (x, y) is a definite form;
and f (, y) has no extreme when Gn (x, y) is an indefinite form.
When Gn (x, y) is a semi-definite form, no conclusion can be drawn as
to the existence of an extreme of the function f(x,?), as the conditions
contained in the theorem of ~ 370 are not satisfied.
If Gn (x, y) be definite, it is of the form
r-k
r=l
where n = 2k. Let us assume that A is positive; then
r=k
Gn (x, y)A HI 8.2. X1",
r=l
for all values of x and y; it follows that the first condition of the theorem is
satisfied.
The case in which A is negative may be treated in a similar manner.
Again
(y-._.c)2 + 2  (cX2 =  (x7J2 +             22-)_.>-  2Y )+
+        >,2 1/y2 +  2 Y2
hence      | G ( (y), y) > I (n  (y), y)  A y n 1,
I2 + tn. 2
and therefore the second condition of the theorem is satisfied.




371, 372]


Maxima and minima


519


Next let G (x, y) be an indefinite form; in which case G, (x, y) has
neither a maximum nor a minimum at (0, 0). Let (x', y') be a point at which
Gn (x', y') > 0; and first suppose that I y'  x ' x, so that x' I > 0.
Let x, y be such that y/x = y'/x', and let x, x' have the same sign; we have
G, (x, y) > 0, and it follows that
G (, x  (x)) > Gn(,  )      n > 
G^n (X,  })^   n       Ix (  > 0.
Next suppose that |x ' I | y' 1, so that j y' > 0; we then shew in the same
manner that
Gn ((y), y)   n(lY i )    n>0,
where y has the same sign as y'.
Since there are also values of (x', y') such that Gn(x', y')< 0, we can
shew as before that
Gn (x, (x))  Gn (   x', Y')  n < 0,
Gn (X, 0 (X)) -5  l / I ~
where x and x' have the same sign, and that
an (# (y), y)  G (     yx, )  < 
where y has the same sign as y'. It has thus been established that, when
Gn (x, y) is an indefinite form, the conditions of the theorem of ~ 370 are
satisfied.
The following general result has now been obtained:If f (x, y) -f (0, ) be of the form Gn (x, y) + Rn+i (x, y), where Gn (x, y)
is a homogeneous function of degree n, then, if n be odd, f (O, ) is not an
extreme of f (x, y). If n be even, and Gn (x, y) be an indefinite form, f (0, 0)
is not an extreme of f(x, y). If Gn (c, y) be a definite form, f(O, O) is a
proper minimum, or a proper maximum, of f (x, y), according as Gn (x, y)
is positive or negative. If G, (x, y) be a semi-definite form, no conclusion can
be drawn from the consideration of G, (x, y) by itself, as to the existence or
non-existence of an extreme of f(x, y) at the point (0, 0).
372. When those terms in the expansion of f(x, y) in powers of x and y,
which are of the lowest degree, give a semi-definite form, it is necessary to
take a value of n greater than this lowest degree; we have therefore to
consider the case in which G,, (x, y) is not homogeneous. We have then, in
order to apply the theorem of ~ 242, to G,,(x, y), to determine the four
functions Gn (x,  (xc)), Gn(x, p (x)), Gn ( (y), y), Gn (4 (y), y). The values
y =  (x), y = (x), may be either in the interior, or at the ends of the
interval (- x, x). In the former case they must be such as to satisfy the




520


Functions defined by sequences


[OH. VI


condition dG, (x  )=0; in the latter case they will in general not satisfy
dy
this condition, although they may do so. The method of procedure, by
which G, (x, ) (x)), Gn (x, 4 (x)) may be obtained, is to obtain the various
solutions of the equation dGn (x, ) = 0, in which y is expressed as a series of
dy
fractional or integral powers of x; only such values of y need be considered,
as vanish for x = 0.
Let y = P1 (x), y = P2 (x),... y = Pr (x) denote these series; we then
form the expressions Gn (x, - x), Gn (x, x), G, (x, P1 (x)),... Gn (x, P (x)).
It is certain that the two expressions G (x, +(x)), Gn(x, ((x)) must
both occur amongst these r + 2 expressions, and a comparison of the leading
terms of these expressions will enable us to identify the two expressions
required.  If the indices of the leading terms in G (x, 4 (x)), G (x, ) (x)),
are not greater than n, the first condition of the general theorem is satisfied.
A  similar method, in which    the equation   dGn -   ) = 0 is employed,
dx
will lead to the determination of Gn ( (y), y), Gn (* (y), y).
The details of the investigation have been fully carried out by Scheeffer,
who employs the somewhat more symmetrical, but practically less simple,
method, in which    x and y are expressed as series involving a single
parameter.
When, for any value of n, the result of this process is that Gn (x, y) is such
that the conditions contained in the theorem of ~ 370 are not satisfied, a
larger value of n in which more terms off(x, y) are included in Gn (x, y)
must be taken, and the process repeated until a definite result is obtained.
EXAMPLES.
1. Let f(x, y)-f(0, O)=ax2+ 2xy + by2 + R3 (x,?). The form a2 + 2hxy+ by2 is
definite if ab-h2 is positive; in this case f(O, 0) is a proper minimum or a proper
maximum of f(x, y), according as a is positive or negative. If ab - h2 is negative, then
ax2+ 2hxy+ by2 is an indefinite form, and in that case f(O, 0) is not an extreme off(x, y).
If ab- 2= 0, the form ax2 + 2hxy + by2 is semi-definite, and no conclusion can be drawn as
to the existence of an extreme of f(x, y). It will be necessary in the last case to consider terms of order higher than 2 as included in G, (x, y). By taking n=3, 4,... a
function Gn (x, y) may be determinable which satisfies the conditions of the theorem
of ~ 370.
2.* Let f(x, y)= ay2 + 2bx2y+ cx4+R5 (, y), where a is positive; in this case we have
4= 2 (ay+ bx),
* See Stolz, Grundzuge, vol. i, p. 234.




372]                      Maxima and minima                                521
b
and this vanishes for y =  - x2. We have
a
G4 (x, -x)=ax2 - 2bx3 + cx4,  G4 (, x) =ax2+2bX3+c C 4,
and                         &4Q.    b 2)   ac- b2 4
and                          G4 ', - x2 = --- x4.
\  a   I    a
It follows that G4 (x, -x) or G4 (,.X) is the value of G4(x, q) (x)), and that G4 (,-  x2)
is that of G4 (x, q (x)). If ac - b2 be negative, the two expressions G4 (x, q (x)), G4 (, q (x))
have opposite signs; therefore f(O, 0) is not an extreme of f(x, y). If ac - b2 be positive,
the two expressions are both positive, and the first condition of the general theorem is
satisfied, since the indices of x in the leading terms are not greater than 4.
We find that -G4     has for roots x=   / by, and x=0; we thus form the
ax                             c
expressions           G4(0, y) =ay2, G4( + y, y) = ay2 + 2by3 + cy4,
It is unnecessary to consider the roots x = ~  -, because, for sufficiently small
values of y, xl>lyi, and thus these roots. could not give the extremes for jx_|lyi.
Remembering that a and c are both positive, let b >O, then the value of G4 ( (y), y) is
ay2+2by3 + cy4, and that of G4 (+ (y), y) is ay2; these values being both positive, we see
that G4(0, 0) is a proper minimum  of G4 (,?/). The same conclusion may be made
when b_0. Therefore, when ac-b2>0, a>0, since the conditions of the theorem   of
~ 370 are satisfied, f(x, y) has a proper minimum at (0, 0). If ac-b2>0, a<0, there
is a proper maximum. If ac-b2=O, we have
1
f (x, y)=a (ay +bx2) + R(x, y);
a
hence G4 (x, y) has an improper extreme at (0, 0), and no conclusion can be drawn as
regards f(x, y).
GaC
3.*  Let f(x, y)=y2+2y +R4 (x, y). We find - 3=2 y+x2, and thence we have
G3 (X, - x2)= - x4;
also                  G3 (x, x) =-2 + x3,  G3 (,  - x )=x2 - X3.
It is clear that, in this case, G3 (x, q(x)), G3 (x, 0(x)) have opposite signs, provided x
be sufficiently small, therefore G(x, y) has no extreme at the point (0, 0). Since
G(x    x ()) =-4x
it is not the case that | G3(x, +(X)) I _c x 3, for any value of c, in a neighbourhood of x=0;
the theorem of ~ 370 is therefore not applicable. No information is obtained as to whether
f(x, y) has an extreme at (0, 0), or not. It will in fact be shewn, in the next example,
that y2+ x2y + x4 has a minimum at (0, 0).
4. Let f(x,y)=y2+x2y +X4 + R (, y). We find a4=2y+X2, hence
a~~~aG4y
4=0 gives y=- x,2;
hence                        G4 (x, - x2)= x4 +...;
also               G4 (x, x)=x2+.3+x4,  G (, - X -x)=x2- X3+x4.


* Scheeffer, loc. cit., p. 573.




522


Functions defined by sequences


[CH. VI


In this case G4 (x, p(X)), G4 (x, (.v)) are both positive, and are greater than c)x 4 for a
fixed c. It can be shewn that the other condition is also satisfied. It follows that f(x, y)
has a minimum at (0, 0).
5. Let     f(x, y) = 24- 3xy3x + 6y2 -3xy7 +y8 _ lOOy+ 512 + R13 (X,).
In this case -2=0 has the three roots
y=5X4 +..., y= _  2_ 47  +..., 4+ y2  + 2f4..+
On substituting these values in G12 (, y), and forming also G12 (x, x), G12 (x, - ), we
find that 12 (x, -  (x)) is G(x, - ) or G (, x) according as x is positive or negative; and
the expression commences with the term x6. We find for G12 (, ( (x)) an expression
-4x0+.... Since G12(X, +(x)), G12(x, q(x)) have opposite signs, it follows that (0, 0) is
not a point at which G1 (x, y) has an extreme. Since the indices of the leading terms of
G12(x, +(X)), G12(x, (x)) are both less than 12, the condition of the theorem of ~ 370 is
satisfied, and we can therefore infer that f(x, y) has no extreme at (0, 0).
FUNCTIONS REPRESENTABLE BY SERIES OF CONTINUOUS FUNCTIONS.
373. Before proceeding to consider the most general class of functions
which are representable as the limits of sequences of continuous functions,
and therefore by infinite series of which the terms are continuous functions,
we shall first examine the case in which the function to be represented is
itself continuous in a given interval.
The following general theorem is due to Weierstrass*:If a function f(x) be continuous throughout a given interval (a, b), and
if 3 be an arbitrarily chosen positive number, a finite polynomial P (x) can be
found such that If(x)- P (x) <, for every value of x in (a, b).
In order to prove the theorem, it is convenient first to consider certain
special cases. Let a function y be defined, for the interval (-a, a), by the
specifications y =mx, for 0     x   a, and y=-mx, for - a         x  0; thus
y is the continuous function which is represented geometrically by portions
of two straight lines which meet at the origin and are equally inclined to
the x-axis.  The function is represented in the whole interval (-a, a) by
y =  a L1 + (2      - 1
where the positive value of the radical is to be taken; the expression for y
32
can be expanded by the Binomial Theorem       in a series of powers of - - 1,
* Sitzungsber. of the Berlin Acad., 1885. The proof given here is substantially that due to
Lebesgue; see Bulletin des Sciences Math., ser. 2, vol. xxII (1), p. 278, 1898. Other proofs have
been given by Picard, Trait d'Analyse, vol. I, p. 258; by Volterra, Rend. del Circolo mat. di
Palermo, 1897, p. 83; by Mittag-Leffler, Rend. del Circolo mat. di Palermo, 1900; by Lerch,
Acta Math., vol. xxvii, p. 339. See also Borel's Leqons sur les fonctions de variables reelles,
p. 50.




372, 373]


Series of continuous functions


523


and this series converges uniformly in the whole interval. In this manner,
by taking one, two, three, &c. terms of the series, we obtain a sequence of
polynomials in x which converges uniformly to the value of the function; and
thus this particular case of the general theorem has been established.
Next, let the function y be defined for the interval (a, b) as follows:Let y= O, for a - x - c, and y = n (x- c), for c: x: b, where c is a fixed
number between a and b; this function is represented geometrically by the
portion of the x-axis between the points a and c, and by the portion of the
straight line y = m (x - c) between the points c and b. The function may
be represented by
y=2(a-o) +    2 (x-c) 
and since, as has been shewn in the last case,
- (X - c)
is representable as the limit of a sequence of polynomials, the same is true
of the function now considered.
Next, let (a, b) be divided into a finite number of intervals
(a, xi), (x1, X2), (x2, 3),... (xn-1, b),
and let ordinates to the x-axis be erected at the points
a, xl, x2, XI. X -1, b,
the extremities of these ordinates being denoted by
P, P, P2,... P_-1 Q.
Let the consecutive pairs of these points be joined by straight lines, an open
polygon PPiP2... Q being thus formed; it will be shewn that the continuous function ( (x) defined by the ordinates of this open polygon is such
that a polynomial P (x) can be found, such that
I (x)- P(x) < 
for every value of x in (a, b). It is clear that the function  (x) can be
expressed as the sum of n functions
01 (x),  b,2 (x),... Pn (X),
which are such that 01 (x) is linear in the whole interval (a, b); 02 (x) vanishes
in the interval (a, xI), and is linear in the interval (x1, b); 3, (x) vanishes in
the interval (a, x,), and is linear in the interval (x2, x,); and generally b,. (x)
vanishes in the interval (a, x,-), and is linear in the interval (x,-_, b). Since
polynomials satisfying the prescribed conditions can be found for each of the
functions
0l(),te  (i), nn  (x),
the theorem is established for the function 0 (x).




524


Functions defined by sequences


[CH. VI


In the general case in which f(x) is any function continuous in (a, b), it
follows from the known theorem that f(x) is necessarily uniformly continuous in (a, b), that, if e be a prescribed positive number, the interval can
be divided into parts (a, x1), (x', x2),... (x,_-, b), such that the fluctuation of
f (x) in each of these parts is < e.
If b (x) denotes the function considered above, which we take to be equal
to f(x) at each of the points a, x1, x2,... b, and to be linear between each
consecutive pair of these points, we see that
If(x)-   (X) <<e
in the whole interval; and as it has been shewn that a polynomial P(x)
exists, such that
I (x) - P () I< I,
it follows that         I f(x)- P (x) I < e +;
hence, since e, q are both arbitrarily chosen, Weierstrass' theorem has been
established.
If 81, 2,... 8,... be a diminishing sequence of positive numbers which
converges to the limit zero, then a sequence of polynomials
P (x), P2 (x),
can be found such that! f(x) -Pl()l < 8, f(x)-P,() < 82,,... ]f (x)-P () I < n,,...
for all values of x in (a, b).
Since the sequence of polynomials Pi (x), P (x),... P. (x),... converges
uniformly to f(x) as their limit, f(x) may be regarded as the sum-function
of the uniformly convergent series
P, (x) + [P2 (x)-P P (x)]... + [Pn ()-Pn-1 (x)] +...;
thus the following theorem has been established:If f (x) be a function which is continuous throughout the interval (a, b), the
function is expressible as the limiting sum of a uniformly convergent series, of
which the terms are finite polynomials.
Weierstrass' theorem may be extended to the case of functions of any
number of variables. The general result may be stated as follows:A function defined for any closed domain D, and continuous in that
domain with respect to (x., x2,... xn), can be represented in that domain by
a series of polynomials which converges uniformly and absolutely in D.
For a proof of this theorem, and for a discussion of the methods of
Lagrange and Tchebicheff for the approximate representation of functions by
series of polynomials, reference may be made to Borel's Legons sur les
fonctions de variables reelles, Chapter Iv.




373, 374]


Series of continuous functions


525


374. The question which arises as to the nature of the most general
finction that can be represented in a given interval as the sum     of a series
of continuous functions which converges at every point in the given interval,
has been completely answered by Baire, whose result is contained in the
following general theorem*:The necessary and sufficient condition that a function may, in a given
interval, be representable as the sum of a series of continuous functions which
converges at every point to the value of the function, is that the given function
shall be at most point-wise discontinuous with respect to every perfect set of
points in the given interval.
It will be in the first place assumed that the given function is limited in
its domain.
To shew that the condition stated in the theorem is necessary, let
(X) +. U () 2 +... + Un (X) +...
be a series, such that the functions un(x) are all continuous in a given
interval (a, b), and such that the series converges for every point in (a, b).
Instead of the function s, (x), we may consider the transformed sum-function
s (x, y), as defined in ~ 349. This function is defined for every point in the
rectangle bounded by the four straight lines x = a, x = b, y = 0, y = 1, and is
everywhere continuous with respect to y; it is also everywhere continuous
with respect to x, except upon the axis y = 0, upon which its properties are
to be investigated. The function s (x, y) falls under the case investigated in
~ 244, where it is shewn that in every portion of a continuous curve
Y =   (c),
there exist points at which s (x, y) is continuous with respect to (x, y). In
particular, on the axis y = 0, there must in every interval be points at which
s (x, y) is continuous with respect to (x, y), and a fortiori with respect to x;
thus the function s (x, 0) which represents the limiting sum of the series is
at most point-wise discontinuous with respect to x. The same result holds
if only such values of x are taken into account, as correspond to the points
of any perfect set in the interval (a, b); hence it has been shewn to be
a necessary property of the sum of a convergent series of continuous
functions that it is at most point-wise discontinuous with respect to every
perfect set.
* The theorem is fully developed in Baire's memoir, " Sur les fonctions de variables reelles,"
Annali di Mat. ser. III. A, vol. III, 1899; also in his treatise Lemons sur les fonctions discontinues. He has proved the sufficiency of the condition more succinctly in the Bulletin de la Soc.
Math. de France, vol. xxvIII (1900); this is the proof given in the text. Another proof of the
theorem, of a very instructive character, has been given by Lebesgue in Borel's Lemons sur les
fonctions de variables reelles, pp. 149-155. See also Lebesgue's memoir " Sur les fonctions
representables analytiquement," Liouville's Journal, ser. 6, vol. I, 1905, where the theorem is
proved, and also generalised.




526


Functions defined by sequences


[CH. VI


In order to prove that the condition stated in the theorem is sufficient,
let us suppose that a function f(x) is defined for the domain (0, 1), and is at
most point-wise discontinuous with respect to every perfect set of points
contained in that domain; it must then be shewn that a sequence
f (X),/2 ( ),... / n (x)...
of functions can be determined which converges at every point of (0,1) to the
value off(x).
If p be a positive integer, the points
a,
X '=
~~..,~~ $'~2P)
where a, is a positive integer, will be called principal points of order p;
the interval
(a-i    a    )+1
V 2P '   2P 
of which the centre is -, will be called a principal domain Dp of order p;
the domains with centres 0, 1, of order p, will be taken to be the intervals
(0 2P), (1- 2) 2) respectively.
A point P of (0, 1) belongs to two domains of order p; the centres of these
domains will be called the principal points of order p associated with P.
The problem of constructing the required sequence of functions is reducible to the following problem:(A) To determine a number b (D), corresponding to each principal
domain D, which is such that, having given any point P and a positive
arbitrarily chosen number e, there exists an interval with centre P, such
that, for every principal domain D entirely contained in the interval, and
containing P, the number 4 (D) differs from f(P) by less than e.
If the numbers  (D) have been determined, the functions fp can be
constructed as follows:-At each principal point Q of order p, let fp have for
its value that of b (D) corresponding to the domain of which Q is the centre.
Then let fp be made continuous, and such as to have at every point P
a value intermediate between the greater and the lesser of its values at the
principal points of order p, associated with P; for example fp may be taken
to be linear with respect to x, between each consecutive pair of principal
points. Now take an interval with centre P satisfying the condition stated
above; when p exceeds a certain value, the principal domains of order p,
which contain P, are interior to this interval; hence the values of fp at the
principal points associated with P differ from f(P) by less than e; and the
same is true of the function fp (P). Hence it follows that f(P) is the limit
offp (P), when p is indefinitely increased.




Series of continuous functions


527


Before we proceed to shew that the problem (A) can be solved for the
case of any function f which is at most point-wise discontinuous relatively to
every perfect set of points, it will be shewn that the problem is capable of
a very simple solution in the case in which f is a semi-continuous function.
Let us suppose thatf is an upper semi-continuous function. In each domain
D, let / (D) be the maximum of f in that domain, then the conditions of (A)
are satisfied; for an interval with P as centre can be found in which, at every
point P', f(P') <f(P) + e. If D is contained in this interval, and contains P,
the maximum   of f in this interval, that is b (D), lies between f(P) and
f(P) + e, hence the conditions of (A) are satisfied.  It has thus been
established that every function f which is semi-continuous throughout (0, 1)
is the limit of a sequence of continuous functions, and is therefore representable
as the sum of an infinite series of continuous functions.
To solve the problem (A) in the general case, let us take a descending
sequence e1, 62,...,... of positive numbers which converge to zero; let D
be a principal domain, then we have to define b (D).
Let e- be the greatest number in the sequence fe}, such that in D there
are points at which the saltus off is i o-1; let PI denote the set of points in D
for which the saltus w (f) -o-. If the perfect component PI" of P1 exist, and
if f be not continuous relatively to Pi,, let -, be the greatest number of the
sequence {e}, such that there are points of D at which the saltus o (f, P1,) of
frelatively to Pic, is  c-2; let P2 be the set of such points. In this manner
we obtain closed sets
PI, 2 P3,    PaP,... P     a... P..(1),
all contained in D, and a corresponding set of numbers
1,, O   a-3,.. **   *,... -* *  0a.......................... (2),
all belonging to the sequence {e}. If a be a non-limiting number, (a and Pa
are derived from o-a-, Pa-, just as -2 and P2 are derived from no, P,. If a be
a limiting number, Pa is the set common to all the sets of which the index is
less than a, and -a is the inferior limit of all the -'s of which the index is less
than a.
It is clear that, if / < a, then Pa _ Pa, and aos _ o-a.
It has been shewn in ~ 74, that, for such sets, there exists a number a of
the first or of the second class, such that Pa = Pa+,. In the present case we
must have Pa = 0; for if Pa existed, we should have
P   - =  -  a+= pl),
and there would be a positive number -a+,, such that, relatively to Pa0, the
saltus of f at every point is -_ oa+l; and thus f-would be at every point
discontinuous relatively to the perfect set Pa", which is contrary to the
hypothesis made as regards the nature of the function f. It has thus: been




528


Functions defined by sequences


[CH. VI


shewn that a number /3 exists, such that either P, is enumerable, in which
case P2 =Pp+1 = 0; or else such that Pga exists, but f is continuous
relatively to it, in which case also P+ = 0.
If P, be enumerable, there exists a number 7, such that PsY consists of a
finite number of points; we then take for 4 (D) some number between the
extreme values off at the points of Pe; it having been assumed that f is a
limited function. If f is continuous relatively to Ppa, we take for ( (D) any
number between the extreme values off in Pp,.
The number 4 (D) having now been defined for each domain, it must be
shewn that the conditions of (A) are thus satisfied.
If P be a point of (0, 1), let T1 be the greatest of the numbers {e} such
that the saltus off at P, op (f) _ 71; and let Q, be the set of points in (0, 1)
at which Co (f) - Tr. If Qua exists and contains P, and if the function is not
continuous at P relatively to Qa, let T, be the greatest number of the set {e},
such that wp(f, Q1a) _ -r. Let Q2 be the set of points in (0, 1) at which
(f (f, Q1a) _ r2; proceeding in this manner we obtain a set of closed sets
Qi, Q2  Q, n.. Q,... Q..................... (3),
all of which contain P, and a set of numbers
TI, 72,...,. - t.... T.........(......... (4).
The set Qa and the number T7 depend upon Qa-i, Ta-i just as Q2, 72 depend
upon Q, T,, in case a is not a limiting number. If a is a limiting number, Qa
consists of all points common to all the Q's with indices lower than a, and Ta is
the inferior limit of all the T's with indices inferior to a. As before, Qa must
vanish after a certain index, and thus there is a number V, such that, either Qua
does not exist, or such that it exists and does not contain P, or else such that
it contains P, but P is a point of continuity of f relatively to Q&.
Each of the numbers of (4) belongs to the sequence {e}, except perhaps
the last one T7; these numbers can be arranged in groups, the numbers of the
same group being equal to the same number X of the sequence {e}. The index
of the first number in each group is necessarily a non-limiting number; for if
a be a limiting number, as Ta is the inferior limit of all the numbers Ta', such
that a' < a, and as there exists only a finite number of values of the T,, there
are numbers a', such that T,, = T,. We can write
T1   =2     =..    = Xi,
fTa,+l -Taw+2 =X*j. =..a2 =..  (5-1......................................................... ).
ak-1+1= Tak-1+2 =.* * *   Tak X 
The number of groups of the r's may be finite, say k, in which case ak =;




Series of continuous functions


529


or else the number of groups is infinite, in which case Xi, X2,.. k,... have
their limit zero.
Let X', X',... Xk,... be the numbers in the sequence {e} which immediately precede the numbers Xi, X2,... Xk,.... Let R, be the set of points of
(0, 1) at which o (f) - X'; let R, be the set of points of Qa&, for which
o(f, (Qa1) - X';
and generally let Rk be the set of points of Qak,,_ for which
o (f Qak,) - Xk.
It may happen, if X be the first number of {e}, that no number X' exists, and
thus that R, may not exist. In accordance with the definition of the sets Q,
the point P does not belong to any of the closed sets
R1, R2,... Rk, Bk.;
hence, whatever h may be, there is an interval with centre P which contains
no points of the sets
R1, R2,... Rh.
Let D be a principal domain containing P, and contained in the interval; if,
in this domain, the sets Pi, P2,... Pa are defined, all those of these sets of
which the index is equal to or inferior to ah coincide, in D, with the sets
Qi, Q2,.. Qa,
having respectively the same indices; in fact, there is in D no point at which
c (f) - X/,
whilst there is at least one, the point P, at which
(f ) -hX, (= 7),
therefore al = T,; and consequently, in D, Pi coincides with QI. It will be shewn
that, in D, the sets Pa, Qa coincide, if a  ap; it is sufficient to establish
the proposition for a non-limiting number.  There are two cases to be
considered.
(1) If a be not the index of the first term of a group in (5); let us
assume a-_ = T,_i, and that Pa-_, Qa- coincide in D. According to hypothesis Ta = Tai, so that at P the saltus relatively to
sa-1-, a-I
is _ Ta; therefore a-a, which cannot exceed
'a —1 = Ta-I = Ta,
is identical with Ta. Therefore Pa is, in D, identical with Qa.
(2) If a be the index of the first term of one of the groups in (5), say
as + 1, (8 < h), we assume Pa, = Qua; in the set Qa the saltus at P is greater
than or equal to Xh+1. As D contains no point of Rs+1, the set of points where
co (f, Qa) s- X'+1,


H.


34




530


Functions defined by sequences


[CH. VI


we have                    oa8+il = X+l = -aa+l;
hence                        Pas+l, QaS+l
coincide in D. It has now been proved by induction that the two sets are
coincident in D.
The final stage of the proof falls into two cases:(1) For the point P, the number of groups in (5) may be finite, say k;
then ak = s. The set Q, exists, but Q,+, does not exist. There are then two
sub-cases.
(1)a  If Qua does not exist, or exists but does not contain P; the point
P  belongs to a certain set Q"v, but not to Qq"+1; P   is therefore an
isolated point of Q,. An interval can be determined, with P as its
middle point, containing no point of Q,1 except P, and containing no
points of R,, R2,... Rk. If D is a domain containing P, and contained in
the interval, in this domain P, and Q, coincide; whence P,, Q, consist of the
one point P, and P,+l == 0. In accordance with the definition of < (D), we
have c (D) =f (P).
(1)b It may happen that P belongs to Q,n, and that the function f is
continuous at P relatively to Q a. An interval can be determined, with its
centre at P, containing no points of' R1, R,... Rk, and such that the saltus
of f relatively to the part of Q a in the interval, is < e. If D is contained in
the interval, and contains P, we have P,, Qa coincident in D; the set Psy
or Po which occurs in the definition of h (D), is certainly contained in Qa;
hence b (D) differs from f(P) by less than e.
(2) The groups in (5) may be infinite in number; if e is fixed, h can
be found such that Xh < e. Relatively to Q,,, the saltus at P is < e; thus an
interval, with P as its middle point, can be determined, which contains no
points of R1, R2,... Rh, and such that the fluctuation of f in the portion of
Qaah which is contained in the interval is < e. As in (l)b, the set PsY or Ppa
which serves to define  (D), is contained in Qa,,; thus 0 (D) differs from
f(P) by less than e.
It has now been shewn that the conditions of the problem (A) are satisfied.
Thus Baire's theorem has been completely established*, on the assumption that
the given function is a limited one.
375. In order to extend the result to the case of a function which is not
limited in its domain, let us suppose that the function y =f(x) has points of
infinite discontinuity; the case may also be included in which there are
* The above proof, as given by Baire, is applicable to the case of a function of any number of
variables. The only modification required for this extension is an obvious generalisation in the
definition of the principal points and principal domains.




374-376]


Series of continuous functions


531


values of x for which y has the improper value oo, or the improper value - o.
Let a new function z = f (x) be defined by means of the relations
1Y    for 0  y  o o,
and                   z, for - o o   y < 0.
It is then clear that, corresponding to the unlimited function y=f (x), we
have a limited function z =  (x), in which the dependent variable z has
1 and -1 for its upper and lower limits in the whole domain of x. It is
easily seen that two values of y correspond to values of z in which the
relation of order is conserved, and conversely; and further, that, to a
convergent sequence of values of y, there corresponds a convergent sequence
of values of z, and the converse.  A point of continuity of f(x) is also
a point of continuity of b (x), and the converse is true; and this also holds of
the points of continuity with respect to any perfect set of points in the
domain of x. It follows that, if one of the two functions f(x), f (x) is pointwise discontinuous with respect to every perfect set, then the other function
has the same property. Iff(x) is the limit of a sequence f, (x), f2 (x),... of
continuous functions, then the functions <qb (x), 02 (x),... obtained by applying
the above transformation are also continuous, and they converge to q(x).
Therefore, if f(x) is the limit of a sequence of continuous functions, b (x) has
the same property. If + (x) is assumed to be the limit of a sequence of
continuous functions, we can assume these functions, (x) to have + 1, - 1
for upper and lower limits, as these are the upper and lower limits of b (x).
Let e1, e2, e3,... be a convergent sequence of increasing proper fractions, of
which 1 is the limit.
The functions ejj (x), e22(x),... are continuous, and converge to the
limit p (x); moreover eno, (x) has + en,-en for its upper and lower limits,
and these numbers lie within the interval (-1, + 1). The function fn(x)
which transforms into en4n (x) is then continuous and limited; and since the
limit of the sequence enn,(x)} is + (x), it follows that the limit of the
sequence {fn ()} is f(x). The condition that k (x) is the limit of a sequence
of continuous functions is that it should be at most point-wise discontinuous
with respect to every perfect set in the domain of x; it follows that, when
this condition is satisfied,f(x) is also at most point-wise discontinuous with
respect to every perfect set. The general theorem has now been completely
established.
376. It may be shewn that any function f(x) which satisfies the condition
contained in Baire's general theorem can be expressed as the limit of a
sequence of finite polynomials. Since, under the condition stated, f(x) is the
limit of a sequence {f,(x)} of continuous functions, if e1, 2,,... be a sequence
of decreasing positive numbers converging to zero, we can, in virtue of
34-2




532


Functions defined by sequences


[CH. VI


Weierstrass' theorem  (~ 373), determine a polynomial Pn(x) such that
Pn (x) -fn (x) < en, in the whole domain of x. It follows that the sequence
P (x), P (x),... of polynomials so determined, converges to the limit f(x).
We have, in particular, the following theorem:The necessary and sufficient condition that a function f(x) can be represented by a convergent series of polynomials, each of finite degree, is that f(x)
should be at most point-wise discontinuous with respect to every perfect set of
points in the domain of x.
377. A theorem relating to those continuous functions which in any
interval of the independent variable are either differentiable, or at least
possess everywhere a definite derivative on one and the same side, can be
deduced from Baire's general theorem. The derivative may at any point
be infinite with a definite sign.
f f(' + h) -f(x)
If b$ + -  -h, for positive values of h have a definite limit, finite or
infinite, for each value of x, as h has a sequence of positive values converging
to zero, it follows from Baire's theorem, that the derivative so defined is at
most point-wise discontinuous with respect to every perfect set in the
interval. Thus the following theorem has been established:If f(x) be continuous in any interval, and possess at every point a
differential coefficient, or else a definite derivative on one and the same Side,
then the differential coefficient, or the derivative, is either continuous in the
interval, or else is point-wise discontinuous with respect to every perfect set of
points contained in the interval.
BAIRE'S CLASSIFICATION OF FUNCTIONS.
378. A classification of functions of a real variable has been suggested
by Baire*, based upon the properties of the functions in relation to their
representation as limits of sequences of functions. Continuous functions
form the class 0; and functions which are not continuous but are the limits
of sequences of continuous functions belong to the class 1. Functions of class
2 are those which can be represented as the limits of sequences of functions
of class 1, and do not themselves belong to either of the classes 0 or 1. In
general, a function is of class n, if it can be represented as the limit of a
sequence of functions of class n- 1, provided it does not belong to any of
the classes 0, 1, 2,... n- 1.
It can be shewn by means of an example, that functions of class 2
exist. Consider the function f(x) which has the value 1, for all rational
values of x, and the value 0, for every irrational value of x. This function
does not belong to either of the classes 0 and 1, for it is totally discontinuous;
* loc. cit. (above, p. 525). See also Comptes Rendus, December 4 and 11, 1899,




376-378]


Classification of functions


533


but it is easily seen to be the limit of a sequence of functions all of which are
of class 1. For let us define f,(x) to be zero at every point except those
points at which the value of x is rational and has for its denominator an
integer not exceeding n; this function f,(x) has only a finite number of
discontinuities in any given interval, and therefore belongs to class 1. The
function f(x) is the limit of the sequence {f,(x)}, and is therefore of
class 2. This function is capable of the analytical representation
f(x) =  r l im li  (cos m! rrx)2n,n=ci r2= oo
A function belonging to class 2 can be represented by a double series
oo  00
X Pm,nl(x), where Pm,n denotes a finite polynomial. This double series
m=l n=l
cannot be reduced to a single one, the terms of which are continuous, for the
function would then not be of class 2. In general, a function of class p must
be representable by a p-fold series of finite polynomials.
A function which is the limit of a sequence of functions belonging to the
classes 1, 2, 3,... 1,..., and which does not itself belong to any of these
classes, is said to be of class o, where o denotes the first transfinite ordinal
number. A function which is the limit of a sequence of functions of class o,
and which is not itself of class w, or of any class inferior to o, is said to be of
class co+l. Proceeding in this manner, we may attach a meaning to the
statement that a function is of class /, where 3 denotes any prescribed ordinal
number of the second class. In case /3 be a limiting number, a function is
said to be of class /3 when it is the limit of a sequence of functions each of
which is of some class ordinally preceding /3, provided the function be not
itself of any of the classes preceding 3. If / be a non-limiting number, a
function which is the limit of a sequence of functions of class 3 - 1 is said to
be of class 3, provided it be not of any class inferior to /3. A finction
belonging to any class is necessarily a summable function.
The question has been discussed by Baire* and by Borell-, whether it be
possible effectively to define functions of all classes, the numbers of the classes
being finite or transfinite.  Those functions which belong to the classes
of which the numbers are less than some prescribed number form an
aggregate of cardinal number c, which is less than the cardinal number f
of all functions. This would indicate that functions cannot be exhaustively
classified under numbers less than some fixed number, either finite, or transfinite of the second class, but this does not settle the question whether it
be possible to define a particular function which does not belong to one of
the classes in question. The question also arises whether it be possible to
define functions which do not belong to any of Baire's classes.
* Annali di Mat. ser. 3 A, vol. III, p. 71.
t Lemons sur les fonctions de variables reelles, p. 156.




534


Functions defined by sequences


[oI. VI


These questions have been fully discussed by Lebesgue in his memoir*
"Sur les fonctions representables analytiquenment."  It is there pointed out
that all functions that are representable analytically belong to Baire's
classes. A function is said to be representable analytically when it is
constructible by effecting, according to a determinate norm, a finite, or
enumerably infinite, number of additions, multiplications, and passages to
the limit, upon variables and constants. The other operations of analysis
are reducible to those here enumerated. Lebesgue has advanced proofs that
functions of all Baire's classes exist, in the sense that it is possible effectively
to define a function of any prescribed class. He has also shewn that functions
can be defined which do not belong to any of Baire's classes, and which are
therefore incapable of analytical representation. In some of Borel's reasoning,
correspondences are assumed to exist which are defined only by means of an
infinity of separate acts of choice. The objection to such reasoning is
recognised by Lebesguet, who has endeavoured to avoid this difficulty. In
view of the grave difficulties connected with the necessary nature of an
adequate definition of a function, and which form the subject of controversy
at present, it would perhaps be premature to assume that the questions here
referred to have been finally settled.
THE INTEGRATION OF SERIES.
379. Let u, (x), u, (x),... Un (x),... be limited integrable functions defined
for the interval (a, b), and such that, at each point of the domain, the series
u, (x) + u, (x) +... + u, (x) +... converges to the sum s (x); the conditions will
be determined that the function s (x) has a proper integral in the domain (a, b).
The term "integrable" is here used in the sense employed by Riemann.
Since the functions s, (x), s (x),... s, (x),... are limited functions, I s (x),
| s (x)\,... s,(x),... have upper limits u,, u,,... u,,... in the domain (a, b).
If u,, z,,... u,,... have a finite upper limit U, it can be shewn that s (x) I has
a finite upper limit in (a, b); for if the upper limit of s (x) I were indefinitely
great, a value of x would exist such that s(x) I= U+a, where a is some
positive number; now n can be taken so great that s (x)- s (x) \< e, where
e is arbitrarily small, hence s(x) < ls(x) + e < U+ e, and since e can be
chosen to be < a, it is impossible that s (x) I = U + a; and therefore it is
impossible that the upper limit of s (x) l be not finite.
The condition just stated, that Is (x) may have a finite upper limit, is a
sufficient one but not a necessary one; in fact we know that at the point
(x, 0), the function s (x, y) may have an infinite discontinuity, whilst s (x) has
only a finite discontinuity, or is continuous, at the point x. In what follows,
* Liouville's Journal, ser. 6, vol. I, 1905.
t Also by Borel himself; see Bulletin de la soc. math. de France, vol. xxxiII, 1905, pp. 272, 273.




378-380]


Integration of series


535


it will be assumed that I s (x) I has a finite upper limit in (a, b), so that in case
s (x) is integrable, the integral is a proper one.
Let E be a set of points in (a, b) of measure zero. Let e be an arbitrarily
chosen positive number, and n, an arbitrarily chosen positive integer. Let
us suppose that, for each point x, of (a, b) which does not belong to a certain
component E, of E, an integer n (> n) can be determined, and also a
neighbourhood (x1 - 8, xI + 8'), such that the condition I Rn (x) < e is satisfied
for every point x in that neighbourhood and lying in (a, b). Then, provided
this condition is satisfied for every value of e, and E is such that each point
of it belongs to Ee, for some sufficiently small value of e, the convergence of
the sequence s( (x), s, (x),... to s (x) is said to be regular in (a, b) except for
the set E of zero measure.
It will be observed that, for a fixed e, the integer n (> n) depends in
general upon the particular point x, which does not belong to Ee. Moreover,
since n1 is arbitrary, there exists for a particular point x, an infinite number
of values of n; the neighbourhood (x-8, x-, x+ 8') depending however in
general upon the value of n chosen.
In the particular case in which every Un (x) is positive or zero, for every
value of x and n, so that the sequence s (x), s2(x),... is a non-diminishing
sequence, when the condition I R,(x) < e is satisfied for a particular value
of n, it is also satisfied for every greater value. In the general case this does
not hold; the condition is satisfied for an infinite number of greater values
of n, but not necessarily for every such value.
It is easily seen that the set E, must, for each value of e, be a non-dense
closed set, although the set E is not necessarily non-dense, and may be
everywhere-dense in (a, b). For, if I be a limiting point of the set E,, then
every neighbourhood of: contains points of E,, and it is impossible that the
condition I Rn (x) < e can be satisfied for every point of such a neighbourhood.
Therefore ~ must itself belong to E,, which must consequently be a closed
set; and since it has the measure zero, it cannot contain all the points of any
interval (a, /), and is thus non-dense in (a, b).
380. The following theorem will now be established:The necessary and sufficient condition that the limited function s (x) may be
integrable in (a, b), in accordance with Riemann's definition, is that the sequence
of integrable functions s, (x), s (x),... shall converge to s(x) regularly, except
for a set of points E of zero measure, and of the first category.
To prove that the condition stated is necessary, let it be assumed that
s (x) is integrable in (a, b). The number e, and the integer nl being fixed, let
it be assumed that, if possible, the set E, of points of (a, b), for each of which
it is impossible to fix a value of n(>ni), and a neighbourhood such that
i R (x) I < e, for all points of that neighbourhood, has a measure greater than




536


Functions defined by sequences


[OH. VI


zero. Since s(x), and s, (x), s, (x),... are all integrable, the set of points at
which one of these functions is discontinuous has the measure zero, and it
follows that the set of all points at which one at least of these functions is
discontinuous has the measure zero. Remove from E, every point at which
one or more of the functions s(x), s (x), s2(x),... is discontinuous, we then
have left a set Fe, of the same measure as Ee, which is by hypothesis greater
than zero. At every point of Fe, the functions s,,(x); and the function s (x)
are all continuous.
If: be a point of F,, the number n (> n1) can be so chosen that
I (8)-sit() | < 36;
also 8 can be so chosen that, for every x in the interval (I- 8, + 8) the
inequalities
| S(t)-S (X) | < 36, | Sn (t)-Sn(x}) <  6
are both satisfied. This follows from the fact that s (x), s, (x) are continuous
at I. From these inequalities we deduce that the inequality
| (X) - Sn (X) I < E
is satisfied at all points x in the interval ( 8 - 8, +- 8). But this is contrary
to the hypothesis that ~ belongs to the set Ee, for the points of which no
neighbourhoods in which the last condition is satisfied can be determined.
It therefore follows that it is impossible that the set Ee can have its measure
greater than zero.
The set E, having been now shewn to have the measure zero, we may
consider a descending sequence e1, e, e3,... of values of e converging to zero.
The sets Ee1, E2, E3,... have their measures zero, and they determine a set
E of the first category, consisting of all points which belong to any of these
sets. It follows from the theorem of ~ 82, that the set E has zero measure;
and it has thus been established that, if s(x) be integrable, then the convergence is regular except for the points of this set E.
To shew that the condition stated in the theorem is sufficient, let e and n1
be fixed, then the set E, is a non-dense set of zero content. The points of
E, can therefore all be enclosed in the interiors of intervals of a finite set, the
sum of whose lengths is an arbitrarily small number a. The remainder of
(a, b) consists of a finite set of intervals; and for each point xA of any one of
these intervals, a neighbourhood (x1 - 8, xi + 8') can be determined, and also
a number n (>n 1), not necessarily the same for all points x1, such that the
condition I Rn (x) | < e is satisfied for all points of (x1 - 8, xa + 8') which are
in (a, b). To the set of all such intervals we may apply the Heine-Borel
theorem; and consequently a finite set of intervals can be determined, such
that each point of (a, b), not in the interior of the excluded intervals, is in the
interior of at least one of the intervals; and in each one of this finite set of
intervals the condition R,, (x) < e is everywhere satisfied for some one value




380]


Integration of series


537


of n, greater than n,.  When the set of intervals of which the sum is q is
excluded from (a, b), the remainder may be divided into a finite number of
parts such that, in each part, the condition Rn (x) I < e is satisfied for a value
of n belonging to a finite set nh + p,, n, + p2,... ni + pr of numbers all > n,.
To shew that s(x) is integrable in (a, b), we now apply Riemann's test of
integrability.  Divide (a, b) into a number of parts h,, h2,... hs, so chosen
that all the end-points of the excluded intervals, and also all the end-points
of those finite parts for each of which I Rn (x) I < e for a single value of n,
are end-points of the parts AI, h2,... h,. For an interval h in the excluded
set, the product of h into the fluctuation of s (x) is less than (M-m) h, where
M and m are the upper and lower limits of s (x) in (a, b). For an interval
h, for the whole of which  Rn,+p (x) < e, we see that the fluctuation of s(x)
cannot exceed that of Snlp(x) by more than 2e. It follows that the sum of
the products of each h into the corresponding fluctuation of s(x) is not
greater than
(M - r) V +   SZ h {2e + fluctuation of s,+ (Xc)}
where, in the double summation, the first summation refers to all those of
the h's which are in an interval for which p has one and the same value, and
the second summation refers to the values pl, p2,....  Since Sn+p(x) is
integrable through the interval to which it belongs, and for which p has a
fixed value, we see that when the number s is sufficiently increased, and the
greatest of the h's is sufficiently small, E 2 h x fluctuation of Sn+p (x) becomes
p
arbitrarily small.  Since r and e are arbitrarily small, it follows that
Riemann's test of integrability of s(x) is satisfied.
The general theorem having now been completely established, it is seen
from the foregoing proof that it may be stated in the following form:If u6 (x) + u2 (x) +... converges to a definite value s (x) at every point in
(a, b), and if all the functions au (x), u2 (x),... have proper integrals in (a, b),
then the necessary and sufficient conditions that s(x) may have a proper
integral are (1) that the upper limit of s(cx) in (a, b) be finite, and (2) that,
corresponding to two arbitrarily small positive numbers v, e, and to any positive
integer n,, a finite number of intervals whose sum is less than qr can be excluded
from  (a, b), so that, in the remainder of (a, b), Rn,+p () I < e, for every x,
where p has one of a finite number of values which depend on x, but are such
that the same p is applicable to all points x in a certain continuous interval.
The condition (2) contained in this theorem was obtained* first by Arzela,
and is expressed by him in the form, that there must be a certain mode of
* "Sulle serie di funzioni," Part ii, Meem. della R. Accad. d. Sci. di Bologna, ser. 5, vol. viiI,
1900. A proof different from that in the text was given by Hobson, see Proc. Lond. Math. Soc.
ser. 2, vol. I, where it is shewn that Arzela's proof of his theorem is invalid.




538


Functions defined by sequences


[oH. VI


convergence of the series called uniform convergence by segments in general
(convergenza uniforme a tratti in generale). This mode of convergence
differs from that of uniform convergence by segments, considered in ~ 354,
in that a finite number of intervals of arbitrarily small sum must be excluded
from the domain in order that the condition may be satisfied.
381. The theorem obtained above contains the necessary and sufficient
conditions that the limit of a sequence of integrable functions is itself
integrable, the conception of integration being that of Riemann.    The
corresponding theorem, when the conception of an integral in the extended
sense introduced by Lebesgue is employed, is of a simpler form:If u1 (x), u2 (x),... Un (x),... be limited functions, defined for (a, b), which
are integrable in accordance with Lebesgue's definition, and if the series
u1 (x) + u2 () +... + Un (x) +... converge to the function s (x), limited in the
whole interval (a, b), then the function s (x) is also integrable in accordance with
Lebesgue's definition.
Let A, B be any two fixed numbers, such that A < B, and let e be an
arbitrarily chosen positive number. Let us consider the sets of points Gn (e),
Gn+1 (e),... Gn+m (e),., where n is a fixed integer, and Gn+,, (e) denotes the
set of points x, such that A- e <sSn+ (x)< B + e; these sets are all
measurable, since all the functions s,+, (x) are integrable in accordance
with Lebesgue's definition of integration. The set of points x for which
A < s (x)  B, is such that any point x of the set belongs to all the sets
Gn(e), Gn+j (e),... Gn+m (e),..., from and after some fixed one of the sets.
Denoting by G (e) the set of all points each of which belongs to all the sets
Gn (e), Gn+i (e),..., from and after some fixed one of the sets, we see that the
set of points x such that A < s (x)  B, is a component of the set G (e), which
set G (e) is measurable (~ 82). Now take a descending sequence e1, 62,... es,...
of values of e, which converges to zero; the set of points such that A _ s (x) < B,
is the set which is common to all the measurable sets G (e1), G (2),... G (e)...;
and this set is therefore itself measurable. Since the set of points for which
A _ s(x) --  B is measurable whatever values A and B may have, and s (x) is
a limited function, it follows that s(x) satisfies Lebesgue's condition of
integrability.
The theorem may be also stated as follows:If the function s (x) be the limit of a sequence {sn (x)} of limited summable
functions, defined for an interval (a, b), then s (x) is also a summable function,
and therefore has a Lebesgue integral in (a, b), in case it be limited.
382. When the sum-function s(x) of the series ^u (x) is integrable in
(a, b), and therefore in any interval (a, x), where x < b, the question arises
whether the sum S     u (x) dx is finite and continuous, and if so whether it
=1 a




380-383]


Integration of series


539


is equal to  s (x) dx, which is necessarily a continuous function. Sufficient
a
conditions will be obtained that such a term by term integration of the series
Eu (x) is valid.
The following theorem will be first established:If the series Zu (x) converge to s (x) for every value of x in (a, b), and
if e be a fixed positive number, the set of points Gn for which I Rn (xc) > e,
is such that its interior measure has the limit zero, when n is indefinitely
increased.
To prove this theorem we observe that, if the limit of the interior
measure of Gn is not zero, there must be an infinite number of values of n
for which the interior measure of Gn is greater than some number a. It
follows from the theorem of ~ 93, that a set of points of interior measure _ a
exists, each of which belongs to an infinite number of the sets G,. This
is inconsistent with the condition that the series u (x) converges for each
value of x; for any fixed point x, there can only be a finite number of values
of n such that Rn (x) > e. It follows that the interior measure of G,, must
have the limit zero, when n is indefinitely increased.
383. Let us now assume that s (x) has a proper integral, and further
that I Rn (x) \ is, for every value of n and of x, less than some fixed number C;
then, Rn (x) being integrable, I R, (x) I is also integrable, and therefore the set
of points for which [ Rn (x) > e is measurable. It follows from the theorem
proved above, that the measurable set of points for which I Rn (x) >, is
such that its measure has the limit zero, when n is indefinitely increased.
We see therefore that
|Rn(x)dx <(x - a) e +    C,
where n is chosen so great that the measure of the set of points, at each of
which I Rn (x) > e, is less than the arbitrarily chosen number;: Since e,
v are arbitrarily small, it follows that
lim   Rn (x) dx = O.
q=0 so a
and thus that           s (x) dx =lim  sn (x) dx.
*J a      n-=oo J a
The following theorem has now been established:If u, (x) +   u2 ((x) +... be a series of limited integrable functions which converges at every point in the interval (a, b) to the integrable
function s (x), then if I Rn (x)  s (x) - Sn (x) I is, for every value of x




540


Functions defined by sequences


[CH. VI


and of n, less than some fixed number C, the series may be integrated term
by term  in any interval (a, x), the stm of the integrals converging to
fs (x) dx.
The condition stated in this theorem may be replaced by the condition
that s,, (x) | be less than some fixed number, for all values of n and x.
If the transformed remainder function   R (x, y) I have no upper limit in
its domain, there must be at least one point such that the saltus of the
function is indefinitely great; it is clear that such a point must be in
the x-axis, and is a point at which the measure of non-uniform convergence
is indefinitely great. Conversely, if there is no such point, the upper limit
of I R (x, y), or of Rn (x) ], is finite.  The foregoing theorem  may thus be
stated as follows*:A sufficient condition for the term  by term  integrability of a series of
limited integrable functions which   converges in a given    interval to an
integrable function s(x), is that there be no points at which the measure
of non-uniform  convergence of the series is indefinitely great.
A special case of the general result which has been obtained, is that
in which the series converges uniformly. This condition is sufficient to
ensure both that the sum-function s(x) is integrable, and that its integral
through any interval (a, x) is the sum of the corresponding integrals of
the functions u (x); thus we have the theorem:If a series  Zu (x) of integrable functions converge uniformly in the
interval (a, b) to the sum  s(x), then the sum  Z| u(x)dx converges to the
r/
fa
value  s (x) dx, where a    x < b.
If it be assumed that the convergence of   u (x) is simply uniform only,
this is sufficient to ensure that s(x) is integrable, but it is then not
necessarily true that z  un (x)dx is a convergent series.   It can however
be shewn that whenever this series is convergent, it converges to the value
j s (x) dx.
In fact we know that, by bracketing the terms of the simply uniformly
convergent series fUn (x) in a suitable manner, the series is thereby converted
' This theorem was obtained first by Osgood, for the case in which s (x) and all the u (x) are
continuous; see Amer. Journal of Math., vol. xix, 1897. The case in which s (x) is not necessarily
continuous was obtained by Hobson, Proc. Lond. Math. Soc., vol. xxxIv, p. 245, and the general
case was investigated by W. H. Young, Proc. Lond. Math. Soc., ser. 2, vol. i, and also by Arzela,
loc. cit.




383, 384]


Integration of series


541


into a uniformly convergent series Evn(x), and the above theorem is then
applicable to the new series, and thus  ' v,, (x)dx converges to the value
1    x
s (x) dx. It is clear that whenever        u, (x) dx converges, it must
J a                                    1 a
converge to the same value as does the series:     v|m  (x) dx. We thus
1 Ja
obtain the following theorem:If a series Su (x) of integrable functions converge simply-uniformly in
the interval (a, b) to the sum s(x), then* (1) if the series  u (x) dx be
convergent, it converges to the value  s s(x)dx, and (2) if the series be not
convergent, it may by suitably bracketing the terms, and amalgamating the
terms in each bracket, be converted into a series which converges to the
value   s (x) dx.
384.  It has been shewn in ~ 383, that if R,,(x) is < C, for every
value of n and of x in the interval (a, b), then
R, (x) dx < (x - a) e + fC,
where n is so great that the measure of the set of points in (a, x) at which
Rn (x) I> e, is < V, where e, q denote arbitrarily chosen numbers. Since
the set Gn of points of (a, b) at which ] R  (x) > e, has the limit zero when n
is indefinitely increased, we may fix a number n, such that, for every value
of n that is _ n, the measure of Gn is less than j. We have then
IRn () dx < (x -a) e +vC < (b- a) e 1 C,
provided n ~ n,, for every value of x in the interval (a, b). Since (b - a) e + rC
rf
is arbitrarily small, it follows that j Rn(x)dx converges uniformly to zero
for all values of x in the interval (a, b), as n is indefinitely increased. We
have therefore the theoremt:When there are no points at which the measure of non-uniform convergence
of the series 2u (x), of integrable functions, which converges to the integrable
function s(x) in (a, b), is indefinitely great, then the convergence of the
rx                     rx
series  J fu (x) dx to the value  s (x) dx is uniform in the interval (a, b).
* The first part of this theorem was given by Bendixson, for the case in which the functions
u (x) are continuous; see Stockholm Ofv. vol. LIV, p. 609.
+ This theorem was stated, and proved otherwise by W. H. Young; see Comptes Rendus,
vol. cxxxvI, p. 1632.




542


Functions defined by sequences


[CH. VI


The proofs of the two theorems in ~ 383 are still applicable when the
functions are integrable only in the extended sense employed by Lebesgue.
We have therefore the following theorem which includes the former ones
as special cases:If,u (x), u, (x),... un (x),... be a sequence of limited functions integrable
in (a, b) in the extended sense of the term, and if the series Vu (x) converges
to s(x), then in case I Rn (x) I has a finite upper limit for all values of n
rx
and x, the series, Un (x) dx, converges uniformly for all values of x in (a, b)
Ja
to the sum fs (x) dx, the integrals being taken to be Lebesgue integrals.
This theorem may also be stated as follows:If s (x) be the limit of a sequence {sn (x)} of summable functions defined for
the interval (a, b), and if I sn (x) have a finite upper limit for all values of n
and x, then  s (x) dx converges uniformly, for all values of x in (a, b), to
f s (x) dx, which has been shewn in ~ 381 to exist.
385. We proceed to consider the case in which the condition that
I Rn (x) I has a finite upper limit for all values of x in (a, b), and all values
of n, is not satisfied. In this case there is a set G of points at which the
measure of non-uniform  convergence of the series is indefinitely great; it
has been shewn in ~ 350 that the set G is closed. If we assume that the
00  rx
series    un (x) dx is everywhere convergent, and has U (x) for its sum,
1 a
and that the integral    s (x) dx everywhere converges to S(x), it may
happen that U(x) is discontinuous, and is consequently not everywhere equal
to the essentially continuous function S (x). It may however happen that
U(x) is continuous, and yet is not equal to S (x). It will however be
shewn that, in case U(x) is continuous, it is a sufficient condition for the
equality of S (x) and U (x), that the set G should be enumerable.
Let x be a point which does not belong to G; then a neighbourhood
(x-61, x+62) of x can be found, such that, for all points in this neighbourhood,
and for all values of n, I Rn (x) I has a finite upper limit. Denoting the sum
f us (x) dx, by Un (x); since U (x) = lim Un, (x), a value N of n can be found,
1J a                              n=oo
such that, for n  N,
U(x)- Un(x) 1, I U(x + h)- Un (x + h) 
are both less than an arbitrarily chosen number 8, where x +h is a fixed




384, 385]


Integration of series


543


point in the neighbourhood (x-e, x+62) of the point x already chosen.
We now have
(x + h) - U ()    Un ( + h)- Un(x)    28
h                  h            h
Now, since the interval (x, x + h) contains no points at which the measure
of non-uniform convergence of the series  u, (x) is indefinitely great, for a
sufficiently great value of n,
Sl (c) dx -  S (x) < 
where 8' is arbitrarily chosen. Therefore, if n - N', where N' is some fixed
integer, we have
Un (x + h) - Un (x)  S (x + h)-S (x)  8'
h                 h          h'
From the two inequalities which have been obtained, we deduce that
U(x+h)-U(x) _S(x+h)-S(x)            28+ 8'
h                h             h
Since 8, 8' are arbitrarily small, and independent of x and h, we have
U(xc+h) - U(x)    S(x+h) - S(x)
h                h 
and this holds for any point x which does not belong to G, and for any
point x + h, in a neighbourhood of G which does not contain points of G.
It follows that any one of the four derivatives D+U(x), D+U(x), D-U(x),
D_U(x), at x, is equal to the corresponding one of the four derivatives
of S(x). Since one of the four derivatives of the two functions S(x), U(x)
is such that its value is the same for the two continuous functions at all
points except at those of the enumerable closed set G, it follows (~ 206) that
the two functions differ only by a constant; and since both vanish at x = a,
they must be everywhere equal. It has thus been shewn that*:If the series (u,(x) converges to s(x) in the interval (a, b), and if the
rx                                              o rx
integral  s(x) dx have everywhere a definite finite value, and 2j un (x) dx
have everywhere a definite finite value, and be a continuous function, it is a
sufficient condition of the equality of the two, that the set of points at which
the measure of non-uniform convergence is indefinitely great, should be an
enumerable set.
When the set G is not enumerable, it contains a perfect component;
and in that case the sum of the integrals of the terms of the series is not
* This theorem was given by Osgood, American Journal of Math., vol. xix, in the case in which
the terms and the sum of the series are continuous. The general theorem was given by Arzela,
Mem. di Bologna, ser. 5, vol. vIIi, 1900.




544


VFunctions defined by sequences


[CH. VI


necessarily equal to the integral of the sum, even when both exist and
the condition of continuity of ZS   un(x)dx is satisfied.
It will be observed that, in accordance with the theorems which have
been demonstrated above, the term by term integration of a series may
fail to give the integral of the sum, either (1) when the set G contains a
finite, or enumerable, set of points, but the condition that the sum of the
series     u (x) dx is a continuous function of x is not satisfied; or (2) when
G contains a perfect component; or (3) when the condition that the
convergence of the series Yu (x) is of the kind called uniform convergence
by intervals in general, is not satisfied, so that s(x) is not integrable in
accordance with Riemann's definition.   In case (3), the term   by term
integration may, however, give the Lebesgue integral of s (x).
386. We have hitherto assumed that the terms u,, (x), of the series
u,,,(x), are all limited in the interval (a, b), and that the same holds as
regards the sum-function s (x).   It is however possible, under certain
restrictions, to  remove these  conditions.  We shall assume that the
functions s, (x), and therefore also the functions Sn (x), are not necessarily
all limited in the interval (a, b). In this case there may be values of x
for which the series   zun (x) is divergent; and    such points will be
regarded as points of infinite discontinuity of s (x), although s(x) is not
properly defined at such points. Let us assume that there is an enumerable
closed set of points G, in (a, b), such that, in any interval (a, 8) which
contains, in its interior and at its ends, no point of G, the condition, that
s,n (x) i is less than some fixed number, is satisfied for every value of n, and
for the whole interval (a, /3). Let us further assume that all the functions
Un,(x) possess improper, or proper, integrals in (a, b), and that the series,f /Un(x) dx is convergent for all values of x, and that its sum   U(x)
= 1 a
is a continuous function of x, for the whole interval (a, b), their ends being
included. Also let it be assumed that s (x) has an improper integral in (a, b);
then the function js(x) dx = S(x) is a continuous function of x. The
enumerable set G contains every point of divergence of the given series,
and also every point of non-uniform convergence of which the measure is
indefinitely great. It is now clear that, with these assumptions, the proof
of the theorem in ~ 385 is applicable, without modification, to establish the
legitimacy of term by term integration of the series  un (x).  We obtain
therefore the following theorem:If the series Zun (x) converges to the function s (x) at every point which
does not belong to a reducible set of points G, and the functions s,(x),
although not necessarily limited in (a, b), satisfy the condition that, in any




385-387]


Integration of series


545


interval (a, /) which contains in its interior and at its ends no point of G,
I n (x) I is less than some fixed finite number, for every value of n and x;
and if further   Un (x) dx exist as an improper or proper integral, for
J a
every value of n, and the series E  Un (x) dx, for a - x  b, is convergent
=1 a
and represents a continuous function of x; and if s (x) have an improper
integral in (a, b), then the theorem
rb             rb
s (x) dx =lim   s, (x) dx
Jc an       =ooJ a
holds, and thus term by term integration is applicable to the series.
387. Lastly, the case will be considered in which the interval (a, b)
is unlimited; we may assume that b has the improper value oo. Let us
suppose that term by term integration is applicable for every finite interval
(a, C); and thus that
lim   Sn (x) dx   s (x) dx,
w=oo Jf          a
the integrals being either proper ones, or improper ones, subject to the
conditions of the theorem in ~ 386. It follows that, if C'> C,
lim   Sn (x) dx =  s (x) dx.
n=ro, C           C
Let us now assume that, if e be an arbitrarily chosen positive number,
an integer n1, and a number C> a, can be determined, such that
Sn (x) dx <e,
for every value of C' > C, and for every value of n _ ni. It then follows that
fs (x) dx - e; and since e is arbitrary, it follows that    (x) dx is
convergent. We assume that all the integrals Isn (x) dx exist. We
have now
s (x) dx -  Sn (x) dxa   s (x) dx -  Sn (x) dx
+    s(x) dx + s,, (x) dx
and by taking a sufficiently great value of n _- n,, and a sufficiently great
value of C, the expression on the right-hand side is _ 3e. It thus
appears that
s (x) dx = lim   Sn (x) dx;
hd therefore= the following00heorem has been established
and therefore the following theorem has been established:

H.


35




546                Functions defined by sequences             [OH. VI
If the series  un (x) have as its sum-function s (x), in the sense previously
2 =1
defined, then if  un (x) dx exists for each value of n, and in every interval
rC         X rC
(a, C) the condition that    s (x) dx = E   n (x) dx, be satisfied; and
J    a     ==l*/ a
further, ijf corresponding to an arbitrarily chosen e, an integer n~, and a
pG
number C>a can be so determined that I    s (x) dx < e, for every value
of C'> C, and for all values of n _ nl, then the integral fs (x) dx exists, and
is equal to  fj     n(x) dx.
n=l a
It may also be shewn that, on the assumption that the condition
s (x) dx ==    Ul (x) dx holds for every value of C > a, then provided
S    un (x) dx be convergent, and that   un (x) dx converge to the value
=   a                               n =-l I
of Y    Un (x) dx, when C is indefinitely increased, it follows that the integral
n=l a
r 00                            oo  roo
Is (x) dx exists, and is equal to  J u (x) dx.
J a                            1=1 a
co rC
For, on the assumption that  J u, (x) dx converges to a definite limit,
n=1 a
as C is indefinitely increased, we see that, if e be fixed, C may be so chosen
that
S    Un (x) dx <e,
n=l C
for C'> C; and from this it follows that
's (x) dx < e,
for C' > C; and since e is arbitrary, it follows that  s (x) dx exists.
a
Also since            s (x) dx =    u  x) dx,
J a,      n=l a
we see that   s (x) dx is equal to the limit to which. a
Un (x) dx
n=1 a
converges when C is indefinitely increased; and this limit is
00 ~=
e f Un (x) dx.
Therefore the theorem is established.




387, 388]


Integration of series


547


388. When s(x) is the sum-function of a series u (x) + u2(x) +... in
an interval (a, b), it is frequently desirable to know whether, for a function
F(x), defined for the interval (a, b), the series
r=1
X   f F(x) )n(x) dx
rb
converges to the limit        F (x) s (x) dx.
If we assume that s(x), F(x) are limited integrable functions, it is
clearly sufficient for the validity of the term by term integration, after
multiplication by F(x), that ISn(x) I should be less than some fixed finite
number, for all values of x and n. For it then follows that IF(x) s, (x)
is less than a fixed finite number, for all values of n and x, and the result
then follows by applying the theorem  of ~ 383 to the series EF(x)un((x),
since F(x)s(x) is an integrable function.
Again, it is sufficient that s, (x) I should be less than a fixed number C,
for all values of x and n, and that F (x) should have an absolutely convergent
improper integral in (a, b).
We may choose the integer n so that the measure of the set H, of those
points at which
I (X) - Sn (x) > e,
is arbitrarily small, say v; we have then
fF (x) { (x) - S (x)} dx < e  F (x) dx + 2C   F (x) I dx,
and since e and q are arbitrarily small, it follows that
rb                  rb
F (x) s (x) dx = limf F (x) n(xo) dx;
and therefore the sufficiency of the criterion is established.
EXAMPLES.
1. Let s,n(x)=nxe-~x2, when n is odd, and =0, when n is even. In this case the
series is simply-uniformly convergent; the sum s(x) is the continuous function 0.
| Sn(X)(-(1 -e-" 2), or O,
0o
according as n is odd or even; thus
lim l s, (x) dx
n=co o
has no definite value, but      s(x)=0.
35-2






_ __ _ _


548                  Functions defined by sequences                      [CH. VI
The term by term integration fails in this case, because there is one point x=0, at which
the measure of non-uniform convergence is indefinitely great, as may be seen from
Rn (  )=-    ne-1, (n odd),
rx
the limit lim  s, (x) dx not being a continuous function of x.
Jo
2. Let s, (x)=2n2xe- n22, then s(x)=0; at the point x=0, there is a point of
indefinitely great measure of non-uniform convergence, since
n () = 2ne - 
x,(x) d = e - no'- ejn2b2) x,<O.
S o
If x be different from zero, lim  s (x) dx'= 0, but at x=0 the limit is -1; thus, in any
n= ooJ Xo
interval which contains the point 0, the function lim f s (x) dx is discontinuous, and
1-=oo 20
fx
therefore cannot equal   s(x) dx, which is zero.
xo,n-I
(n - 1i)
3. Let         8u (x)-= (n1) +n2_e-nx   (-n-1)2xe-(n-1)2x~;
we find s(0)=1, and s(x)=ex, for Ix >0.
We have                f  (x)dx=ex -l. Also lim / sn (x)dx
J                     n=- ooJ o
is discontinuous at the point x=0, which is a point at which the measure of non-uniform
convergence is infinite; it converges to ex -  if x>O, and to zero if x=O.
~~4. Let             k~ On' (x)  kn IO + 1+' (x)
4. Let               un (x)-  1 + {b4n (x)}2 - 1+ (n + 1 ()}2
where kn is a function of n, and,, (x),,n' (x) are finite and continuous in the interval
(a, b), and vanish for x —a. Further let it be assumed that On (x), qn'(x) increase
indefinitely with n, for every value of x except a, but so that lim un (x) is zero.
nXX
We have      f sn (x) dx= - k +1 tan-1 {( + 1 (x)} + kl tan -1 (1 ()},
J a
fs (x) - = kl tan-1 {(1 (x)};
a
these are not identical unless  k, +1 tan-' {0n +1 (x)}
has the limit zero. If         q () = hn (x- a)2,
where hA is positive and increases indefinitely with n, we have
lim kn + 1 tan-' {n, +1 (x)} = 27r lim kn +1.
Hence, if lim kn+i have a finite value, the two expressions have different finite values;
if k,,+ increases indefinitely with n, the series of integrals of the terms of the series
zu( (x) diverges. The series of integrals has in this case a point of discontinuity
at x=a; we find that
$n a      + +     A,)-   h,  n -kn+l h,+, I,
— A    n12. _A~+l2 1 -  1 - 1,




388]                       Integration     of series                        549
and this increases indefinitely as n increases, and thus the point a is a point at which the
measure of non-uniform convergence is indefinitely great.
5. Let  (x —a)  2k+ lh+ 1  (x - a)
5. Llet            ~unv(X^)-=] —T —(  ))2 ---        2)
1 + h. (x - a)2      h + (x - a)2
where A,, increases indefinitely with n, and kn = - -_.
(log h]n)
In this case s (a+       ) increases indefinitely with n, and thus a is a point of
infinite measure of non-uniform convergence.
lim   S (x) dx = l log 1 + hi (x-a)2}-limk,+1 loghn+, x>a,
n=J a                             n=oo
fx
and                       lim  s n(x)dx=O, when x=a;
n=ooj a
also                        s (x) dx=k-l  log {1 +hi (x- a)2}.
a
If j3 1, lim k + 1 log h,, +1 is not zero, hence the term by term integration fails; but
%-=oo
if 3>1, this limit is zero, and the integral of s(x) is equal to the sum of the series of
integrals, although in either case the point a is a point of infinite measure of non-uniform
convergence.
6. Let r be a perfect set of points constructed as follows:-In the middle of the
interval (0, 1) lay off an interval (1) of length 11=X-IX, where X is a positive number
not greater than unity. In the middle of each of the free end intervals, lay off an
interval (2), both of these intervals to be of the same length 12, and such that the total
length of the intervals (1), (2) is 11+2l2=kX —\. Proceeding in this manner, in the
middle of the equal free intervals, after n-1 such steps, lay off an interval (n), all
these intervals to be of the same length l,, and such that the total length of all the
intervals (1), (2),..., (n) is
11+ 22 2213... J ~2n - l1 l = X
When n is indefinitely increased, the set of end-points of the intervals, and the limiting
points of these end-points, form the perfect set r. Let, (x) -=nxe- nx2, x  0;
then form the function
n (x, )=   sin —.      cos-,   O   x   l,
TT.          ( TX\       I
=-      sin -. n (cos-,  — xO,
1     1 -        /'     -2 
= 0, for all other values of x.
Let the middle points of the above intervals (n) be denoted by al(n), a2(n),..,, a(n)2n_l,
and let s, (x) be defined by
s, (x) =, (x - al(l), 11) + fn (x - a1(2), 12)
+ (,, ((x - al(3), 13) +....+,n (x- a4(3), 13)
+ q). (x -a,("), gn) +... + +n (X - a(n-1), In)'




550                 Functions defined by sequences                    [CH. VI
s, (x) is continuous in (0, 1), and converges to 0 for every value of x; for, if x0 be a point
of any interval (i), at most one term in the expression for s, (x) is different from zero,
and this term converges to zero. If x0 does not lie in any interval (i), all the terms
of s, (x) are zero. Every point of the perfect set r is a point of infinite measure of
non-uniform convergence of the series of which s, (x) is the partial sum. In this case
the series n2 f   (x) dx is uniformly convergent, and thus has a continuous sum,
a
which does not however coincide with the value of fs (x) dx.
J a
We find that           (7s  7 (x - a(j), Ij) dx = 1 - e n
J0s (x) dx=-2P  (1-e-n),
where p,<2n-1, is the number of the intervals (n) which fall within (0, x).
It can now be shewn that lim    s (x)dx is a continuous function of x which
0increases from  to 1 s  inrease  erea
increases from 0 to 1 as x increases from 0 to 1, whereas f lim s, (x) dx=O 0, for every
value of x.
If any perfect non-dense set of points G be given, and a,... be the middle point of
the complementary interval of length 1,,, nthe function
i=nj=ki
S (x) =2         - j( x  a 4j,, j)
i=1 j=1
will have, at every point of G, an infinite measure of non-uniform convergence to its
limit s (x). The intervals li,j are here arranged in enumerable order, so that if
e1, 62,... ei,...* be a descending sequence of positive numbers which converges to
zero, li,,  2,..., i, ki are those of which the lengths are < e- 1 and > ei.
THE FUNDAMENTAL THEOREM OF THE INTEGRAL CALCULUS
FOR LEBESGUE INTEGRALS.
389.   Let +(x) be a continuous function defined for the interval (a, b),
and suppose that b(x) has at every point of the interval a differential
coefficient f(x); let us further assume that f(x) is limited in (a, b).   The
function  f(x) is definable    as the limit of a sequence       of continuous
functions c~ (x + h) - 0 (x)
functions       (      (, where h has the values in a sequence of which the
limit is zero. It follows that f (x) is of class 1, unless it be continuous, and
it is consequently a summable function; in fact the theorem    of ~ 381 shews
that f () has a Lebesgue integral, since 1 (x + h)     (x) has a finite upper
Ab




388-390]   The fundamental theorem of integration


551


limit for all values of x and h. In accordance with the last theorem of ~ 384,
we have
f(x) d    =7 m J 'f(x + A) - 0 (x) 
f (x) dx = lim        h       dx
/ a        h=O a                d
1 U x+h   A   dfa+h
= limr -n            (x) da   )dxh=O h (
= k (x) - f (a),
since p (x) is everywhere continuous. We have therefore established the
theorem:If b (x) be a function which possesses a differential coefficient f (x), limited
in an interval (a, b), then f(x) always possesses an integral F(x), in an
interval (a, x), which differs from  ) (x) by a constant only.
This theorem corresponds to the theorem (B) of ~ 258. An example due
to Volterra has been given in ~ 264 of a function which possesses a limited
differential coefficient that is not integrable in accordance with Riemann's
definition.
The above theorem shews that this differential coefficient possesses
a Lebesgue integral. It thus appears that the part (B) of the fundamental
theorem of the Integral Calculus, as stated in ~ 258, holds without limitation,
if Lebesgue's definition be employed, so long as the differential coefficient is
a limited function.
390. Lebesgue has established* the following general theorem:In order that one of the four derivatives of a function may be integrable,
that derivative being supposed finite at every point, it is necessary and sufficient
that the function be of limited total fluctuation. Its total variation is the
integral of the absolute value of the derivative.
The indefinite integral of such a summable derivative is the function of which
it is the derivative.
This theorem affords a solution of the problem of the determination
of a function when either its differential coefficient, or one of its four
derivatives, is a given function, for the case in which that given function
is limited, or also when it is known that the function to be determined
must be of limited total fluctuation. This problem has been already considered in ~ 264.
In order to prove these theorems, we observe that, if + (x) be a continuous function, defined for an interval (a, b), the derivatives D+0((x),
D+ f (x), are the upper, and the lower, limits of indeterminacy of lim I (x, h),
h=O
* Leqons sur l'integration, p. 123.




552


Functions defined by sequences


[CH. VI


where I (x, h) denotes the incrementary ratio  ( + h) - (x)  It will first
h
be proved that a sequence of positive values of h can be determined, such
that the upper and lower limits of I(x, h), for h = 0, are for every value
of x the same when h has the successive values of the numbers in
this sequence as when h is not restricted to have such values. Let
h1', h2',... h',... be a sequence of diminishing positive numbers, converging
to zero, and let e1, 2,...,... be another such sequence.  Since I(x, h)
is continuous with respect to (x, h), for all values of x in (a, b), and for
positive values of h greater than zero, it follows, from the uniform continuity
of I (x, A), that the interval (h',,,, h,) can be divided into a definite number
r. of parts, such that j I (x, h) - I(x, h') < e,, for every value of x, provided
h and h' both lie in one and the same part of the interval. Let this subdivision be made for each value of n, and let hA, h,, h3,... denote the end-points
of all the parts of all the intervals (h'n+, ha'). The sequence h,, h2, h,...
converges to zero, and it is a sequence such as satisfies the required condition;
for we have I I(x, h,) - I (x, h) < ce, provided h-m  h - h+,,, the integer s
being determinate, corresponding to each value of mn. It follows that the
upper and the lower limits of the sequence I(x, h,), I (x, h,),... 1(x, h,1)... are
identical with those of any other sequence I (x, h,'), I (x, h2'),... I (x, h')...,
where h, _h' _ h~, h2  h2' > A,..., and  generally h >- h'm _- hm+l; and
this is the case for every value of x in (a, b). Therefore the sequence [hn} has
the required property. Next, it will be proved that D+ b (x) is a measurable
function, and that it is at most of the second class.  Let u, (x), u2 (x),...
UN (x),... denote a sequence of continuous functions, defined for an interval
(a, b); and let u (x), u (x) denote the upper and lower limits of indeterminacy
lim Un (x), lim u, (x). Let v,, (x) denote that function which, for each value of
2= ~~    n =co
x, has the value of the greatest of the functions ul (x), I2 (x),...  (x), for that
value of x. It is easily seen that the functions vn (x) are all continuous
in (a, b). The functions vl (x, v2 (x),... v, (x)... form a sequence, which for
each value of x is non-diminishing; let w, (x) denote its limit. The function
Wi (x) is measurable, and at most of the first class. Let the function w2 (x)
be formed in the same manner as w, (x), by leaving out the function u, (x),
and proceeding as before. The function w,, (x) is formed by leaving out the
first n- 1 of the functions u (x), and then proceeding in the same manner as
that in which wI (x) was formed from the original sequence. The functions
Wv (x), w2 (x),... Wn(x)... form  a non-increasing sequence of measurable
functions, of the first class at most; their limit is u (x). It follows that
ui (x) is measurable, and of the second class at most. In a similar manner it
can be shewn that u (x) has the same property. If we identify the functions
un (x) with the functions I (x, hA,), where the sequence {hn} is formed as has
been explained above, we see that the two derivatives D+ 0 (x), D+ 9 (x) are




390]     The fundamental theorem of integration


553


measurable, and of the second class at most. The derivatives D-     (x),
D_   (x) clearly have the same property.
We now assume that, for each value of x in (a, b), the derivative
D+ > (x) has a finite value. Let the unlimited interval (- oo, x ) be divided
into intervals (a,, a,+l), where the integer l has all positive and negative
values, and so that, for each value of l, a,+l - a, < e, where e is a fixed positive
number. Let e, denote that set of points x in (a, b), for which
a, < DI ~ (x) - a,+1;
and arrange the sets e, in the order eo, el, e_1, e2, e-2,... en, e_-,....
Let ko, k1, k_-1, k2, k_2,..., kn, kn... be a sequence of positive numbers so
chosen that the limiting sum of
ko I ac  + k, I a, I + k-_ | a- 1  +... + kn I a I + k-, I an I +...
is less than e. Let the set e0 be enclosed in a set of intervals A0, and the
complementary set-C (eO) in a set of intervals A0', so that the measure of that set
of intervals which is common to the sets A0, A0' does not exceed k0. Enclose
e, in a set of intervals Ai, and C(eo + el) in a set of intervals A/'; where the
sets Al and A,/ are both interior to Ao', and have in common a set of intervals
of measure not exceeding kI. Proceeding in this manner, we enclose ep, where
p is positive or negative, in a set of intervals Ap, and C (e0 + el + e_1 +... eq),
where eq immediately precedes ep, in a set A'p, so that Alp, A' are both interior
to A',, and have in common a set of intervals of measure not exceeding kp.
We have now m (A) - m (ep) - kp, and A, has in common with all the other
sets A, a set of intervals whose measure is less than kp.
Since      ap m (Ap)-    ] ap I m (ep), where p = 0, 1, -1, 2, -2...,
is less than  ckp[ ap, or than e, we see that
p
a I m (Ap),    I ap I m (e.)
P             p
are either both divergent or both convergent; and in the latter case the
[b
difference of their sums is less than e. If  D+   (x) dx exists, we have
rb
I  D+   (x) I dx < (b - a) e + S I a li m (e.) < (b - a - 1) e + Z l a. I m (A.);
P                          p
and the integral exists if E I al t m (Ap) is convergent. Similarly, the necessary
p
and sufficient condition that  D+   (x) dx should exist is that 2 ap m (Ap)
should converge; and then the two differ by less than (b-a-1)e.     Any
point x in (a, b) belongs to one of the sets ep; let 8p be that interval
of the set Ap which contains x. Let (x, x + h) be the longest interval with h
positive, contained in 8p, which does not exceed e, and is also such that
aP   I (x, h) ^ a+, + e.




554


Functions defined by sequences


[CH. VI


Starting from the point a, we define in this manner an interval (a, x);
then we take the interval (x,, x2) corresponding to x1, and so on. The point
b will be reached, either as the end-point xg of an interval (x-1, xp), where 3
is some number of the first, or of the second class, or else it is the limiting
point of the end-points of a sequence of intervals. The value of the sum
(xa.+   )-    (x.)i
taken for all the intervals (x,, xaa+i), by which the points a and b are joined,
lies between 2 I ap I 8'Pa ~ 2e (b-a), where 8'P is the interval (x,, xa+i), and
pa denotes the corresponding value of p. Also     lap  S'Pa differs from
p                       p
which do not belong to intervals 'Pa, necessarily belong to one of the sets A.,
where q is different from p, and their measure accordingly does not exceed kp.
Therefore 2 4 (x~a+) - ) (xa) I lies between the two numbers
X l aa | m (A)- e' + 2e (b - a), where ' < e;
5b
and these numbers are finite if   D  4D ) (x) dx exists.
- a
It follows that the necessary and sufficient condition that D+q (x)
should be integrable is that the function ( (x) should be of limited total
fluctuation, in which case its total variation in (a, b) is limited.
We have also
(b) -  (a) =    {a+i (x) -  a (x)} =f D+  (x) dx,
in case D+ 4 (x) is integrable; the reasoning being the same as before, and
remembering that the number e is arbitrarily small. The theorem may be
proved for the case of the other derivatives in a precisely similar manner.
If then ) (x) be of limited total Iluctuation, and have its four derivatives
finite at each point, we have
/:'b           /b            /b
(b)-) (a) =f D+(x)dx=      D+ (x) dx=     D-   (x)dx= fD_    (x)dx.
391. The following theorem, also due to Lebesgue, will now      be
established:
A function 4 (x), of limited total fluctuation in (a, b), and of which one of
the four derivatives is limited, has a differential coefficient 4' (x) at every point
of (a, b), with the exception of points belonging to a set of measure zero. The
theorem also holds in case the derivative has points of infinite discontinuity
belonging to a closed set of points, of zero content.
Let e be a measurable set of points in (a, b), and let e (x) denote the part
of e in the interval (a, x).  It will then be shewn that, me(x) denoting
the measure of e(x), the function me (x) has a differential coefficient equal




390, 391]   The fundamental theorem of integration


555


to 1 at a set of points contained in e, and of measure m(e); and that it
has a differential coefficient equal to 0 at a set of points not belonging
to e, and of measure b - a - n(e).  Thus the set of points at which me(x)
has no differential coefficient is of measure zero. All the points of e may be
enclosed in the interiors of the intervals of sets D1, D2,... Dn,..., such that
rn (Dn) converges to m (e), as n is increased indefinitely. Let En (x) denote
the part of Dn which is in the interval (a, x); we then have
D+ m En (x) _I D+ me (x) - 0.
But D+ mE, (x)= 1, at all interior points of Dn; and it follows that
D+ m E (x)= 0 at all points not interior to Di, except at points of a set
of measure zero; for otherwise
D+ mE (x) dx
would exceed mD,.   As this holds for every value of n, it follows from
the above inequality that D+ mle (x) must be zero at every point not belonging
either to e, or to a certain set of zero measure. Since D+me (x) = 0, at
a set of points of measure b- a - m(e), and since
rb
f D+ me (x) dx= m (e),
it follows that D+ me (x) is equal to unity at a set of points of measure
mr(e), belonging to the inner limiting set defined by the sequence {D~}.
Also this inner limiting set has the measure m (e); therefore D+ me (x)= 1,
at a set of points all belonging to e, and of measure m (e). A similar theorem
can be established for each of the other derivatives of me (x). It then follows
that me(x) has a differential coefficient equal to 1, at a set of points of
measure m (e), belonging to e, and also a differential coefficient equal to 0, at
a set of points of measure b - a - m (e), belonging to C (e).
Next, let f(x) be a limited summable function defined in (a, b), and
of which L, U are the lower and the upper limits.  Divide the interval
(L, U) into parts
(a0, a), (a,, a2)... (a,_,, a,)... (an-_, an), where a0 = L, an = U;
and where a, - a_- does not exceed e, for any value of t. Let 0b (x), 0b (x) be
two functions defined as follows:-For each value of x such that
a, f (x) < a,+1, let, (x) = a,;
and for each value of x such that a,_ <.f(x)  a,, let 0. (x) = a,.  Thus
P0 (x), (2 (x) are defined for the whole interval (a, b), each of them having
only a finite set of values. Let
F    (x) =  x)dx, Fi(x)     1 ) (=  d (   F2 ()=  0 (x) dx;
a                 a                  a




556


Fuanctions defined by sequences


[CH. VI


we then have D+' F (x) - D+ F(x) - D+ F (x), at each point x in (a, b).
Also, we have
YF (x) = ao, me, (x) + a, me, (x) +.. + an men (x),
where e, (x) is the part, contained in (a, x), of the set e, of points of (a, b) at
which a,  f (x) < a,+,. The function F1 (x) has a differential coefficient equal
to 0, (x), at all points of (a, b) not belonging to a set of measure zero. For
me, (x) has a differential coefficient at every point not belonging to a set
of zero measure, and this holds for every value of t; therefore F1 (x) has
a differential coefficient at every point not belonging to a set of measure zero.
Also - me, (x) is unity at all points of e, except at points of a set of zero
dx
measure, and is zero at all points of C (e,) except at points of a set of zero
measure; and this holds for each value of t. A similar result holds for
the function F,(x).
At any point x, we have
(x +   ) - F, (x)  F(x + h) - F () <  (x +h)-F2 (x)
h                 h                  h
for positive or negative values of h. Therefore the four derivatives of F(x)
all lie between, (x), and,2 (x), at each point x which does not belong
to that set of points of zero measure at which F1 (x), F2 (x) do not possess
differential coefficients equal to 0b (x), b0 (). Now  b1 (x),,0 (x) differ from
one another, and from f(x), by not more than e; therefore, at every point
not belonging to a set of measure zero, the four derivatives of F(x) do
not differ from one another by more than e. By taking a sequence of values
of e converging to zero, we then see that F(x) has a differential coefficient
equal to f(x) at every point of (a, b) not belonging to a set of zero
measure.
If the function f(x) be unlimited, but summable and possessing a
Lebesgue integral, the functions F1(x), F (x) are each defined by series,
infinite in both directions. The term by term differentiation of this series
would then require justification. It will be, however, sufficient* for the
present purpose to consider the case in which the points of infinite discontinuity of f(x) form a closed set of zero content. By enclosing this
set of points in a finite set of intervals of arbitrarily small sum, and applying
the result obtained above to each of the complementary intervals in which
f(x) is limited, the theorem may be extended to the case of such a function
f ().
* Lebesgue applies reasoning similar to that in the text to the case of any summable function,
loc. cit., p. 124. As each of his functions 0 is the sum of an infinite number of functions 1, when
f is unlimited, the process would then appear to require some further justification.




391, 392]   The ftndamental theorem of integration


557


It has now been established that, if f (x) be a summzable function which
is either limited in (a, b), or has points of infinite discontinuity belonging to a
closed set of points of zero content, then  f (x) dx has a differential coefficient
equal to f(x), at a set of points whose measure is equal to that of the whole
interval (a, b).
The function + (x), being of limited total fluctuation, is equal to the
integral   D+ q (x) dx, the integrand being supposed finite at each point.  It
has now been shewn that, if the derivative D+ 0 (x) is limited, or at most has
indefinitely great values in the neighbourhoods of points of a closed set of zero
content, then 0b (x) has a differential coefficient equal to D+ + (x) at each
point of a set of measure b-a.   At such a point the four derivatives are
of course equal to one another.    Therefore the theorem   stated at the
beginning of the present section has been established.
Lebesgue has also established* the following theorem:Every function with limited total fluctuation, and in particular, every
monotone function, has a finite differential coefficient, except at the points of a
set of which the measure is zero.
This differential coefficient is summable in the domain which consists
of those points at which it exists and is finite, but the integral is not necessarily
the given function, unless one of the four derivatives of the given function be
everywhere finite.
392. Besides the Lebesgue integrals of unlimited functions, as defined in
~ 291, which integrals are necessarily absolutely convergent, there is a class of
non-absolutely convergent improper Lebesgue integrals, which may be defined
by extending Harnack's definition of improper integrals given in ~ 271. The
extension consists in taking the integrals in the intervals V employed in ~ 271,
to be Lebesgue integrals, and not necessarily Riemann integrals. In case the
integral so defined, of an unlimited function, be absolutely convergent, it has
been shewn in ~ 291, to be in agreement with the ordinary Lebesgue integral
of the same function, as defined in ~ 291. If, however, the integrals taken
through the set of intervals {I} have a limit, but the limit of the integrals
of the absolute values of the functions do not exist, we have then a nonabsolutely convergent improper Lebesgue integral.
An example of such an integral is the following:-Let f(x)= 0, for all
rational values of x in (0, 1); and for irrational values of x, let f(x) = -sin - 
x x
then    f(x) dx exists only as a non-absolutely convergent Lebesgue integral,
Lecons sur 'int6gration, pp. 123 and 128.




558


Functions defined by sequences


[CH. VI


being defined as lim  f(x) dx.  The integral   f(x) dx does not exist;
for  If(x) ldx is not convergent, for  = 0.
393. The theorem proved in ~ 282 that, if f(x) dx exist as an improper
integral, whether absolutely convergent or not, in accordance with Harnack's
rb'
definition, then  f (s) dx exists, where a - a' _ b' - b, and that the cona'
vergence of this integral is uniform for all values of a' and b', is applicable
without change to the case of improper Lebesgue integrals, whether absolutely
convergent or not. The proof in ~ 282, is valid in this more general case.
Also the proof that
f (x) dx = j  (x) dx +  f(x) dx
a          a          x
given in ~ 282, is applicable without change.
It may be proved that, for an improper Lebesgue integral,  f (x) dx is
a continuous function of the upper limit x.
Using the notation of ~ 282, we have
f (x) dx-J  fs ( x)dx <  e,
and                   f (x) dx-  f(x) dx <,
provided the set of intervals {8} is properly chosen. We have also, since
fs (x) is a limited function,
f (X) dx-    fs (x) dx <,
provided I h is less than some positive number a. It follows that
I    +() dxz-  f(x)dx <Eif{ hI <.
Therefore, since e is arbitrary,  f (x) dx is continuous.
It has been shewn in ~ 390, that if + (x) have limited total fluctuation in
(a, b), and if D  ( (x) be everywhere finite, then
(b)-   (a)=)  D   (x) dx.
Now let D  ( (x) be indefinitely great at points of a reducible set G; then
if (a', x) be interior to one of the complementary intervals of the set G,




392-394]   The fundamental theorem of integration


559


and 4 (x) have limited total fluctuation in the complementary interval,
we have
4()- () (a) =    4)(x)dx;
and if D   (x) have a Lebesgue integral, or a non-absolutely convergent
improper Lebesgue integral in (a, b), we have
D     (x) dx -  D   (x)dx =  x)- ) (a).
J a         J a
Therefore  D   (x) dx -p (x) is constant throughout the interior of the. a
complementary interval considered.  The function 4 (x) being continuous,
this difference is continuous throughout (a, b); and since G is reducible,
it follows from the theorem of ~ 206, that f D 4 (x) dx-  (x) is constant
throughout (a, b), and therefore = -  (a). The following extension of the
theorem of ~390, has therefore been established:If ) (.x) be a continuous function such that D b (x) is finite at every point of
(a, b) which does not belong to a reducible set G, and if (D 4 (x) dx exist as a
Lebesgue integral, or as an improper non-absolutely convergent Lebesgue
integral, then
fD ) (x) dx =    (x) -  (a)
for every point x in (a, b).  Dof (x) denotes any one of the four derivatives
of 4 (x).
If the set G were not reducible, but contained a perfect component,
then   D   (x) dx would in general differ from ( (x) by a function with
a
an everywhere-dense set of lines of invariability.
INTEGRATION BY PARTS FOR LEBESGUE INTEGRALS.
394. If u, v be two continuous functions with limited total fluctuation in
(a, b), then the product uv has the same property.  For if u=u,- -,,
v = v -v2, where 21t, u2, v, 2 are monotone non-diminishing functions, then
u1v1 +u2v2, u1 v2 +u2v have the same property, and therefore (u.-u2) (v -v 2)
is of limited total fluctuation.
Let us assume that u, v both have limited derivatives in (a, b); then in
du dv
accordance with the theorem of ~ 391, d' d  both exist at all points of
(a, b), except points of a set with zero measure.




560


Functions defined by sequences


[CH. VI


d (uv)   du    dv
We have                     = v   + u —,
dx     dx    dx'
at each point of the set E   of points where the differential coefficients
exist. We have then
d (uv)        du          dv
d     dx d- J v      d J  dx.
Also           v - dx =I    DU dx,    u $t dx=   u Dv dx,
since Dzu, Dv are limited, and m (E) = b- a. Also
d -V) dx= Lvi;
E dx    dx    uva
b,       r    ~}b  rb
therefore            b u Dv dx = [u    v    Du dx.
a               Jat
Now let            =I       U dx, v=   V x dx,
where U and V are limited in (a, b); then U only differs fiom Du at points
of a set of zero measure, and V differs from Dv only in the same manner.
We thus obtain the formula for integration by parts,
fv    U dx  dx    (        f )(      ) -x) fU( f  V dx) dx,
where a, /3 are arbitrarily fixed points in the interval (a, b).
If Du, Dv be not limited, but have points of infinite discontinuity which
belong to an enumerable closed set G, let X (x) denote
uv   -     u Dv dx -   v Du dx.
_   a  Ja          Ja
The function X (x) is constant in any interval contained in an interval
complementary to G. The functions u, v being continuous in (a, b), the
function % (x) is continuous in (a, b), and therefore, since G is enumerable,
X (x) is constant throughout (a, b); and it is zero, since it vanishes at the
point a.
Therefore we have
u Dv=     uv   -   v Du dx,
J a         a J^  J a
where u, v are continuous, and Du, Dv have points of infinite discontinuity
belonging to a reducible set of points.




394-396]


Differentiation of series


561


THE DIFFERENTIATION OF SERIES.
395. If s (x) denote the sum-function of a series u (x) + u (x) +..., and
it be assumed that, either at a particular point x, or in a continuous interval,
all the terms iu (x), u2 (x),... are continuous and differentiable, it is a subject
for investigation under what conditions s (x) possesses a differential coefficient
which is the limit of the sum ut1'(x)+u2'(x)+..., the series of which the
terms are the differential coefficients of the original series. It may happen
that (1) s(x) possesses no differential coefficient, or (2) that the series
u' (x) + u' (x) +... is not convergent, or both (1) and (2) may happen
simultaneously, or (3) that s' (x) exists and the series is also convergent, but
that its limiting sum is not s' (x).
Writing s (x) = s, (x) + Rn (x), we have, at any point of convergence of the
series, lim Rn (x) = 0; further we have
n= 00
s (x + h) - s (x)  sn(x +h) - S (x)  Rn (x +h)-Rn (x)
h                h                 h
On the hypothesis that all the terms of the series have finite differential
coefficients at the point x, we have lim s (x + h) -   = ()  (); if then
h=O        h
Rn (x) possesses a differential coefficient, so also does s (x). If Rn' (x) exists,
and converges to the limit zero, when n is indefinitely increased, we have
s' (x) = lim Sn' () = lim {u' (x) + tU2 (x) +... + Un (x)'.
n=co       n=oo
In case Rn,'(x) either does not exist, or exists but does not converge to
the limit zero, when n is increased indefinitely, the term by term differentiation of the series is inapplicable.
396. Let us assume that, in a given interval (a, b), the terms of the
convergent series u1 (x) + tu (x) +... + ~u (x) +... are differentiable, and that
their differential coefficients are everywhere finite, and are integrable in
(a, b), and in case they are unlimited have a reducible set of points of
infinite discontinuity, so that  un' (x) dx = Un (x) - t~, (a).  The integral
may be either a Riemann integral, or an improper one in accordance with
Harnack's definition, or a Lebesgue integral, or a non-absolutely convergent
improper Lebesgue integral. Let it be further assumed that the series
u/ (x) + u2' (x) +... + Un' (x) +...
is convergent everywhere in (a, b); then denoting the sum-function of this
latter series by p (x), we may apply the theorems in ~ 379-386, to obtain
rX
conditions that b (x) possesses an integral fx (x) dx, where a - x  b, and
Ja
that the series
{a1 (x) - i1 (a)} + {Ji2 () - 2 (a)} +...
H.                                                           36




562                Functions defined by sequences               [OH. VI
converges to the sum    ) (x)dx. If these conditions are satisfied, we have
(x) dx= s (x) - s (a); from which it follows that, at least at any point
of continuity of b (x), the differential coefficient s' (x) exists, and is the sum
of the series
U/ (x) + U, (x) +... +  (x) +....
It follows from  a theorem  in ~ 383, that if the series   u,n'(x) be
uniformly convergent, the function +(x) is integrable in (a, b), and that
f   (x) d = s (x) - s (a); and thus s' (x) = ) (x), at any point of continuity of
) (x). In particular, ) (x) is everywhere continuous if all the terms u,' (x)
are continuous functions.  We have therefore established the following
theorems:If the series Su, (x) converge in (a, b), and the terms of the series u,'n (x)
be all finite and continuous in (a, b), and the latter series converge uniformly,
then s' (x) exists, and is the sum of the series Su,/ (x), at all points in (a, b).
If the series Szu (x) converge in (a, b), and the differential coefficients
un' (x) have all definite finite values everywhere in (a, b), and are integrable
in the sense explained above, and the series  Un' (x) be uniformly convergent;
then at every point of continuity of the sum of the series  Un' (x), that sum
is s'(x).
It is a known theorem that the sum-function 4 (x) of a uniformly
convergent series of point-wise discontinuous functions is at most pointwise discontinuous; and in the present case the points of discontinuity form
a set of zero measure, provided the integrals of the terms u' (x) are
Riemann integrals.
Exactly similar theorems hold for derivatives of the terms u, (x), on one
side.
The condition of uniform convergence contained in these theorems is a
sufficient, but not a necessary, condition for the validity of the process of term
by term differentiation.
A less stringent, but sufficient, condition would, in accordance with the last
theorem of ~ 383, be obtained by replacing the condition that Zu,' (x) should
converge uniformly in (a, b) by the condition that its convergence should be
simply uniform*.
Still wider conditions for the validity of the process are obtained by
applying the theorems of ~~ 384, 385.    We thus obtain the following
theorems:If the series Sun(x) converge in (a, b), and the differential coefficients
Un '(x) everywhere exist, and are limited, and the series  un' (x) be every* See Bendixson, " Sur la convergence uniforme des s6ries," Stockholm Ofv. vol. LIv, 1897.




396, 397]


Di&ferentiation of series


563


where convergent; and if firther  un' (x) be, for every value of n and x, less
'1
than some fixed positive number, then S  (xt) = d-w 1  (x), at least at every
1        dx 1
point of continuity of Su,' (x).
1
I/ the series Xun (x) converge in (a, b), and the functions u,' (x) everywhere
exist, and are limited, then provided the set of points, in the neighbourhood
if
of which the condition that lUn' (x) is for every n and x less than a fixed
positive number is not satisfied, be an enumerable set, and if also  u (x) be
1
continuous in (a, b), then Yut' (x) =- Sun (x), at least at every point of con1        dx 1
00
tinuity of:un' (x).
1
A particular case of this theorem is that in which all the terms u' (x), and
00
1un,'(x), are continuous. In the general case, it will be observed that, in
1
accordance with the theorem of ~ 389, the terms u,,'(x) are all integrable,
possessing at least Lebesgue integrals. Also, in virtue of the theorem
proved in ~ 391, the points at which the term by term differentiation does
not hold, form at most a set of points of measure zero.
By employing the theorem established in ~ 386, the above theorems can
be extended to the case in which the series 2'u' (x) fails to converge at points
belonging to a reducible set of points.
397. The condition of the validity of term by term differentiation of
the convergent series lu (x), at a particular point a of the domain of x, is
identical with the condition that the two repeated limits of
s(a + h, y) - s (a, y)
h
for h = 0, y = 0, should exist, and have one and the same value. By applying
the theorems of ~ 234, 235, which contain the necessary and sufficient conditions for the existence and equality of repeated limits of a function at a
point, we obtain the following theorems:If the series szun (x) everywhere converge in a suficiently small neighbourhood of a point a, and the differential coeficients u,, (a) exist, and are finite,
then the necessary and sufficient conditions that d- s (  at x = a, may exist
00
and be equal to u n' (a) are (1) that Zu n (a) be convergent, and (2) that, e being an
1
arbitrarily chosen positive number, and no an arbitrarily chosen positive integer,
36-2




564


4Functions defined by sequences


[CH. VI


a number q7, positive and > 0 can be found, and also a positive integer n > no,
such that the condition R (a + h)- R(a) < e is satisfied for this value of n,
h
and for every value of h such that 0 < I h I < 7, and for which a + h is interior
to the given neighbourhood of a.
If the series (Un (x) converge everywhere in a sufficiently small neighbourhood of a point a, and the differential coefficients Un' (a) exist and are finite,
dx
then the necessary and suacient condition that dxxs() at x —a, may eist
and be equal to Sun' (a) is that, corresponding to any arbitrarily chosen positive
1
number e, an integer no exists, such that corresponding to each integer n > no, a
positive number q7, in general dependent on n, can be found, such that the
condition  Rn (a + h)- R (a) <  is satisfied for every value of h such that
h
0 < I h j < 7, and for which a + h is interior to the given neighbourhood of a.
It is clear from ~ 234, that the uniform convergence of  ( + h)-   ()
to the limit (a + h   (a), for all values of h, except 0, in a fixed interval
(-, 3') for h, is a sufficient condition that s'(a) exists, and that the series
lu,' (a) converges to s' (a).
398. The following theorem * is frequently more convenient than the
theorems of ~ 397, for the purpose of ascertaining whether a function defined
by a convergent series of functions is differentiable or not.
If the series Uel, (x) converge in (a, b), and the differential coefficients u.' (a)
d
exist, and are finite, then the necessary and sufficient conditions that  s (x)
may exist at x = a, and be the sum of the series EuS' (a), are (1) that the series
2;u'(a) be convergent, and (2) that, corresponding to an arbitrarily fixed
positive number e, and an arbitrarily fixed integer m', a positive number 8
can be determined such that, for each value of h numerically less than 8, and
for which a + h is in (a, b), an integer m (> m'), in general varying with h, can
be found, for which the three numbers
2 {Un (a +h)-Un(a)     (a)     R  (a + h)  R,} (a)
n=l 1  -                   )       h          h
are all numerically less than e.
The convenience in application of this theorem arises from the fact that


* Dini, Grundlagen, p. 152.




397, 398]


Differentiation of series


565


it provides a test in which only a single value of h is employed. To prove
that the conditions stated in the theorem are sufficient, we have
s(a +h) -s(a)    Go               (a+h)-     (a)
h         n=1                    h 
R(a + h)     R  (a)   R
+ —.-R
+     h          h
where Rm' denotes the remainder, after m terms, of the series EUn (a).  The
number m' can be so chosen that I Rn' < e, for n? m', since the series,Un'(a)
is convergent. If m be chosen > m', and such that the second condition
in the theorem   is satisfied, we see that  s( + h) - s()_ -,(a) < 4e,
h         1n=1
provided | h <8; and therefore lim s    +  )        is E un(a). Therefore
h         sTh
the conditions are sufficient.
To shew that the conditions stated are necessary; it is clear that
(1) must be satisfied, and therefore that mn' can be determined so that
IRm,  <  e, if m  nm'. Moreover, a positive number 8 can be determined
such that     +h)s (- h      - a  u'' (a) is numerically less than I e, if hl < 8.
h         n=1
Also since ]~n (x) is convergent, for each value of h, a corresponding value
of m (min') exists, such that  m(+) Rm() are each numerically <           e.
h         h                            4
It then follows that, for these values of h and m, the condition
Un (a + h)- un (a)        )
un'(a) <
is satisfied.  Therefore the conditions in the theorem  are necessary.
EXAMPLES.
1. Let, (x) = - sin nx; the series 2u,, (x) converges everywhere in any interval, but
the series 2 cos nx does not converge. The term by term differentiation of the given series
is therefore inapplicable.
sW  Xng + 1
2. Let u,, (x)= - = -  -; the series 2u,, (x) converges to the sum-function s (x) = x, in
n   n+1'
the interval (0, 1). The series E (x-l - xn) converges to s' (x)=1, for all values of x in the
interval (0, 1), except for x=0, when it converges to 0, which is not equal to s' (0). The
series 2 (Xn-l- xn) has the point x=0 for a point of non-uniform convergence, and thus
the convergence is not uniform in the interval (0, 1).
3. The series 2 b cos (a'x), where 0<b<1, converges uniformly in any interval. The
n=1
series - (ab)" sin (alx), for ab>l, is not convergent. It will be shewn later that the
function defined by the given series is not differentiable for any value of x, provided ab
exceeds a certain value.




566


Functions defined by sequences


[CH. VI


REPEATED IMPROPER INTEGRALS.
399. If f(x, y) be an unlimited function, defined in the fundamental
rectangle bounded by x = a, x = b, y = c, y = d, it is an important case of the
problem of the reversal of the order of repeated limits, to investigate
conditions under which the repeated integrals
rb   rd            Id   rb
dx      f (x, y) dy,  dy   f (x, y) dx
when they both exist, have the same value. In the case in which the
improper double integral   f(x, y) (dxdy) exists, it is also a matter for
investigation whether, or under what conditions, this double integral can
be replaced by one or other of the corresponding repeated integrals. These
problems have been investigated* by de la Vallee-Poussin, who has obtained
a number of results of a general character. An investigationt will be here
given, in which the questions are considered in a still more general manner,
free from some of the restrictions introduced by de la Vallee-Poussin.
The definition of a double integral will be taken to be that of ~ 321; that of
a single integral will be taken to be that of Riemann, or that of de la ValleePoussin, in the case of an unlimited function.
An extension of the definition, given in ~ 379, of the regular convergence
of a sequence of functions to a limiting function will be first given.
Let 0, (  x), 02 (x),... n (x),... be a sequence of functions defined for the
interval (a, b). We shall suppose that, for each value of x, any one of these
functions,,b (x) has either a definite value, or is multiple-valued; and in the
latter case it is regarded as indeterminate between limits of indeterminacy,
either of which may be finite or infinite, of which the upper limit may be
denoted by   n (x), and the lower limit by, (x). For any value of x for
which bn (x) is determinate, we have, (x) = On (x). When either On (x) or
fn (x) is to be taken indifferently, we may use the notation 6 {(x).
The consideration of a function /n (x) which, for a particular set of values
of x, is indeterminate, as a single function, involves an extension of Dirichlet's
definition of a function which is justified by its convenience for use in
investigations such as the present one. This extension, which has been
already referred to in ~ 166, is convenient when the functional value, (x)
* His investigations are contained in three memoirs, the first in the Annales de la Societe
scientzfique de Bruxelles, vol. xvii; the second in Liouville's Journal, ser. 4, vol. vIII, 1892; and the
third in Liouville's Journal, ser. 5, vol. v.
It See also Hobson, Proc. Lond. Math. Soc., ser. 2, vol. iv, 1906.




399]


Repeated improper integrals


567


at a point x is defined by means of a limit, say f, (x) = lim  fn (x, in), which
m= oo
may be such that, for a particular value of x, lim * (x, m) has no single
value, but may be multiple-valued between limits On (x), On (x). The
function,,,(x), for such a value of x, may be capable of having a finite
number, or an infinite number, of values, and possibly of having all values
between 0pn (x) and ~n (x); but in the application of the theory we need only
attend to the upper and lower limits of indeterminacy, it being indifferent
whether,n (x) has all values between these limits, or some values only. The
fluctuation of d/n (x) in any interval (a, /3) is the excess of the upper limit of
the numbers b,{ (x) for all points in (a, /), over the lower limit of the numbers
'fn (x) in the same interval. The saltus of 4, (x) at the point x is the limit
of the fluctuation in an interval (x -, x + ), when 8 is indefinitely
diminished, and this saltus is _ 0Pn (x) - dp (x). Riemann's theory of integration is applicable to such a function bn(x), when it is limited, just as
in the case of a single-valued function. For any fixed value of x, the numbers
q01(x), b2(x), *.. n(X)... 01(x), 022(x), 1.. b,(x)... form a set which we may
denote by G. Let us consider the derivative G' of G; then, if G' be limited,
since it is a closed set, it has a greatest value A, and a least value B. These
numbers A and B are such that, for a given e, there are an infinite number
of values of n such that |I n (x)- A I < e, and also an infinite number of values
of n such that I n(x)- B < e. If G' be unlimited in one direction, or in
both directions, either A, or B, or both, may be regarded as having one of the
improper values o,, - oo.
We now define a function + (x), for the interval (a, b), in the following
manner: —When, for a particular value of x, the numbers A and B are equal
and finite, their value is taken to be that of b (x). If A and B are unequal
and finite, we regard 0 (x) as multiple-valued, with  ( (x) = A, ( (x) = B. If
either A or B have one of the improper values oo, - oc, the point x is taken
to be a point of infinite discontinuity of b (x). The function b (x) is regarded
as a single function, not necessarily limited, and it may have either a proper
integral, or an improper integral in (a, b), in accordance with Harnack's
definition of the improper integral of an unlimited function. This function
b (x) is said to be the limiting function defined by the sequence J{q, (x)}; and
the functions fcn (x) are said to converge, in an extended sense of the term, to
the function p (x); and thus we write b (x) = lim c/ (x).
= 0oo
In case the sequence {n (x)} be non-diminishing, so that, for every value
of x and n, the condition c((x) -  c/Kn+, (x) is satisfied, the sequence {,n(x)}
has, for each value of x, either a definite upper limit A, or else the improper




568


Functions defined by sequences


[CH. VI


limit + oo. If (, (x) _ btn+, (x), for every value of x and n, the sequence
{cn (x)} has, for each value of x, either a definite lower limit B, or else the
improper lower limit - oo.
Let a positive number e, and a positive integer n1 be arbitrarily chosen,
and let E be a set of points in (a, b), of which the measure is zero. Let us
suppose that, for each point xa, in (a, b), which does not belong to a certain
component E, of E, this component depending on e, an integer n (> n,), and
also a neighbourhood (x, - 8, x, + 8') can be determined, such that the four
inequalities I b (x) -  n (x) < e are all satisfied at every point in the interval
(x1 - $, x1 + 3') which is in (a, b). Then, provided this condition be satisfied
for every value of e; and also E be such that each point of it belongs to Ee
for some sufficiently small value of e, the convergence of the sequence {(n (x)}
to b (x) is said to be regular, in the extended sense, in (a, b) except for the
set E of zero measure.
In case, for each value of x, the sequence {n (x)} is an increasing one,
so that Sbn(x) < Sn+i(x), and also,n (x) <,n+1 (X), when tlhe conditions
0 (x)-   (x) l  <  are satisfied for a particular value of n, they are also
satisfied for every greater value. In the general case however this is no
longer true.
As in ~ 379, it is seen that the set E, must be non-dense, and therefore
that the set E is of the first category.
The set E contains every point at which b (x) has not a definite finite
value, for since S (x) - n (x), (x) - n (x) are both numerically less than e,
at a point which does not belong to E, for some value of n, it follows that
b (x) - b(x) is less than 2e; and since e is arbitrarily small, it follows that
(b(x)= b(x). It is clear that the points of infinite discontinuity of p (x)
belong to the set E~, whatever be the value of e.
As in ~ 380, it can be shewn that, if all the points of E, be enclosed in
the interiors of intervals of a finite set, of which the sum is V, the conditions
I ((x)- fn (x) < e are satisfied at every point of (a, b) not interior to the
intervals of the finite set, where n has one of a finite number of values
nz +p, 1+, n      -+p2,. ' +p.. The particular number n +p which must be
taken for a point x depends upon the value of x, but the same number
nl+p is applicable to all the points of one or more continuous intervals.
400. It will now   be assumed that the improper double integral
ff(, y)(dxdy) exists, in accordance with the definition of ~ 321; and a
neesary condition  ill b fud that d(      eists.
necessary condition will be found thatJ dx  f(x, y)dy exists.
Ja  Jc




399, 400]          Repeated improper integrals                      569
We shall consider a sequence f, (x, y), f (x, y),...f(x, y)... of functions
obtained from f(x, y) as in de la Vallee-Poussin's definition of the improper
double integral, given in ~ 321.
rd
The integral  fn(x, y)dy will be denoted by b,,(x), where 0,,(x) may
either have a determinate value, or may have as limits of indeterminacy
jn(x), A(x), the upper and lower values of the integral   fn(x, y)dy, in
accordance with Darboux's definition of the upper and lower integrals of
a limited function (see ~ 252). The existence* of fn, (x, y)(dx dy) does not
ensure the determinacy of obn(x) for all values of x. The integral
f(x, y)dy
will be denoted by   (x); a similar remark applies to the determinacy of
+ (x) as in the case of qfn(x).  Moreover 0(x) may have the improper
value o, or -oo, or may have one of these as a limit of indeterminacy; for
f(x, y) does not necessarily possess for each value of x either a proper or an
improper integral in the interval (c, d). When, for a fixed x, the function
f(x, y) has points of infinite discontinuity with respect to the variable y, in
the interval (c, d), the value of  f(x, y)dy, or +(x), is the upper limit
of the derivative of the set of numbers,,(x); also b(x) is the lower limit
of the derivative of the set of numbers Oj(x).  It may happen that ((x),
or (b(x), has an improper value oo or - oc.  In the present case all the
functions Ob(x) are limited functions.
Since f(x, y) is integrable in the fundamental rectangle, all the functions
fn(x, y) have proper integrals in that domain. The proper integral
f f (x, y) (dx dy),
is, by the theorem of ~ 314, replaceable by the repeated integral
rb   rd
Jdx     fn (x, y) dy,
a    c
and thus 4 (x) is integrable in the linear interval (a, b). It follows that the
points of discontinuity of fn(x) form a set of points of linear measure zero.
The set of all points of discontinuity of any of the functions
01(x), 02 (X),.*..  (x)..,
* In de la Vallee-Poussin's investigation in Liouville's Journal, ser. 4, vol. vI1I, the restrictive
assumptions are made that on (x) and o (x) are everywhere definite and finite.




570


Futnctions defined by sequences


[CH. VI


is consequently also a set of zero measure. If + (x) be integrable in the
interval (a, b), its points of discontinuity must form a set of zero measure.
Let us suppose that ( (x) is integrable in (a, b), and thus that
rb    CZ
f dx; f(x, y)dy
exists; and let us assume, that, if possible, the set E,, referred to in the
definition of regular convergence, has its measure greater than zero. Remove
from E, those points at which one or more of the functions
<41 ( ) (X, ),..  ( )...
is discontinuous, and also remove all those points at which O(x) is discontinuous. We have then left a set Fe, of measure equal to that of E,,
and therefore by- hypothesis greater than zero.  At every point of Fe all the
functions fn (x) are definite and continuous, and + (x) is also definite and
continuous. If f be a point of Fe, the number n(> n1) can be so chosen
that
also 8 can be so chosen that, for every x in the interval ( -, + $), the
four inequalities
I  (~)-  (*) I < 36
[ On(E>  n(X) i < 36
are all satisfied. From these inequalities we deduce that the four inequalities
Ib (x)-  n(x) 1< e are all satisfied for all points x in the interval
(~S-, ~+s).
But this is contrary to the hypothesis that ~ is a point belonging to E,.
It therefore follows that, on the assumption that f(x, y) has an improper
integral in the fundamental rectangle, the repeated integral
rb   rd
dx ff(x, y)dy
cannot exist unless E, has the measure zero. Since this holds for every e,
we have obtained the following theorem:If f(x, y) have an improper (absolutely convergent) integral in the
fundamental rectangle, a necessary condition for the existence of the repeated
rb   rd
integral  dx   f (x, y) dy is that the convergence of
fn(x, y) dy to   f (x, y) dy
e               V e' c




400, 401]


Repeated improper integrals


571


should be regular, exceptfor a set of points E, of the first category, and of zero
measure.
401. Let it now be assumed that f(x, y) _ 0, throughout the fundamental
rectangle. It will be shewn that, in this case, the condition of regular convergence of [{n(x)} to 0 (x), at all points except a set of the first category,
rb   Cd
and of zero measure, is sufficient to ensure that  d  / f(x, y)dy exists, and
that it is equal to f (x, y) (dx dy); it being assumed that the double integral
exists.
In this case the four inequalities i (x)- bn(x) I < e, are equivalent to
the one p (x) - bn(X) < e; also if, at any point x, this is satisfied for a value
of n, then it is also satisfied for all greater values of n. Including all the
points of E, in the interior of intervals of a finite set, such that the sum of
these intervals is the arbitrarily small number s7, we see that the condition
b (x) - On(x) < 6 is satisfied for one and the same value of n (> n1) at all
points x not interior to the intervals whose sum is q. For we have only to
take for n the greatest of the numbers n1 + p, ln + 2,... ni +p. defined
in ~ 399.  The number e being fixed, we can choose 7 so small that the
double integral ff(x, y)(dxdy) over those rectangles of which the height
is d - c, and the sum of the breadths 7, is less than an arbitrarily fixed
positive number g; this follows from de la Vallee-Poussin's definition of an
improper integral. For m may be chosen so great that
(xn, y) (dx dy) -  fin (x, y) (dxdy) < -2the
when the integrals are taken over the fundamental rectangle, and therefore
also when they are both taken over any prescribed part of that rectangle; now
s can be so chosen that     fm (x, y) (dx dy) <, and therefore so that
f(x, y)(dxdy) <, the integrals being taken over the rectangles of total
breadth j. The number V being thus fixed, a number m exists, such that for
n - m, we have + (x)- 6n(x)< e, except in the interiors of the intervals
which enclose E,. We have therefore
(x) dx -   (n (x) dx < e (b-a - -) < e (b - a),
* This theorem was established by de la Vall6e-Poussin only for the case in which f (x, y) O,
and only under a restrictive hypothesis, that f f(x, y) dy and  (x, y) dy both have definite
finite values at all points x not belonging to a set of points of zero content; whereas this set need
only have zero measure.




572               Functions defined by sequences            [OH. VI
the integration being taken along the parts of (a, b) which remain when the
enclosing intervals are removed. Hence we have
fi (x) dx -f fn(x, y) (dx dy)< e (b-a),
where the double integral is taken over the fundamental rectangle with the
exception of those rectangles of which the breadths are the enclosing
intervals of sum q. Also, if 4' be an arbitrarily chosen positive number, we
can choose n so great that
ff(x, y) (dxdy) - ffn(x, y) (dx dy) < <,
where both double integrals are taken over the same region as before. We
now see that
((x)dx -     j(x, y)(dxdy) <   +e (b-a),
and hence we have
ff(x, y) (dx dy)-f (x) dx <  + +    e (b-a),
where the double integral is now taken over the whole fundamental rectangle,
and the single integral over (a, b) with the exception of those parts whose
sum is -. Now 4' is arbitrarily small, and 4, V converge together to zero.
(b
It follows that  b(x) dx, whether definite or not, lies between
f/(x, y)(dxdy) + (b- a);
rb
and since e is arbitrarily small, it follows that  ( (x) dx exists as a
definite proper or improper integral, and is equal to
If (, y) (d dy).
The following theorem has now been established:If f (x, y) _ 0, and the function have an improper double integral in the
rd
fundamental rectangle, then the condition that  fn(x, y) dy converges reguc, d
larly to  f (x, y) dy, except for a set of points E, of the first category, and
rb   rd
of zero measure, is a sufficient condition that  dx f (x, y)dy exists, and
is equal to f (x, y)(dxdy).




401, 402]


Repeated improper integrals


573


The sufficiency of the same condition, for the case in which f(x, y) is not
restricted to have one sign only, does not appear to be capable of establishment, because it is in this case impossible to shew that the conditions
are satisfied at all points except in the enclosing intervals, for one and the
same value of n; it having been only established that they hold when n has
one of a finite number of values.
Combining the present result with that of ~ 400, we see that:If f(x, y) - 0, and have an absolutely convergent improper integral in the
fundamental rectangle, the necessary and sufficient condition that
rb   rd
dx  ff(x, y) dy
should exist, and be equal to ff(x, y)(dxdy), is that  fn(x, y)dy should
converge regularly to  f (x, y) dy, except for a set E of the first category
and of zero measure.
It has also been established that, when Jf(x, y)(dx dy) exists, then if
rb   rC
dx    f (x, y) dy have a definite meaning, it is equal to the double integral.
a    c
For it has been shewn in ~ 400, that the repeated integral cannot have
a definite meaning, 4 (x) being integrable in (a, b), unless the convergence
is of the kind specified; and when f(x, y) > 0, this is sufficient that the
repeated integral may be equal to the double integral.
402. Returning to the case in which f(x, y) is not restricted to be of
one sign, the following theorem will be established:If f(x, y) have an absolutely convergent improper integral in the fundarb   rI
mental rectangle, a sufficient condition that  dx  f (x, y) dy may exist, and
may have the same value as the double integral ff(x, y) (dx dy), is that
\fn (x y)I dy
rd
shall converge regularly to  f f(x, y) I dy, except for a set of points of the
first category and of zero measure.
Employing f(x, y) =f+(x, y) -f-(x, y), fn(x, y) =f?~(x, y) -f-(x, y), and
d              rd                             respectiv
denoting +fn+, y) dy,    fn-(Ox, y)dy by if (x), On-(x) respectively, we




574


Functions defined by sequences


[CH. VI


see that the condition stated in the theorem   is that f,,+(x) +- ^,(x)
converges regularly to 6+(x) + (-(x).  In order that this condition may
be satisfied we must have ++(x) + f-(x) -,+(x)-,n-(x) < e, for a sufficiently
great value of n, at every point of (a, b) not interior to a finite set of intervals
of arbitrarily small sum v enclosing the points of E,, a set of zero content.
From this condition we deduce that
(+(x) - On(x) < e, and )-(x) -  n,-(x) < e,
at every point not in the interior of the intervals, for all sufficiently great
values of n.
Hence it follows that qn+ (x) converges regularly to ++(x), and also ~,-(x)
converges regularly to +-(x), at all points except those of a set E, of zero
measure. It follows fiom  the theorem of ~ 401, that the two repeated
integrals
rb   rd             rb   rd
fdx     f +(x, y)dy, f dxf f (x, y)dy
exist, and are equal to
f +(x, y) (dx dy), f-(x, y)(dx dy)
respectively; and from this it follows that
rb   rI                       in
dx    f(x, y) dy exists, and = f(x, y)(dxdy).
The condition stated in the theorem, though sufficient, is not necessary;
rb
for the integral f {++(x) -?-(x)} dx may exist only as a non-absolutely
convergent improper integral, in which case
f {+(x) + +-(x)} dx
does not exist.  In this case
rb   Ed
Idx  If(x,y) dy
not being existent, the convergence of
rd                rd
I fit(X, y) dy to   If(x, y) dy
cannot be regular.
403. Whether the double integral ff(x, y) (dxdy) exist or not, the
proof of the theorem in ~ 400, suffices to shew that, if all the double integrals




402, 4.03]


Repeated improper integrals


575


nf,,(x, y) (dxdy) exist, then it is a necessary condition for the existence
rb   r(
of the repeated integral f dx ff(x, y) dy, that the integrals
fjfl(x, y) dy
c
rd
should converge regularly to  f(x, y) dy, except for a set of points of the
first category and of zero measure.
Moreover, if it be known that  f(x, y) dy is a function of x, which
is limited in the interval (a, b), we can infer the existence of the double
integral f (x, y) (dx dy). For since
r                 rb  [d
ff(x, y) (dx dy)=   dx f;(x, y)dy,
we have
jf(x, y) (dxdy) < (b - a) U,
d 
where U is the upper limit of  f fn(x, y) dy in the interval (a, b). It is
thus seen that ff,(x, y)(dx dy) cannot increase indefinitely in numerical
value, as n is increased indefinitely. The following theorem has thus been
established:If all the functions f,(x, y) have double integrals in the fundamental
rd                      rd
rectangle, and  f, (x, y)dy converges to  f (x, y) dy regularly for all
values of x, except for a set of zero measure and of the first category, then,
rd
if   f f(, y) dy be a limited function of x in the interval (a, b), the double
integral ff(x, y)(dxdy) exists, and is equal to
rb   rd
dx ff(x, y) dy.
Combining this theorem with that of ~ 402, we obtain the following
theorem:If all the functions fn(x, y) have double integrals in the fundamental
rectagle,  either,   dy is liite   n te intervl (a, b) of, or
rectangle, and either l f(, y)dy is limited in the interval (a, b) of x, or
Jc




576


Functions defined by sequences


[CH. VI


Ib
f(x, y) dx is limited in the interval (c, d) of y, then if the conditions are
satisfied, that the sequences
rd              rb
I f (x, y) I dy,!f (x, y) | dx
converge to the limits
rd             rb
1 f(x, y) dy,  I f(x, y) dx,
in each case regularly, except for a set of points of zero measure and of the
first category, then the double integral exists, and
r                rb   rd           rd   rb
f(, y) (dx dy) =  dx   f (x, y)d dy  dy  f (x, y) dx.
Further investigations have been given by de la Vallee-Poussin of
sufficient conditions for the equality of the two repeated integrals, without
it being assumed that the double integral exists. In these investigations
somewhat complicated restrictions are made as to the mode of distribution
of the points of infinite discontinuity of the function. Moreover other
assumptions are implicitly made as to the definiteness of the single integral
rd                                 rb
/(x, y) dy for all values of x, and of j f(x, y) dx for all values of y.
J c                               -) a
REPEATED LEBESGUE INTEGRALS.
404. The subject of repeated integration will now be considered for
the case of a function which possesses a Lebesgue double integral. Some
further definitions will be first required.
The function f(x, y) being a summable function, defined as in ~ 287,
for the points of the rectangle bounded by x = a, x = b, y = c, y = d, it is to
be observed that we have no assurance that the function f(xo, y), defined for
all values of y on the straight line x = x0, is a summable function of y; unless
indeed we assume that all functions are summable.  For it is not certain
that the section E (x), of a measurable two-dimensional set of points E,
by the straight line x =x0, is linearly measurable. It will be convenient to
denote the linear measure of a set e of points on a straight line by m (e), in
order to distinguish linear measure from the plane measure m (E) of a set in
the plane.
Let F(x) be a limited function defined for the points of a measurable
set e of points in the x-axis.  If F(x) be a summable function, then the
Lebesgue integral of F(x) taken over the set e exists, in accordance with the
definition given in ~ 287.  For the case in which F(x) is not summable,
Lebesgue has defined upper and lower integrals in e, which are however




403-405]


Repeated Lebesgue integrals


577


quite distinct conceptions from the upper and lower integrals as defined in
~ 252, in connection with integration as defined by Riemann.  For any summable function, defined for the interval (a, b), the upper and lower integrals
defined by Lebesgue and denoted by
F(x) dx,     F (x) dx,
sup         inf
have one and the same value, whereas the upper and lower integrals
j F(x)dx, J     F(x)dx,
in accordance with the definition of ~ 252, have different values, unless F(x)
possesses a Riemann integral in the interval (a, b).
If )b (x) denote any limited slemmable function defined for the set e, for
which F(x) is defined, and such that q (x) > F (x) for every point of e, then
the lower limit of the Lebesgue integral f   q (x) dx, for all possible functions
b (x) which satisfy the prescribed condition, is defined to be the upper Lebesgue
integral of F(x) taken over e, and is denoted by
f F(x) dx.
sup
The lower Lebesgue integral F (x) dx is the upper limit of the integrals
inf
of all limited summable functions b (x), such that F(x) > f (x).
It has been shewn in ~ 83, that any linear measurable set e contains
a set el of the same measure as itself, and such that el consists of all those
points which are common to all sets of intervals belonging to a sequence of
such sets.  It has also been shewn that e is contained in a set e,, of measure
equal to ml (e), of the same type as the set e,.
Exactly similar reasoning to that in ~ 83 suffices to shew that a plane
set E contains a set E1, such that m (E) = m (E1); where E1 is that set of
points which is common to all sets of rectangles belonging to a sequence of
such sets of rectangles.  The successive sets of the sequence may be taken
each to contain the next; and in each set the rectangles do not overlap. The
set E is also contained in a set * E2, such that m (E2) = m (E), and such that
E2 is a set of the same type as E1.
405. Let the function qb (x, y) be defined to be = 1, for all points of the
measurable set E, and to be zero at all other points of the fundamental
rectangle. If E1, Ef be the sets of measures equal to that of E, E1 being
* The sets el, ea, El, E, are of the kind described by Lebesgue as "measurable (B)."
H.                                                          37




578


Functions defined by sequences


[CH. VI


contained in E, and EI containing E, as explained in ~ 404; then the sections
E1 (Xo), E2 (xo) of E, and E2, by the straight line x = xo, are both measurable.
It is clear that
rd
nzl [El (o0)]  0 <(xo, y) dy,
c
inf
m[ [E2 (Xo)]   0 f   (xo, y) dy,
c
sup
the integrals being taken over the interval (c, d) of y.
If DI, D,... Dt,... be a sequence of sets of rectangles, such that each
set contains the next, and if Dj denote a single rectangle of the set D,, we
may denote the section of Dy by the straight line x = xo, by Dj (xo); also we
may denote by D (x0) the section of the set D,. We have then
ml [D, (xo)] = E m1 [Dj (ox)].
j=1
Since X mn [DLA (xo)] is limited, for all values of j and x0, we may integrate
j=l
term by term; hence we have
b
m (DA) =     m1 [DLj (x)] dx
j=1 a
rb
=      mn [D, (x)] dx,
in virtue of the theorem of ~ 384.
Again, we have
mr (E) ==n (E) = lim m (D,)
L=00
= lim i  m [D, (x)] dx;
= 00 a
and since mn [D, (x)] is limited, for all values of t and x, we have
nm(E)=f ml [E1(x)] dxc,
if we take the sequence {D,j to be the one which defines E1.
It follows that
rb
m (E)  Mi, inm t [E (x)] dox,
inf
where ml, int denotes the interior measure of the set; and therefore
~~r          ruinb  rd
J     (x, y)(dxdy)  dxf    (x, y) dy.
inf  inf




405]


Repeated Lebesgue integrals


579


In a similar manner, it can be proved that
f                 rb   rd
(x, y) (dx dy)     dxf fd (x, y) dy.
sup  sup
It follows from the two relations, that
I  (x, y) (dxdy) =  dxf  (x, y) dy =  dx ~ (x, y) dy,
sup  sup          inf inf
where / (x, y)= 1, at the points of E, and = 0, at the points of C(E).
Denoting by U and L, the upper and lower limits off(x, y) in the
rectangle, let (L, U) be divided into parts (a0, a,), (a,, a2),... (a,_, an), where
ao = L, a,, = U; and let a denote the greatest of the differences ap - ap-, for
p=l, 2, 3,... n.  Let Ep be the set of points in the rectangle at which
f(x, y)= ap, and let Ep' denote that set for which ap <f(x, y)< ap+1. Also
let fp (x, y) denote a function which is equal to f(, y) at all points of the set
Ep, and is zero at all other points; and let fp'(x, y) denote a function which
is equal to f (x, y) at points of the set Ep', and is zero at all other points.
We have then
r               n r                n-i r
j/(   y) (dx dy)=  \ ff (x, y) (x dy) +  f'(x, y) (dxdyJ).
p=                  p==O
Now      fp(x, y) (dxdy) = apm (Ep) =  dx  fp(x, y) dy,
inf  inf
by the theorem proved above.
Also fp'(x, y)(dxdy) is between apin (Ep') and ap+lm (Ep'), or between
rb   rd               rb   f C
dx  q (x, y) dy and f dx c  f (x, y) dy;
inf  inf              inf  inf
where c (x, y)= ap, fr(x, y) = ap+, at the points of E', and where both
functions vanish at all other points.  These two repeated integrals differ
rb   rd
from one another by less than am(Ep'); also J dx     fp'(x, y) dy differs
inf  inf
from either by less than amn(Ep'). Hence we have
fp (x, y) (dx dy) -  f dx  fp'(x, y) dy  < am(Ep').
inf  inf
It follows that
r            ff rkn b  d               d   fd
f (x, y) (dxdy)- E    dx a  fp(x, y) dy -   dx     J p(x, y) dy,
A               p=0 a    c            p=O a    c
inf  inf              inf  inf
is numerically less than aA. We next see that
f f(x, y) dy >    fp (x, y) dy     fp'(x, y) dy.
Jc          =o P c           P=o c
inf             inf              inf
37-2




580


Functions defined by sequences


[CIo. VI


For, if possible, let
f f(, y) dy =  I  fP(', y) dy +    fp'(x, y) dy - 
c        p=oJ c           3=0 c
inf             inf              inf
where g is some positive number; then summable functions
fp (.. y).  fp (, y)
such that; (    > y) >fp       (. y)  fp(  y) >fP (y )
can be so determined that
ff   (, y) dy - f (, y) dy < ~/(2n + 1)
inf
with a similar inequality for the case of the functions
X/ (. Y).   fp (X, Y);
we have then
(x, ) dy <      2fL (X, Y) + E j(, ( y) dy,
^~c      ^   2c =o       ~=0
inf
whereas f(x, y) is greater than the summable integrand in the integral
on the right-hand side.  This is contrary to the definition of the lower
integral, and therefore the positive number ' cannot exist. A repetition of
the same reasoning suffices to shew  that
rb   rd            n  rb   rd-1 b                 rd
Jdx  f (x, y) dy _     dx j fp(X, y)dy +E     dx   fp'(x, y) dy.
Jo,  J( c         'p^O J   )cp=0         2=  J a 
inrf  inf            inf  inf                inf
We see now that
f     (x, y)(dx dy) <  dx f (x, y) dy + tA;
inf  inf
and as a is arbitrarily small, this shews that
A   (, y (dy)ddy)_   dx  f(x, y)dy.
inf  inf
It may, in a similar manner, be proved that
ff(x, y)(addy)    dxd  f(x, y) dy.
sup  sup
From these relations we deduce the theoremt
r                'b   rd            rb   rd
f f(  y)(dxdy)=j dxj    f(x, y)dy     dx   f(x, y) dy,
sup  sup           inf  inf
where f(x, y) is any limited function, summable in the plane.
* Lebesgue's statement (loc. cit., p. 279) is the reverse of the inequality here given. It would
appear that this is due to accident, as his result does not follow from the inequality he gives.
+ Lebesgue, Annali di Mat. ser. 3, vol. vII, p. 278.




405, 406]


Repeated Lebesgue integrals


581


406.  From   the theorem   just established, and considering the corresponding repeated integrals taken first with respect to x and then with
respect to y, we have the following theorem:f f (x, y) be any limited sunmmable fuction, defined in the rectangle A,
then
r                  rb   rd             rd    rb
f (, y) (dxdy) =    dx   f (x, y) dy=    dy   f (x, y) dx,
whenever the repeated integrals have definite meanings, in accordance with
Lebesgue's definition of integration.  It may happen that only one of the
repeated integrals has a ceaning.
In some cases the function f(x, y) may be such that both the repeated
integrals exist in accordance with Riemann's definition of integration. In
case the function have a double integral in accordance with the extended
Riemann definition for a function of two variables, then the theorem reduces
to the one established in ~ 314. It may however happen that f(x, y) has
a double integral only in accordance with Lebesgue's definition.  This throws
light upon the questions considered in ~ 316.    We may, in fact, state the
following theorems:If the repeated integrals of a limited sumnmable function f(x, y) both
exist, in accordance with Riemann's definition, then they are equal to one
another, and to the Lebesgue double integral of f(x, y).
If only one of the repeated integrals of a limited summable function exist,
in accordance with Riemann's definition, then the other may exist in accordance
with Lebesgue's definition, and they are then equal to one another, and to the
Lebesgue double integral of the function.
The only possible case in which both repeated integrals can exist and have
unequal values is when the function is not summable in the plane.
EXAMPLES.
1. For the function defined in ~ 316, Ex. 1, only one of the repeated integrals exists,
in accordance with the definition there employed; neither does the double integral exist.
The Lebesgue double integral exists, and =1.  For the set of points at which f(x, y)=1
has the measure zero; and therefore the function has the same Lebesgue integral as that
function which, at every point (x, y), has the value 2y. The functional values at a set of
points of zero measure are irrelevant in a Lebesgue integral. The other repeated integral
also exists, in accordance with the definition of Lebesgue, as may be easily verified.
2. For the function defined in ~316, Ex. 4, both the repeated integrals exist, in
accordance with the definition there employed, and they have the value c; the double
integral, however, does not exist.  But the Lebesgue double integral exists, and =c; for
the points at which f(x, y) = c' although they are everywhere-dense, form a set of plane
measure zero.




582


8Functions defined by sequences


[CH. VI


407. When f(x, y) is unlimited in the rectangle for which it is defined,
U may have the improper value oo, and L may have the improper value - oo.
No essential change is required in the reasoning of ~ 405, which suffices to
shew that
r               rb   rd            rb   rd
f(x, y) (dxdy)=   dxf f(x, y)gdy=   dxf f(x, y) dy,
inf  inf          sup  sup
whenever the integrals have a meaning. The result
r                rb  rd            rd   rb
f(x, y) (dxdy)=   dx   f (x, y) dy=  dy  f (x, y) dx,
also holds whenever the integrals have a meaning.
rf
The integration of  f(x, y) dx with respect to y between limits (c, yi)
is frequently performed by reversing the order of the integrations; thus the
value of J dy   f(x, y) dx is frequently calculated by means of
f dx    f(x, y)dy.
This process is frequently spoken of as "integrating under the integral sign
with respect to a parameter y."  What has been just established enables us
to state the following theorem:Integration with respect to a parameter, under the integral sign, is always
valid, provided the function is summable and integrable in the plane, if the
result of the process have a definite meaning.
It must, however, be remembered that even if |  dy  f (x, y) dx exist,
in accordance with Riemann's definition or one of its extensions,
dx d  f (x, y) dy
may exist, if it exists at all, only when Lebesgue's definition is employed.
The only case in which the repeated integrals of an unlimited function
can both exist, but have unequal values, is when the function is either not
summable in the plane, or is summable and still does not possess a Lebesgue
integral.
REPEATED INTEGRALS OF UNLIMITED FUNCTIONS.
408. In order to find sufficient conditions for the existence and equality
of the repeated integrals of an unlimited function, which shall not depend
upon the necessary existence of the double integral, in accordance with either
the definitions of Jordan and de la Vallee-Poussin, or with that of Lebesgue,
we recur to the wider definition of a double integral which has been given in




407, 408] Repeated integrals of unlimited functions


583


~ 320.  That definition differs from Jordan's definition in ~ 318, in that
rectangles only are employed for the purpose of enclosing the points of
infinite discontinuity; and thus the domains D,O in Jordan's definition are to
be restricted to consist each of a finite set of rectangles with sides parallel to
the axes. An improper integral which exists in accordance with the definition
of ~ 320, we shall speak of as a restricted Jordan double integral.
Assuming that the integral f f (x, y)(dx dy) exists, as a restricted
Jordan double integral, letff (x, y) be that limited function which, in the
domain D,, consisting of a finite set of rectangles, is equal to f(x, y), and is
zero in C (D,), which contains all the points of infinite discontinuity of
f(, y).
We have then
f (,(   )(ddy)=lim           y(d(ddy), 
f A f~xn=oo A
rb   rd
= lim   dx   fn(x, y) dy,
and therefore we have
f    f(x, y)(dxdy)=  dx  f(x, y) dy,
provided           limf dxo      f(x, y)dy = 0,
n== o Ja J A~n (z)
where A,,(x) denotes that finite set of intervals which forms the section
of C (Dn) by the ordinate corresponding to the abscissa x.
From this result, the following theorem, very similar to one given by
Jordan*, and specifying a particular mode of satisfying the last condition,
may be deduced:For the existence and equality of the two repeated integrals
rb   rd             rd   rb
fdxf f(x, y)dy, y)dy  dy     fx y) dx,
it is sufficient
(1)  That the function f(x, y) possess a restricted Jordan double integral
in the fundamental rectangle.
(2) That the points of infinite discontinuity of f(x, y) be distributed on
a limited number of arcs of continuous curves representing monotone
functions.
(3)  That, corresponding to any fixed positive number e, positive numbers
hi, k1 exist, such that
ee Cors 'Anase, vol., p).   y <7,
* See Cours d'Anaalyse, vol. n, p. 67.




584


Functions defined by sequences


[CH. VI


for h I < h, I k I < k, and for every value of x and y in the fundamental
rectangle.
To shew that, under the conditions stated,
lim   dx      f(x, y)dy=O,
n=oo  a c,( x)
it is clear that the points of any one such curve can be enclosed in the
interiors of a finite number of rectangles, the height of each of which is
< ki.  Then A,(x) consists of a number of intervals not exceeding the
number r of the curves on which the points of infinite discontinuity lie.
We have then         f /(x, y) dy  <re; and therefore
dx       f (x, y) dy
is less than the arbitrarily small number re(b -a). Thus
rb   rd
dxf f(x, y)dy
exists, and is equal to the restricted Jordan integral.  Similarly it can be
shewn that the other repeated integral has the same value.
EXAMPLES.
1. Let f (x, xy)=  2 - Y and let the domain be bounded by x=0, x= 1, y=, y=1.
Neither the double integral, nor the restricted Jordan integral, exists in this case. For, if
the rectangle bounded by x =0, x= a, )y =0, y=/ be excluded from the domain, the double
integral over the remainder is equal to
1          yf C_ _ 2  i1   I9   2 -y 2
J    J (X2+Y2)2      aJ dxj 0v4
or to               f (I 1    x+)          x + 
which is r}r-tan-l-; and this has no definite limit as a, /3 converge independently
to zero.
The repeated integrals exist, and have unequal values; for
n    n   2_ X2 -n 1
Jo   Jo     2 (2+y2)2  Jo X2+1
Jdy J (2+y2)2 d   — J         = kT
2. It has been shewn in the Example, ~ 320, that the double integral of - sin - over
-C  -ove




408]          Repeated     integrals of unlimited functions                  585
the domain bounded by x=0, x x=a, y=0, y=b, does not exist. In this case however
the restricted Jordan double integral exists.  For
b      1    1 a I1          1
f dy    I - sin - dx = b  - sin - dx
and this has a definite limit, for E=0.
The two repeated integrals exist, and are equal to the restricted double integral. The
condition
1
lim   dy.            xf- sin - dx
J~   J  (x)   $x
of ~ 408, is satisfied, for Al (x) is in this case independent of x, and consists of an interval
(0, E).
3. Let f/(, y)=( - - y, and let the domain be bounded by x=a, x=b, y=0, y=c,
where c>a. In this case the double integral exists.
For f (v -          y)-   dy =3x- +3 (c -  )  - 3 lim; - 3 lim E'a, and this is 3xs+ 3 (e-x);
JpO,,~~                 e=0     e'O=
therefore  f ( - y)-  6 dy has a finite upper limit for all values of x in (a, b). It then
follows from the theorem of ~ 403, that the double integral exists. Also since
b    c   (XY) -   2dy
has a definite meaning, its value is the same as that of the double integral. Clearly the
other repeated integral exists, and has the same value as the double integral.
4. Let the function* +(x) be defined for the domain bounded by
=O0, x=1, y=O, y=1,
by the rule that, for every rational value of x of the form
2m + 1                1 ), (  I
and that, for every other value of x, + (x)= 0.
Let f/(x, /)=  1 sin1 -  (,.v), then the improper integral
- 51 i-   r () (dxC
j   /   y I1 )*
exists as a Jordan double integral, and has the value zero. The integral f|  x (x) dx exists
as a Riemann integral, and has the value zero. The repeated integral
I df x f #(x) -sin- dy
does not exist in accordance with Harnack's definition, for
I  +(X)    sill - dy
2m+ I
diverges for the everywhere-dense set of values x=-2-, and therefore the repeated
* Stolz, Grundziige, vol. In, p. 149. The function 4b (x) was first given by Du Bois Reymond,
Crelle's Journal, vol. xcvi, p. 278.




586


Functions defined by sequences


[CH. VI


integral does not exist. It does exist, however, in accordance with Lebesgue's definition,
since the points of divergence of the single integral form a measurable set of measure
zero; and its value is zero, the same as that of the double integral. The other repeated
integral exists, and is zero.
5. Let f(x, y)=0*, at all points in the rectangle bounded byx=0, =1, X =, y =0,  l;
except that at all points x= -2m+1 y <-, it is to have the improper value + oo. The
function f, (x, y) will be given by the condition that it equals NO,, at the points where
f/(, y)= o, and is elsewhere zero.  It can be shewn that
n (X y) (d dy)= 0,
for every value of N,,; and thus Iff(, y)(dxdy) exists, and =0. The condition
f  f(x, y) dy-   fn(x, y) dy<c
is not satisfied for any of the everywhere-dense set of values x = (2m + 1)/2n, if the ordinary
definitions of Harnack and Riemann are employed; and therefore the convergence of
f, (, y) dy to ff(x, y) dy is not regular.  The repeated integral
dcx l     f(, y)dy
is equivalent to jdx lim |fl (x, y) dy  Also, (x, y) cdy is zero, unless
x=(2nm+ 1)/28,
in which case it is Nn/2s, and for such values of x, lim f/2 (x, y) dy is c; and thus the
repeated integral does not exist, in accordance with Harnack's definition.  It does,
however, exist in accordance with Lebesgue's definition, and is equal to zero, the value of
the double integral.
REPEATED INTEGRALS OVER AN INFINITE DOMAIN.
409.   Let the function f(x, y) be defined for all points of the infinite
domain of which the boundaries are the three straight lines x= a, x=b,
y= c; the domain being unbounded in the positive direction of y.           Conditions will be investigated that the repeated integrals
rb   ro              ro     rb
dxf      f (, y) dy, j   dy     f(xc,y) dx,
may exist, and may be equal to one another.
* This example is given by Schinflies, Bericht, pp. 201,202, to illustrate his erroneous theorem,
applicable to Jordan and Harnack integrals, that the improper double integral is always equal to
each of the repeated integrals, and the converse; and that thus the condition of the theorem
of ~ 400 is always satisfied. The example does not bear out his contention. See Hobson, Proc.
Lond. Math. Soc., ser. 2, vol. Iv, pp. 157-159.




408, 409]   Repeated integrals over an infinite domain             587
Let it be assumed that the two repeated integrals
rb   rY             rY    rb
J dx If(, y) dy,     fdy ff(x, y) dx
exist, and are equal, for every definite value of Y ' c.
If   dy f(x, y) dx exist as a definite number, then it is equal to
pb   tY
limr   dx   f(x, y)dy;
Y= oo a     c
and this is equal to
f dx    f (x,?) dy,
provided that
rb   r~
lim   dxf   f(x, y) dy = 0.
Y=J a     Y
This last condition is equivalent to the condition that, corresponding to
an arbitrarily chosen positive e, a value Ye of Y can be so chosen that
Ifdxff(x, y)dy < e, for Y> Y.
We can now state the following result:On the supposition that the two repeated integrals of f(x, y) both exist, and
have equal values in the domain bounded by x = a, x = b, y = c, y = Y, for every
value of Y > c, the necessary and sufficient conditions that the repeated integrals of f(x, y) in the unbounded domain x = a, x = b, y = c, y = co, shall
have equal values are
(1) that j dy f f(x, y) dx shall have a definite value; and
(2) that, corresponding to an arbitrarily chosen c, a value Ye oJ Y can be
fixed, so that  dx    f (, y)dy  <e, for all values of Y> Ye.
It may be observed that the condition (2) does not make it necessary that
f (x, y)dy should have a definite value for all values of x.
Y
It is a sufficient condition in order that (2) may be satisfied, that
f(, y) dy   < e/(b - a),
for every value of x in (a, b), with the possible exception of a set of points
of zero measure.
In this exceptional case, however, f dx  f(x, y)dy, must exist as an
absolutely convergent integral with respect to x.




588


Functions defined by sequences


[cH. VI


410. Next, let us assume the function f(x, y), to be defined for the
infinite domain bounded by x= a, y=c, and unbounded in the positive
directions of x and y.
Let it be assumed that the two repeated integrals
rX   rY          rY   rX
fdx   f (x, y)dy,  dy   f(x, y)dx
exist, and have the same value, for each set of definite values of X and Y,
such that X? a, Y  b. The value of these repeated integrals we denote
by  (X, Y).
We have now
lim  (X, Y)=    dyl f(x, y) dx = lim   dx   f(x, y)dy;
Y'-o         J c     a          Y — o a  J c
it being assumed that this limit has a definite value for each definite value
of X. If now
X   rY            rX       rY
lirm  dx   f(x, y) dy =  dx lim  f (x, y) dy,
Y== so Je a  c        J a  Y= oo c
we then have
lim p (X, Y)=   dx    f (x, y) dy.
Y= GO          a 
The condition that this may be the case is that
rX   rC
lim   dx    f (x, y)dy =0,
Y=oo a    Y
which is equivalent to the condition that, corresponding to an arbitrarily
chosen e, there should exist, for each value of X, a value Y, of Y, such that
Xdx   f (x, y)dy <,
for every value of Y> Ye. When this condition is satisfied, we have
lim lim  b (X, Y)= dfx   f (x, y) dy,
X=o Y= oo         J   J
provided the repeated limit on the left-hand side has a definite value.
Under similar conditions we can shew, in the same manner, that
lim lim b(X, Y)-    dyf (x, y)dx.
= oo X= Oo            J0
The necessary and sufficient condition for the existence and equality of
the repeated limits lim lim r (X, Y), lim lim  ( (X, Y) is obtained from the
X== o Y=o        Y=a X= oo
theorem of ~ 234, by letting X = 1/x, Y= l/y, where x, y are the pair of
variables there employed. Using this condition, we have now established the
following theorem:



410]


Repeated integrals over an infinite domain


589


It being assumed* that the repeated integrals of f(x, y) in a domain
bounded by x = a, x = X, y = c, y = Y, exist, and are equal to one another, for
every pair of definite values of X and Y such that X > a, Y T c, it is sufficient
jor the existence and equality of the two repeated integrals
dx    f(x, y) dy,   dy   f(x, y) dx,
a     c              o   J a
that the following conditions be satisfied:(1)  That f dy    f(x, ) dx,          f (x, y) dy, have definite values
for all definite values of X and Y.
(2)  That, corresponding to an arbitrary e, there should exist, for each
value of X, a value Ye of Y, such that
dx     f (x       <, y) dy <, for every Y > Y;
and also for each value of Y, a value X, of X, such that
dy dy f(x, y) dx < e, for every X > Xe.
(3) That if e be fixed, a number Y > c ccan be determined, such that, for
each value of Y> Yo, a value of X, say Xe, can be found, which is such that,
for the particular value of Y,
dx ff(x, y) dy< e,
for every value of X > X.   This condition may be replaced by the corresponding one in which the integral
dy      f(x, y) dx is employed.
More general conditions could be obtained by not assuming that
lim  b (X, Y),  lim b (X, Y)
Y=oo             X=oO
have necessarily definite meanings, and then applying, instead of (3), the
theorem of ~ 233.
In accordance with ~ 233, if, in condition (3), when X, is determined for
a particular value of Y, the condition be satisfied, not only for that particular
value of Y, but also for all greater values, then the double limit
lim      dx   f(x, y) dy
X=oo, Y=o J a  J c
also exists, and is equal to the repeated integrals with infinite limits.
* Conditions nearly equivalent to these were given by Bromwich, Proc. Lond. Mlath. Soc.,
ser. 2, vol. i, p. 188.




590


Functions defined by sequences


[CH. VI


It has been proposed by Hardy* to employ this double limit as a definition
of the double integral of f(x, y) over the infinite domain. This definition is,
of course, less stringent than the one given in ~ 323.
411. Sufficient conditions of greater stringency, and therefore of less
wide application, can be deduced from the theorem of ~ 410.
The integral   ff(x, y) dx, is said to converge uniformly in the interval
a
(c, Y), if, having given the positive number 7, chosen arbitrarily, it be always
possible to determine X0 so that
f| /(x, y) dx   < v,
for X > Xo, and for every value of y in the interval (c, Y).
If this condition be satisfied for each interval (c, Y) of y, the convergence
is said to be uniform in an arbitrary interval.
If the condition be satisfied for every value of Y > c, the convergence is
said to be uniform in an unlimited interval.
If the condition be satisfied for all points in (c, Y) except those which
belong to a set of points of zero measure, the convergence is said to be
uniform in general in the interval.
The following theoremt will be established:It is sufficient for the existence and equality of the repeated integrals with
infinite limits, (1) that the repeated integrals between finite limits always exist
and are eqtal, (2) that    f (x, y) dx be uniformly convergent in general in an
arbitrary interval, (3) that   f(x, y) dy   satisfy the same condition, and
t rha  fd x,   ) y                    y           ted
(4) that    dxj    f (x, y)dy converge uniformly in an utnlimited interval. The
condition of uniform   convergence in general must be replaced by that of
uniform  convergence, in case the repeated integrals between finite limits exist
only as non-absolutely convergent improper integrals.
To prove this theorem it will be sufficient to shew that if the conditions
given in it be all satisfied, then those in the theorem      of ~ 410 are all
satisfied.
Messenger of Math., vol. xxxII, p. 95.
t This theorem was given by de la Vallee-Poussin, Liouville's Journal, ser. 4, vol. vIII, p. 464,
except that in his definition of uniform convergence in general, the set of exceptional points
is there restricted to be a set of the first species; also the condition (2) is given as that of
uniform convergence.




410, 411] Repeated integrals over an infinite domain


591


It is clear that the condition (2) of ~ 410 is satisfied, if (2) and (3) of the
present theorem are satisfied. For, if j f (x, y) dy be uniformly convergent
in general in the interval (x, X), then for a fixed q, Y0 can be so fixed that
f (,, y) dy <m,
for Y > Y0, and for every value of x in (a, X) except the points of a set of
zero measure.  It then follows that
fdx     f (x, y)dy < j(X-a),
and we may choose    < e/2 (X- a); therefore
fdxf f(x, y)dy <e,
for Y   Y0; this is one part of condition (2) of ~ 410; similarly it may be
shewn that (2) of the present theorem is sufficient to satisfy the other part
of the condition (2) of the former theorem. Again, the condition (4) of the
present theorem  may be stated in the form that
I      +(X, Y)-lim +(X, Y) <e,
X= 00
for every value of Y, and for all values of X which are > a fixed value X0.
Since, on account of (2),
lim  (X, Y) =    dy fr(x, y) dx,
X=oo            c 0
we see that
dy f0 (x, y)dx < e,
for every Y, and for X _ X0, and therefore condition (3) of the former theorem
is satisfied. It has thus been shewn that, if the conditions of the theorem
are satisfied, then those of the theorem in ~ 410 are also satisfied.
EXAMPLES.
1. It will be found that  dx  (2       dy-)2 d=  7r and that
00 dy \      2 - d-.
1l   l (x2 +y2)2 d= 4
In this case the first condition of the theorem in ~ 410 is satisfied, and the second is also
satisfied. For
x      Y   -x2  2  { s   X                ' {  X}
rx   r y xJ2r-y2^^+^^^d --       1     tan    - tanv-1
Ydx   (x2 +y2)   t1uan 




592                   Functions defined        by  sequences               [cn. vi
and hence it is clear that, for a given e, Ye can be so determined that, for every value of
Y> Y,, the absolute value of the repeated integral is less than e. The third condition is,
however, not satisfied; for, since
1X  [    2_ 2             1,X
y d    (2 q+Y)c d~ =tan-l 1  - tan - 1Y'
it is impossible, when Y is fixed, so to choose Xe, that, for every X> X,, the absolute
value of the repeated integral shall be less than E.
2. Let*     ()=  Zp, where p>; then  '(z)=  2zP (1  zp)
The repeated integral f  dy f   'xydx=       but f    d      '(y)dy =-      The
Jo    J o                   Jo      o
second condition of the theorem in ~ 410 is here not satisfied. For, we find
dy r      (xy) dx = -tan -1 (cPP),
Jo   J,Y
and it is impossible to choose Xe so that, for all values of X> Xe, the absolute value
of this repeated integral shall be less than.E
3. We have j      dy f  cosxzxyx=-7r, for a>0; but    dr      cosxyldy does not
exist.
oo    rc          no    ' oo
4. It may be shewn that        dx a  e- Xy      dy e-   xYly  e dx, the conditions of
/o      o          o /o
the theorem in ~ 409 being satisfied.
5. Lett V=- +-) sin Trx sin 7ry.    In this case we find
h    r a27        [     rh a2      /I   1\ a2 V.k 
f dx f   a-   dy=    dy f  a%-  dx= (+ ) sin 7r/w sin wk.
Jir   ~   x aj y   J   I    ay        A    k
f~~   a'v82f V                                    2V
The repeated integral     dx          dy does not however exist; for         dy or
1 axay                                         y o
F V]~ has no definite value, for any value of x.  The double limit
axT    h fh    a2fV 
lim    I dx J      -y dy
h =.oo, k= ool  ~ —t
exists, and is equal to zero.
In this case, as stated in ~ 410, it is possible to employ this double limit as the
definition of the double integral.
6. Let V= (+ )(1 -) sin 7rx, and f(x, y)          - =a  We find, in this case,
f  dx J   f (x, y) dy=0,
but the other repeated integral does not exist, since  f (, y) dx has no definite value.
The double limit
lim      dx    f (x, y)dy=O.
7h=o, Ck=oo 


" See Stolz, Grzndziige, vol. III, pp. 8, 182, where the example is ascribed to Du Bois Reymond.
f Bromwich, Proc. Lond. Math. Soc., ser. 2, vol. i, p. 182.




411]         Repeated     integrals over an       infinite domain             593
a, V        Xt
7. Let f(x, y) =-, V=-l+~;. In this case, the two repeated integrals
da     f/ (, y) dy,  d:       f (, y) dx
Jo    fo              Jo    vo:
exist and are zero.
The conditions of the theorem of ~ 410 are here satisfied. We find that
I   Jo                 Io   JO
f|      f(x, y)    =o= Cf    x   fG(X, y) /d;
and thus the first condition is satisfied.  Again
rx  Y,             XJ           XY E
o dy   f(x, /) dy=1 tX2+     Y2 - +X2+ YE
and, for a given X, this is arbitrarily small for Y> Y,, provided Ye be properly chosen.
Since    dxf f(x y) dy= = + _T   +, it can be at once verified that the third condition is satisfied. It will be found that the conditions of the theorem in ~ 411 are not
satisfied in this case.
8. To prove that the order of integration of
fsinydy f    e-y2 d
may be reversed, the theorem of ~ 410 must be applied.
Let ( (t)=    e -   d$; then  (0)= -'/7r, and  (~) continually diminishes as $ increases. We have
fody /e-y~in=2soin y  =     sod in  y-)dy,
and this is easily seen to be less than
(o) /      0r sin yc
therefore the repeated integral exists.  Again
dx I   e   2sin ydy=    x+1   [ -e- Y(cos Y+2 siY)] dx,
and this is easily seen to be convergent; therefore condition (1) is satisfied.
Next, we have
e-     sin y dy =  4 4 [e- Y2 (cose +x2 sin Ye)-e- Yx2 (cosY+x2 sin Y)],
Je          s     i+n=
and this is less than 2 (1 +x2) - YZ, where Y>Ye. We then find that
dx       e- 2 sin ydy
is numerically less than
i    r (1+;
and it is then seen that condition (3), and the first part of (2) are satisfied.


H.


38




594


Functions defined by sequences


[CH. VI


Also
dy x e-Y siYdx         sin (Y  [ fs/  (XLy)-+ (X^,y)] dy;
since q ($)<-, we have
f sin y< Y                            rr
~   q b (X, N/y) dy < | sl  (X"N/y) dy < 4X;
Jo  V0 o V i
and it is now clear that the second part of (2) is satisfied.
THE LIMIT OF AN INTEGRAL CONTAINING A PARAMETER.
412. When an integral    f(x, y) dx, involving a parameter y, exists
for each value of y such that yo < y < y0 + h, it is of importance to know
under what circumstances the limit of the value of the integral as y
converges to y,, has a definite value which is obtained by integrating
through (a, b) the limit of f(x, y) for y = y,. Expressed symbolically, and
denoting lim f(x, y) by f(x, y, + 0), we require conditions under which
Y=Y0
rb              rb
lim   f(x, y)dx, and  f /(, y+ O) dx may both exist, and have one and
y=/o a                a
the same value.
In order that J f(x, y)dx may be continuous at y = y, on the right, it is
necessary that the further condition be satisfied that f(x, yo + 0) -f(x, y)
should be an integrable null-function in (a, b). In particular, if
f (, y) =f (, yo + 0),
for all values of x in (a, b), this condition is satisfied.
If y1, y2,...yn,... be a sequence of diminishing values of y which converges
to the limit y, we may write s,,(x) for f(x, yn), and s (x) for f(x, y +0),
and then the results of ~~ 383-386, may be applied to obtain sufficient
conditions that, for this sequence of values of y, the expressions
rb                   rb
(x yo + 0) dx, lim  f(, y) dx,
Ja              2/=Z0=o  a
may exist, and be equal, when the particular sequence of values of y is alone
considered.
Criteria of the required character will be obtained by adding the condition that the sufficient conditions so obtained are satisfied for every possible
sequence of diminishing values of y which converges to y,. It is clear that
the case in which h is negative, and f(x, yo- 0) takes the place of
f(x, yo + 0), is not essentially different from the case here considered.
It is clear from the result of ~ 383, that, in case f(x, y) converges to
f(x, Yo +0) uniformly for all values of x in the interval (a, b), sufficient




411, 412] Limit of anr integral containing a parameter  595


conditions for the equality in question are satisfied for every sequence of
values of y which converges to y0. We have therefore the following
theorem:rb
When the proper integral  f (x, y) dx exists, for each value of y such
rb
that yo < y < yo + h, it is a sufficient condition that  f (x, y, + 0) dx may
rb
exist and be equal to lim   f(x, y) dx, that f(x, y) should converge to
f(x, Yo + 0) uniformly for all values of x in the interval (a, b).
The condition of uniform convergence is that, corresponding to an
arbitrarily chosen positive number e, a number k can be determined, such
that!f(x, y + 8)-f(x, yo + 0) I < e, provided 0 < 8 < k, for every value of
x in (a, b).
We may obtain less stringent conditions than the one contained in the
above theorem by applying other results relating to the integration of series.
From the theorem of ~ 380, we obtain the following criterion for the
existence of  f (x, y0 + 0) dx:rb
If f f(x, y) dx be a proper integral, for each value of y such that
yo <y c- yo + h, and f (x, yo + 0) be limited in the interval (a, b), the necessary
rb
and sufficient condition that | f(x, y +0) dx should exist as a Riemann
Ja
integral is that f(x, y) may converge to f(x, Yo + 0) regularly in (a, b), except
for a set of points E, of zero measure, and of the first category.
The condition of regular convergence, except for the set E, is that,
if e be an arbitrarily chosen positive number, and y, an arbitrarily chosen
value of y such that y0 < y1 < y0 + h, then for every point xl which does not
belong to a certain non-dense closed component of E, dependent on e, a value
yx, of y(< y,) can be chosen, and also an interval (x -8, x, + ') can
be fixed, such that If(x, y,) -f(x, y0 + 0) < e, for every value of x belonging
to (a, b) in (x, - 8, xt + 8').
When the conditions stated in the above theorem for the existence of
fb
f (x, yo + 0O)dx are satisfied, we may apply the results of ~ 383 and ~ 385,
to obtain the following theorems:rb
If   f(x, y) dx be a proper integral, for each value of y such that
b
Yo < y  y0 +  I, and the proper integral  f(x, y0 + 0) dx also exist, it is a
38-2




596


Functions defined by sequences


[CH. VI


sutficient condition  that lim  f(x, y) dx should exist and be equal to
Y:=o J cc
I (x, y +0) dx, tha t f(x, y) I should  be less than  some fixed finite
number, for all values of x and y such that a ' x o b, and yo < y 5 yo + h.
rb
If   f(x, y) dx be a proper integral for each value of y such that
yo <y   y - + h, and the proper integral f f(x, yo + O) dx exist, it is saflicient
for the equality of
rb               rb
lim  ff(x, y) dx and   f (, yo + 0) dx,
r.      f
that (1) lim   f(x, y) dx should be a continuous function of x in the whole
interval (a, b), and that (2) the set of points (x, yo) at which the saltus of
f (x, y) considered as a fanction of (x, y), is indefinitely great, should
form  at most an enumerable set.
413. In the case in which the integrals   f(x, y)dx, for values of y
such that Yo < y c yo + h, are not necessarily proper integrals, but may, for
some or all such values of y, exist only as improper integrals, we may
apply the result of ~ 386, to obtain a set of sufficient conditions for the
rb                       rb
equality of f (x, y +  ) dx, and lim J   (x, y) dx.
The result is stated in the following theorem:If f(x, y) converges to a definite limit f (x, yo+ O) for all points x
of the interval (a, b) which do not belong to a reducible set of points G,.
and the functions f(x, y), for y < y < yo + h, although not necessarily limited
in (a, b), satisfy the conditions (1) that, in any interval (a, /3) contained
in (a, b) which contains in its interior and at its ends no point of G, If(x, y) I
is less than some fixed finite number, for all values of x and y such that
b
ax.~1,3 y0<y.yoQ+h; (2) that f f(x,y)dx exists at least as an improper integral, for each value of y such- that yo < y < y0 + h; and (3) that
lim  f (x, y) dx, for a - x - b is convergent and represents a continuous
function of x; and (4) j     f(x, yo+O0) dx exists at least as an improper
rb                    rb
integral; then the equality  f (x, y + 0) dx = lim  f (x, y)dx holds.
414. In the case in which the interval of integration is unlimited,
say b=, we may apply the theorems of ~ 387, to obtain sufficient




412-414]    Limit of an integral containing a parameter              597
conditions that    f (x, yo + 0) dx =  lin  /f(, y) dx. We obtain at once
J             ~a t~21/=Yo  ' a
the following criteria:If the equality  f f(x, yo + 0) dx= lim  f (x, y) dx be satisfied for every
value of C > a, then if, corresponding to an arbitrarily fixed e, a number C > a,
can be determined, and also a value y, of y, such that y0 < y, < yo + h, for which
f(x, y) dx < e, for every value of C' > C, and for every value of y such
th at Yo < y < yi, then  f (x, yo + 0) dx exists, and is equal to limrn  f(x, y) dx.
a                                    V1=90o a
If the equality   f(x, yo + 0) dx = limr  f (x, y) dx holds for every value
of C> a, then, if lim  f (x, y) dx exists, and also lim  f(x, y) dx converges
Y =0 a                           y=Y Jo 
to the value lim rf(x, y) dx, when C is indefinitely increased, these conditions?/=yo a
are sufilcient to ensure that ff (x, y, + 0) dx exists, and  is equal to
lim   f (X, y) dx.
Y=Yo
In order that  f (x, y)dx may be continuous on the right at y0, the
additional condition must be satisfied that f(x, yo + O) = f(x, yo); or more
generally, that f(x, yo +  ) -f(x, yo) should be an integrable null-function in
an arbitrary interval of x.
EXAMPLES.
1. If y>o, we haveh  | inYdx=   r but when s=O f sin x dx vanishes; and
JJo  x                       Jo   x
thus    S ill i dx is discontinuous, for y= +0. It may be seen that the conditions
contained in neither of the above theorems are satisfied.
The condition lim  sn y     dx =  lim si-  x x=0 is satisfied. But
qj=o J   x     Jo =o   x
C'snY         sin yx     2 
f — in   dxf    ld >1 — d> —Cy, if C'y=  r.
IC       = I         >C  7r          2
'c   x      Joy 0     -    r
However Yl and C may be fixed, C' and y (<yl) can always be so chosen that
|sin iXJ dx  >e; and thus the second condition is not satisfied.
2. The equality
2,               b
lim  f (x, y) q (.v) dx =  q (x)f (x, o +0) dx
holds if I, )  les thn some fied nmber, for a vals of    d   sch that
holds if If(x, y)I is less than some fixed number, for all values of x and y such that




598                  Functions defined       by sequences               [CH. vr
b
a _ x  b, O<y -yO+h; provided also f      P(x) dx is either a proper integral, or an
absolutely convergent improper integral with a reducible set of points of infinite discontinuity. This follows from the theorem in ~ 413. We may even suppose b =o;
then under the same conditions the equality holds. For
rel
f  (x) y) 0 (x) dx
rc'
is less than K I    (x) dx, where K is the upper limit of If(x,y), and therefore
for a sufficiently great value of C, we have
f / (X Y) Wx) dx <,
since C may be so chosen that
f(.x)d)(lx <EX
It follows that                 f (x, Y0 + 0) q (x) dx
a
oo
exists, and is equal to      lim fI (x, y)  (x) dx,
q J=Jo a
provided f(x, y) is less than I, and also provided f  (x) dx is absolutely convergent.
ja
rb
3. Consider    e - yx (x) dx, where a is finite, and b is either finite, or + co.
a
It follows from Ex. (2), that
rb              rb
lim   e -   (x) dx =   b (x) dx,
J=O J a             J c
provided   q (x) cdx is absolutely convergent, and  (x) has at most a reducible set of
a a
b
points of infinite discontinuity. The theorem however holds whenever f e -  (x) dx
has a definite value not only for y/=0, but also for all values of y such that 0  y h,
where h is some positive number. If + (x) denote the continuous function if   (x) dx,
Ja
we have
rb                          fb
e-x ) (X) dX = e-y 4 (b) + /   - x + (x) dx,
a                            a
b being taken to be finite. Since  (x) has a finite upper limit U in (a, b), we have
I  [b
I  e-Y +(x) dx < Ue-av (b-a);
[b                     fb
therefore             linm   ey P (x)cd=    (b)=  r    (x) dx.
==O                       J a 
In case b= oo, we have
Ja                a
=.yc fJ|e-Yx + (x) +dxJ   e Yx + (x) dx.
Hence, by applying the first mean value theorem, we have
f e-YX?   (x) dv=+ (xl) (e-"ly-e-VY) + + (X2) eG~
J 




414, 415] Differentiation with respect to parameters.599


where xl is some number between a and 1/N/,, and xs some number greater than 1/V//.
When y converges to the limit zero, the first term on the right-hand side converges to
the limit zero, and the second term to the limit r ( o), or  q (x) ()d. Thus the
theorem is established for the case b=cc.
DIFFERENTIATION OF AN INTEGRAL WITH RESPECT TO A PARAMETER.
415. Let f(x, y) be a function of the variable x and the parameter y,
and which we shall suppose to be defined in the domain of (x, y) defined by
rb
a < x.< b, yo5 y _ y0 + a. Further, let the integral f f(x, y) dx exist as a
a
proper integral, for each value of y in the interval (yo, yo + a). If ut denote
ff(x, y) dx, sufficient conditions will be investigated that
(|) _     fJb{TVa  Y)} Y  dx.
ay =Vo ^         ay I  = g2o
This is the ordinary rule first employed by Leibnitz, of differentiation
under the sign of integration, and is an important example of the process of
changing the order of repeated limits.
In this result (a-)  denotes the derivative at u on one side. No
ay Y=Yo
information is afforded as to the existence of a derivative on the other side.
If, however, the function be defined for values of y on the other side of yo,
the derivative on that side may be treated in a similar manner.
We have
yo+h-____    b f (X,  o +)-f (x, Y) d
o+         f,       + h 
where h < a. If it now be assumed that, for each value of x, f(x, y) has
a differential coefficient with respect to y, at all points interior to the interval
(Yo, o + a) of y, we have from the theorem of ~ 203,
Uy-+h  yo   ff a     Yo + Oh) dx,
h      J       ay
where 0 < 0 < 1, and the value of 0 depends in general upon x and h.
If it be now further assumed that Df(x, ) converges to af(xy), ) as
{y               \        y=yo
y converges to yo, uniformly for the interval (a, b) of x, we have, for a
sufficiently small value of h,
af(x, y + Oh) =af(x, Y)}   +   (
ay     v      y   Y =.
where 1 3 (x) | < e, for every value of x.




600


Functions defined by sequences


[OCH. VI


Under these conditions, we have, since  fi (x) dx < e (b- a),
limY+h -    ''y~  - r;f (X, yY) Z;
ulimrn u.+, - Z.y,  fb i~af(x, l>I,/ dx;
h=O               ay       =
and the limit on the left-hand side of this equation is (l-), in case u has
a differential coefficient at yo; otherwise this limit is the derivative of u at
y, on the right, which has thus been shewn to exist, subject to the conditions assumed.
The following theorem has thus been established:If f(x, y) be defined in the domain by a   x __ b, yo _ y _ yo + a, and
uI- f(x, y) dx exist as a proper integral, for every value of y in that
domain, it is a sufficient condition that u may have a derivative with respect
to y, at yo on the right, and that this derivative be equal to  -f ' J   dx,
Ja (  COf  )Y=Yo
that -f(, y) should exist everywhere in the domain, and should converge to
ay
f (x, y-, uniformly for all values of x in (a, b).
( ay J)v=y o
In particular, the condition stated in the theorem is satisfied if af(x, Y)
ay
is continuous with respect to (x, y), for all points such that a  x  b,
y - y o-   Yo + a.
It is however not necessary for the validity of the process that af (, y)
should exist for values of y which are > yo. We shall assume that f(x, y)
has for y = yo, either a differential coefficient with respect to y, or at least
a definite derivative Df(x, yo) at yo on the right; and this for every value of
x in (a, b).
The function f(x, Y0 + h)-f(x, Y0) is a function of h, of which the limit
h
for h = 0, is Df(x, y,); and we therefore apply the condition in one of the
rb
theorems of ~ 383, to obtain a sufficient condition that  Df(x, yo) dx may
exist and be equal to Du. We thus obtain the following theorem:If * f(x, y) be defined in the domain for which a <s x s b, y < y _ yo + a,
and have a proper integral in (a, b), for each value of y in the interval
(yo, yo + a), it is a sufficient condition that  f f(, y)dx may have a definite
J a


* This theorem was given, with a direct proof, by G. H. Hardy, Messenger of Math., vol. xxxi,
p. 133.




415, 416] Diferentiation with respect to parameters


601


derivative on the right with respect to y at yo, and that this derivative may be
equal to  Df(x, yo) dx, where D denotes differentiation with respect to y on the
f(x, yo 4 h) - f(x, y.)
right at yo, that       h -  x,   ) should converge to D (x, yo) uniformly
for all values of x in (a, b).
It is here not assumed that Df(x, y) has a definite value for any value of
y except yo.
By applying the more general result in ~ 383, we obtain the following
theorem:If f(x, y) have a proper integral in the interval (a, b), for each value of y,
rb
such that yo - y - yo + a, and  J Df(x, yo) dx exist as a proper integral, it is
rb
a sutficient condition that  f(x, y) dx may have a definite derivative at yo on
the right, equal to f Df(x, o) dx, that j f(, o + h) -f(x, Yo) should be less
h
than some fixed number, for all values of x in (a, b), and for all values of
h which are such that 0 < h _ a
For a fixed value of x, the upper limit of f(x' y0 + h) -f(x,  ) cannot
exceed the upper limit of the four derivatives D+f(x, y), D+f(x, y),
D-f(x, y), D_f(x, y) in the interval (y,, yo + a) of y. It follows that the
second condition contained in the above theorem is satisfied if the derivatives
of f(x, y) with respect to y, whether definite or not, are limited in the whole
domain for which a c x   b, yo _ y  yo + a.
416. If we no longer assume that       f(x, y) dx exists as a proper
integral, for all values of y in the interval (y,, yo + h), but may for some,
or all such values of y, be an improper integral, we may apply the theorem
of ~ 386, to obtain sufficient conditions for the validity of the rule of
differentiation under the integral sign.
We thus obtain the following theorem:If the derivative Df(x, y,) at Yo on the right exist as a definite finite
number, for all points x in (a, b) which do not belong to a reducible closed set
of points G, then, under the conditions (1) that in any interval (a, /3) contained
in (a, b) which is free in its interior and at its ends from points of G,
f (, y + h) -f(x, y) is less than some fixed finite number, for all values
h
of x in (a, 8/), and for all values of h, such that O < h - a; (2) that
b,     y)er                            int
f f(x, y) dx exists at least as an improper integral, for all values of y such
a




602


Functions defined by sequences


[OH. VI


that yo < y _ yo + a; and (3) that  f /(, y) dx has a definite derivative on the
right at y0, which is a continuous function of x in the interval (a, b) of x;
rb
and (4) that   Df(x, yo) dx exists, at least as an improper integral; the
rb                                rb
equality of j Df(x, yo)dx with the derivative of j  f(x, y) dx at yo on the
right holds.
417. The validity of the application of the rule for the differentiation of
b
fjf(.x, y) dx with respect to y is, under certain restrictions, dependent on the
equality of two repeated integrals. Let us assume that Df(x, y) has a
proper integral with respect to y in the interval (y,, yo + a), for each value
of x in (a, b); we have then
f(x, yo + h) -f(x, yo)= -  y  Df(x, y) dy.
0
Since now
tUy+7h - UlyO =   /(, yo + h) -f(x, o) dx
h                  h
=I b dxJ        Df(x, y)y;
provided the function Df(x, y) has a proper integral with respect to (x, y) in
the domain bounded by x = a, x = b, y = yo, y = yo + a, the order of integration
in the repeated integral may be reversed, and we find
Uyo+h - UyO I y+=  h dy  Df(x, y) dx.
h       T. j0
rb
If now j Df(x, y) dx be continuous with respect to y at the point yo, the
limit of the expression on the right-hand side of this equation is, for h = 0,
rb
j Df(x, yo) dx. The following theorem has therefore been established:It is sufficient for the validity of the rule of differentiation that Df (x, y)
should have a proper integral with respect to y, in the domain bounded by x = a,
= b, y = yo, y = yo + a, and should also have a proper double integral with
rb
respect to (x, y) in that domain, and further that  Df (x, y) dx should be
continuous with respect to y, for y = yo.
It has here not been assumed that Df(x, y) has a definite value for every
value of y such that yo < y - yo +- a, but only that it is integrable in the
interval (y,, yo + a) of y.




416-418] Differentation with respect to parameters


603


418. In case the upper limit b have the improper value oo, the condition
that Df (x, y) shall have an integral in the domain bounded by y = yo,
y = yo + a, and unbounded in the positive direction of x is not sufficient to
ensure that the repeated integral may be reversed. It may in fact happen
that f Df(x, y) dy does not exist. If however we assume that the conditions
stated in the theorem of ~ 409 are satisfied, the process is still valid, and the
rule of differentiation is still applicable.
In this case, b = oo, we have
rX
yo+7 -   =yo =lim  If ({x, yo + h) -f(x, yo)} dx.T= GO, a
=lim J   dx      Df(x, y) dy,
X=c  a    J0
it being assumed as before that, in the domain bounded by x= a,  = X,
y = Yo, Y = yo + a, Df exists for all values of y, and is integrable with respect
to y; and that this holds however great X may be. On the assumption that
Df(x, y) has a proper integral in the same domain, we have
y0/o+h  fr
+0h - Uy = lim       dy    Df (x, y) dx.        (A)
X= Yo0
If now
r v+h   f X00
lirm     dy    Df(x, y) dy = 0              (B)
X'= ~.
and further if  Df(x, y) dx be continuous with respect to y at y = o, we
have
()      - |  Df (x, y) dx.
Y =Yo
In case    Df(x, y) dx does not exist, or the equation (B) be otherwise
not valid, the equation (A) still holds, and it may in certain cases be applied
to determine the value of -1u)
Let us assume* that     Df(x, y)dx can be divided into two components, so that
jfDf(x, y) dx = b (X, y) + fk (, y) d
(X, y), and here   (, y) is such
where b (X, y) is such that lim j  (X, y) dy =0, and where J (x, y) is such
X=- co 
that
r co ry0+r Y0o+l               co
dx d fo   (x, y) dy =    dy      (x, y) dx.
e  a,      s de  J so    a, 
* De la Vall6e-Poussin, Annales de la soc. scien. de Bruxelles, vol. xvI, 2.




604


4Futnctions defined by sequences


[CH. VI


We find then, provided J   (x, y) dx is a continuous function of y at the
point y = Yo, that
(y     =    *4 (x, yo) dx.
419. In ordinary cases, a special case of the criterion of ~ 409 may be
applied to prove the validity of the inversion involved in the use of the
equation
r30   p o+p ryo+^               rc
dx       Df(x, y) dy =     dy JDf(x, y)dx;
J a  Jo        'J Yo            a
and then, provided   Df(x, y) dx is continuous with respect to y for y = y0,
we have
ay/)y=y =J {Df(x, y)y=yo5 dx.
It is thus established that a suficient condition for the differentiability of
If (x, y) dx at yo, under the sign of integration, is that  Df (x, y) dx shall
converge uzniformly for all values of y in the interval (yo, yo + a), cnd shall be
a continuous function of y at y = yo.
It may be observed that the condition that   Df (x, y)dx shall be a
continuous function of y at yo, may be replaced by the condition that
rx
f  Df (x, y) dx be continuous, whatever value X  may have (>a), it being
assumed that the condition of uniform  convergence of   Df (, y) dx is
satisfied.
For          f Df (x, y) dx =   Df (x, y) dx +  (y)
where I (y) < e, provided X is sufficiently great.
Hence
Df(x, yo + h) dx -  Df (, yo) dx
-=  SDf(x, y, + h) dx -  Df(x, yo) dx + ',
where 1\  < 2e. From this it follows that, for a sufficiently small value of h,
Df(x, yo + h) dx -  oDf(x, yo) dx
f*
is less than 3e; and since e is arbitrarily small,  Df(x, y) dx is continuous
at y = yo.




418-420] Differentiation wzith respect to parameters


605


420. The method employed in ~ 417 may be extended to a wider class of
cases. It may happen that Df(x, y), although limited in the domain bounded
by x = a, x = b, y = yo, y = yo + a, is not integrable with respect to y in the
interval (yo, y,+ h), for all values of x.  In that case, we can employ the
Lebesgue integral j   Df(x, y)dy, which certainly exists for every value
of y, since, f (x, y) being summable, Df (x, y) is also summable, provided it
have at each point a definite value, with the possible exception of a set of
points of zero measure. In accordance with the theorem of ~ 406, the order
of the repeated integrals can be reversed, provided
vo+7^ rb
dy   Df (x, y)dx
J Yo    a
have a definite meaning; and this is certainly the case, since
fDf(x, y)dx
cannot exceed the product of b - a into the upper limit of | Df(wx, y) I in the
two-dimensional domain.  The reasoning of ~ 417 is then applicable, and we
obtain the following theorem:If Df(x, y) be limited and in general definite, in the domain bocnded by
x = a, x = b, y = yo, y = yo + a, then, provided the Lebesgue integral
rb
Df(x, y) dx
be continuonts at yo, on the right, the derivative of
rb
f    f(, y) dx,
at Yo on the right, is       Df (x, Y) dx.
It may happen that Df(x, y) is unlimited in the domain bounded by
x=a, x=b, y=yo, y=yI+a,
but may still possess a Lebesgue integral.
y0o+h
Also j   Df(x, y) dy may exist as a Lebesgue integral, although Df(x, y)
is not necessarily limited in the interval (yo, y, + h). In case the points of
infinite discontinuity form a reducible set of points in the interval (yo, y0+h),
the equation
f Yo+h
f (, Yo + h) -f(x, yo) =  Df (x, y)dy
is still valid.  If further, the repeated integral
yo+7h  rb
dy   Df(x, y) dx
0J  o  J 




606               Functions defined by sequences                  [CH. VI
have a definite meaning, as a repeated Lebesgue integral, then the process of
~ 417 is still valid.  We have accordingly obtained the following theorem,
applicable to those cases in which Df(x, y) is not necessarily a limited
function, but may have points of infinite discontinuity:If the points of infinite discontinuity of Df(x, y), considered as a function
of y, form a reducible set, for each value of x, and
Df(x, y)dy
exist as a Lebesgue integral, for each value of x, and if Df (x, y) have
a Lebesgue double integral in the domain bounded by
x=a, x=b, y=yo, y=yo+a,
fy"o+h  rb 
and if  y   dy   Df (x y) dx exist as a repeated Lebesgue integral, and
Df(x, y) dx
rb
be continuous at y =yo, then the derivative of    f (x, y) dx at y0 on the
-   at
rb
right, is  Df (x, yo) dx.
In any particular case the Lebesgue integrals may exist in accordance
with the older definitions.
EXAMPLES.
1*. Let
f(x, y)=sin (4 tan-1  -     2 cos (4 tan-1;
then                 f  f( y)dx=    sin 4 tan -1 
We find
~a ff(x, y) dx=2   - cos(4tan     );
therefore at the point y=0,
a    f/(x,  ) dx=4.
The value of |  f(x, ) dx is found to be X2   cos  ta       wh    >, and
Jo   y                       2      4tY-      when   > 0, and
zero when y=0. Since this integral is not continuous at y=0, the conditions for the
differentiation under the sign of integration are not satisfied at y =0; in fact we have
X af (, 0) dx =0.
o    ay
The function f(x, y) is discontinuous at x=0, y=0.
* Harnack's Diff. and Int. Calc., Cathcart's translation, p. 266.




420, 421]


The condensation of singularities


607


2. Consider the integral f|   sYdx, where y>0.    This integral is not differ0o                             X
entiable under the sign of integration, for any value of y; for f  cos xydx does not
exist.
3*. The integral f  (x-y)' dx may be differentiated under the sign of integration,
for every value of y. For it has been shewn in Ex. 3, ~ 408, that (x-y)-.~- has an
absolutely convergent double integral in the domain bounded by x=0, x=X, y=0, y=h.
It may be easily verified that the other conditions of the second theorem in ~ 420 are
satisfied.
4. Consider the integral t=  -  s1+   dx, where y >0. The integral
J  x sin xy dx
Jo 1+.X2
satisfies the condition that, for all values of y greater than a positive number yo, it
converges uniformly.  We find, by integration by parts, that
Fi' x sin xxy =   x cos x~' 1     X'  1 - x2
JX   1 + 2        (1 + X2) jx  y    (1 + 2)2 cxydx;
hence if X''>XA>1, the absolute value of the integral is less than
(1 + X2) Yo + - o               ua);
whence the result follows. The condition for differentiation of zu under the integral sign
is therefore satisfied, for any positive value of y.
THE CONDENSATION OF SINGULARITIES.
421. A method of constructing functions which possess, at an infinitely
numerous set of points in a linear interval, singularities in relation to
continuity, derivatives, or oscillations, has been given by Hankel. The
method depends upon the employment of functions which at a single point
possess one of the singularities in question, and consists in building up by the
use of such a function of a simple type, the more complicated analytical representations of a function which possess the singularity at an everywhere-dense
set of points.  To this method, Hankelt has given the name "Principle of
condensation of singularities" (das Prinzip der Verdichtung der Singularititen); the name may however be conveniently applied to other methods of
constructing functions capable of analytical representation which have been
given more recently by other writers.
Let b (y) be a function defined for the interval (- 1, + 1), limited in that
interval, and continuous at every point of the interval, including - 1, + 1,
t See Hardy, Quarterly Journal Math., vol. xxxII, p. 67, where various theorems relating to
the differentiation of integrals will be found.
t See his memoir "Untersuchungen iiber die unendlich oft unstetigen im oscillierenden
Functionen," reproduced in Math. Annalen, vol. xx. The method has been treated in a rigorous
manner by Dini, Grundtlagen, p. 157.




608


Fulnctions defined by sequences


[CH. VI


except at the point y = 0, where however h (0) = 0. The function b (sin rn7x)
is finite and continuous for every value of x which is not a rational fraction
mi/n, with n as denominator, and it vanishes for all points at which x has this
form.
rThe series *   ~  (sin 17xM)
The series    (i S, where s >, is, in accordance with the fact that
n=l1    S
+ (y) is limited, uniformly convergent in every interval; and its sum is a
limited function of x which is continuous for all irrational values of x.
If b (y) were also continuous for y = 0, the function represented by the
series would be continuous also for rational values of x, but when S (y) is
discontinuous at y = 0, the properties of the function
f(  _   0 (sin n?7rx)
n=1 s
in relation to continuity or discontinuity at the points where x has rational
values require investigation.
The series being uniformly continuous, it follows from the theorem of
~ 343, that f(x) is continuous at every point at which all the functions
b (sin n7rx) are continuous, i.e. for all irrational values of x. Let us consider
the values of the function (x) in the neighbourhood of a point x=p/q, where
p and q are relative primes. We may write the value of f(x) in the form
X J(sinn.7rx)   1    b ((sin qm7rx)
nq = 1 -?  q     1=1 -S  -,nq~l  H8       q8    M~ls
where n. has those integral values only which are not multiples of q.
The first of these series is uniformly convergent, and its sum is continuous
at the point p/q; we therefore find that
-pr i  ^/  1  I s(-1 n^ sm qvbrh)
f(p/q + h) -f(p/q) = rl + -    (-          ---,
qs 1T=1      mY
where Nh converges to zero when h does so.
Case I. Let   (y) have an ordinary discontinuity at y =; we then
have
0-   - )     I.   ( 0 )   I
f(p/q + 0) -f(p/q) = (+)                  (o)     1
qS  2E= (2r + 1)s  qs r=1 (2r)s'
/'- 0) / 
f(P/q - o) -/(p/q) = _ (f 0)     1       (-0),    1
qj j  r=O (2r + I)  qS,= (2r)s'
where the upper or lower of the ambiguous signs are to be taken, according
as p is even or odd.
If b (+0), b (-0) are different from one another, and from zero, these relations
shew that, at a point p/q for which p is even, the function f(x) has ordinary
discontinuities both on the right and on the left, the measures of the two
being not identical. Moreover the same statement may be made for a point




The condensation of singularities


609


p/q at which p is odd, unless ( (+ 0), b (- 0) have such values that one or
other of the above expressions vanishes, in which case there is an ordinary
discontinuity on one side, and the function is continuous on the other side.
It is easily seen to be impossible that the two expressions can simultaneously
vanish, and therefore there is an ordinary discontinuity on one side at least.
If p (+ 0) $ 0, c (- 0) = 0, there is discontinuity on the right at the points
x= 2p'/q, and continuity on the left; and at the points x = (2p'+ 1)/q, there
are discontinuities on both sides, with different measures.
If b (+ 0) =  (-0), so that p (y) has only a removable discontinuity at
the point y=0, then the function f(x) has removable discontinuities at all
the points x = p/q.
In every case the function f(x) is a point-wise discontinuous function,
because its discontinuities are all ordinary ones (see ~ 189).
Case II. Let <b (y) have a discontinuity of the second kind, at y = 0, on
one side at least. In this case it will be assumed that s > 2. Denoting by
A the upper limit of  (y) I in the interval (- 1, + 1), we have
p (- 1 mP sin qm7rh)    1  sin
=l        m' — _-" - -    Ip sin  qvrh)
A        1 I    A2 r2 - i
2s-2 m= (m + )2< S-26
and hence
f(p/q + h)-f(p/q) = S (- 1   sin qrh) + h + 2S-2 q
where ' is such that - 1 < '< 1, and is dependent on h.
If b (y) have a discontinuity of the second kind on both sides of the point
y = 0, there are finite oscillations in arbitrarily small neighbourhoods of the
point on the two sides; if then s be chosen so great that A/2s-2 is less than
half the saltus at y=O, we see that f(x) has discontinuities of the second
kind on both sides at all the points x =p/q.
If f (y) have a discontinuity of the second kind at y = O, on the right, and
have a discontinuity of the first kind, or be continuous, on the left, there is at
each of the points x= p/q, where p is even, a discontinuity of f(x) of exactly
the same kind as that of p (y) at y = 0. On the other hand, if s be sufficiently
large, there is at each of the points x = p/q, where p is odd, a discontinuity of
the second kind on both sides. For we may express f(p/q + h) - f(p/q) in
the form
+1    p (-ssin 2r + 1 q7r/)  1 E  (sin 2rq7rh)
rh r=O    (2ry+ 1)      9s r=O   (2r)s


39


H,




610


Functions defined by sequences


[CH. VI


which can, as in the previous case, be reduced to the form
I                I                 A       A 2
vh + -. f (- sin qTrh) + -  (sin 2q7rh) +  -.  + q2-2,,
where   1, ~2 are both in the interval (-1, 1).   We thus see that, if s be
sufficiently great, there are finite oscillations in arbitrarily small neighbourhoods of p/q on both sides.
The existence of the factor 1/q8 in the expression for f (p/q +  ) -f (p/q)
shews that there are only a finite number of points p/q at which the saltus of
f(x) is _ k, where k is an arbitrarily chosen positive number; and thus f(x)
belongs to the special class of point-wise discontinuous functions for which
the set K is a finite set, for each value of k.
EXAMPLES.
1. Let q (y)=sin -, and q (0) 0; the function f (r) is then defined by
f/() == 2 1 sin (cosec nrx),
where, when x-p/q, the terms for which q is a multiple of q are to be omitted. This
function is, at least when s>2, a point-wise discontinuous function which is continuous at
all the irrational points, and has discontinuities of the second kind, on both sides, at the
rational points.
2. Let                (/)=4  2     1sin (21+ 1) 
7r =o (2r + 1)         '
where a>1. For 0<yl1, we have b (y)=l; for -1_y?<O0, we have ((y)-= -1; also
q (0) =0; and thus Q (y) has an ordinary discontinuity at =0.
The fuiction
_(_)= ~       sinnarx}],
f(X)=-=  2 -^ [   9 -   n 2 s   (2 sin (2r+1) sn
/     n7rn=1 L r=o 2+l +I 
is a point-wise discontinuous function, which is continuous for all irrational values of A,
and has ordinary discontinuities on both sides at all the rational points.
3. With the same value of ~ (y) as in Ex. (2), let
x (x)= 2
1 2   S [q (sin nTrx)]s'
1
where s>l. For any irrational value of x, X (x) has the value E-s, and for any rational
1 ns
value of x, the function is indefinitely great. Now let
2 -1 1
f() =then f (x)=1, for all irrational values of A, and f(x)=O, for all rational values of x. The
function f(x) is accordingly totally discontinuous. The values of f(x) are improperly
defined at the rational points.




421, 422]


The condensation of singularities


611


422. Let us next assume that 4 (y) is continuous throughout the interval
(- 1, 1), and has no differential coefficient at the point y = 0, where 4 (0)= 0,
but that, at every other point in the interval (- 1, + 1), it has a differential
coefficient which is numerically less than some fixed finite number A. In
this case  h  has no definite limit for h = 0, either when I is positive, or
when h is negative, or in both cases; or else the two limits both exist, but
have different values.
The numbers    (h)  which are equal to 4 '(0i)1, where 0> 0, have a
definite upper limit U(< A) for all values of h.
Assuming that s > 2, we see that the series
O' (Sl sin 7rx)
W7T 2..-..... COS 9TrTX
converges for all irrational values of x, since the general term is numerically
less than B/nS-1, where B is some fixed number.
Consider the series
4 c [sin {nr (x + ha)] - ( (sin 7rx)
where x has an irrational value. It will be shewn that this series converges
uniformly for all values of h. Unless n and h are such that sin nlr (x + h)
and sin narx are equal, in which case 4 [sin nfr (x + h)] - 4 (sin nrx) = O, we
can write the general term of the series in the form
7r  4 [sin nnr (x + h)] - 4 (sin n7rx) sin }- nrh
--.  - -cos nMr (X +  i- +  -
nS-l    sin nvr (x + h) - sin ncrx  n7roh         + 
It then follows that the general term of the series is numerically less than
7rV,~S_, where V is the upper limit of the absolute values of the incrementary
ratios of the function. Since the series:l/ns-l is convergent, it follows
that the above series converges uniformly for all values of h which are * 0;
and consequently, in accordance with the last theorem of ~ 397, the function
f(x) has a finite differential coefficient for any irrational value of x.
Next let x have the rational value p/q.     We may then express
f (q +  ) -f (P)
q+/ "q-f in the form
l [sin nqrr (x + h)] -      (sin nmrx)          qr1 _ _h)
n,=l            hnqs              qS il-       hmps      )
where itn has all positive integral values which are not multiples of q. In
39-2




612              Functions defined by sequences             [CH. VI
accordance with the above proof we see that 2  (sin  7r) has, for the value
ngq=l  nq
x     = pq, a finite differential coefficient which is the sum
- 4' (sin nqrp/q) 
7r I     ~."1    COS qqrp/9
lq=l    q
We have now shewn that
f   + h)-f()
h
~ 4' (sin nql7rp/q)           1 ~ < (- |'p sin mq7rh)
=     vr       1 — cos nqwp/q + ^ +                       I
nq=l    nq                      q    ' 1=   h9i-S
where Vh is a number which converges to zero when h is indefinitely
diminished.
Case I. Let < (y) have definite derivatives on the right and on the left
when y=O; and thus -(h) has one limit 0' (+0) for positive values of h
converging to zero, and another limit /' (-O) for negative values of h so
converging. We thus have, when p is even,
f ( + h)      (           ' (sinnqrp/q) 
lim -      7=. n-                       cos -qE7p/q
h= +0O     h              l      q1 qg
7Tq (+ 0)    I 1
fS-1 l h=i 8m-1 
flim   / 1)        -/ f (P) =  q' (sin iIqrp/q) cos
lim  vv,  vvr =  2Cros N'p
h=-O       h            n - q==?Tq8.+r (-~2)      1
qS-1  m=1 ms-'
For an uneven value of p, we find
i. n  q ~       (q)     2 hb' (sin zq7rp/q)
lim..... 7r   S-1      cos nqgrp/q
h=+O       h            n-=l    nq
7+( (+ 0)                0 1  r(-O)  1 1
q3-1,=-1 (2r)8-1   9S-l,=o (2r + l)S-'
f ( h) -f ()            GC f  (sin nq7rp/q)
l    -   --    — h=-O  n —l --- cosnqp/q
li=-0  q         q 
_  (+0) _      1       r' (-0) ~    1
9"- -,.=1 (2r + l)s1  qs-    = (2r)s




422]             The condensation      of singularities                613
From  these results it is seen that f(x) has, at the rational points,
definite derivatives on the right and on the left, differing in value fiom one
another, and therefore, at all these points, the function has a singularity of
the same kind as + (y) possesses at the point y = 0.
Case II.   Let   (y) have, on one side of y     0 at least, no definite
derivative.  Unless nmqh is an integer, in which case    I (-I mp sin mqwrh)= 0,
we have
P (- 1 mp" sin mqTrh) _  (- 1 TI' sin mqrrh) sin m.q7rh  -  q
hm8             - 1 Ip sin mq7rh    nzqrrm h        m-1)
and this is numerically less than q —.  It follows that
1 E   ~ (- 1 l'sP sin mqwth) _ 1  (- 1 Psini q7rh)
qs,2=1       hmns          qS                 ~ 
7TU        1
where P is numerically less than    -l   I    S-. By taking a sufficiently
q    M=2 M
large value of s, the number P may be made as small as we please, and
therefore
1  E q (-1 lnP sin mqrh)
qS' m~,hms
g M=1
will, for a sufficiently large value of s, oscillate in the same manner as
1   (-1 P sin qrh)
q         Ah       '
as h is diminished indefinitely. It is thus seen that -     rq       has, on
one side, or on both sides, of h =0, no definite limit; and thus f(x) has no
differential coefficient at any of the rational points, provided a sufficiently
large value of s be chosen.
EXAMPLES.
1. Let (b(y)=y or -y, according as y is positive or negative. The corresponding
function f(S) is  1 /sin2 nr x, where the positive value of the square root is to be taken.
This function is continuous, and has a differential coefficient for all irrational values of x.
At the rational points it has no differential coefficient, but has definite derivatives on both
sides.
o  sin n?7rx [log sin2 nZrx]1
2. Let        q (y) =y sin (log y2), then f (x)=  sin  [log sin2 nsx ]
The function f(x) is continuous, and has a finite differential coefficient for all irrational
values of x. If s be sufficiently large, it has no definite derivatives either on the right or
on the left, for rational values of x; the four derivatives at such a point are all finite.




614


Futnctions defined by sequences


COH. VI


423. Let it next be assumed that + (y) is continuous in the interval
(- 1, 1), and has a finite differential coefficient at every point except at y = 0,
but that this differential coefficient has no upper limit to its absolute magnitude in any neighbourhood of the point y = O. In this case 0 (y) may either
have a differential coefficient at y = 0, which is finite or indefinitely great;
or it may have indefinitely great derivatives, on the right and on the left,
of opposite signs; or it may have no definite derivatives. When b (y) is
a function of this type, it is not certain that f(x) has differential coefficients
for irrational values of x; for the differential coefficients 0' (sin n7rx) are not
all numerically less than a fixed finite number, for such a value of w, and for
all values of n; and thus the argument of ~ 422 does not apply.
For a rational point x = p/q, we have as before,
i? q  A) <  \q  ) s <^ [sinl,7rV (w + It)] - ) (sin uq7rx)
h  qk=1i A]tl "H'
1      (-    lsin mq'lh)
The theorem   of ~ 398 will be applied to shew that the function
E  (sin 7% ) has, for the value p/q of x, a differential coefficient obtained
ity =l  1iq
by means of term by term differentiation of the series.
The first condition required by the theorem  in question, viz. that the
terms of the series
E   f(Y (sinl r q7Trj/q)  p
S        -- 1  I Cos  - W7
nQ=l     1 q            qq
shall be definite, and that this series shall converge, is certainly satisfied.
To shew that the conditions relating to
RAn (P    It    R,M1.. )    _. U
are satisfied, we observe that R, (x) <  s-, where U denotes the upper
limit of    (y) | in the interval (-1,.).   Let t be so chosen that
1 > t >, (s > 2), and let m be that integer next greater than | h -t which
is not a multiple of q; we then have I hrm-l > h ]-(-) t. It follows that, for
each fixed value of h, m has been so chosen that
hjlY+) j      h )
h     I '     h 




423]            The condensation of singularities                615
are both less than U h (s-') t-, and are therefore both less than e, provided
ih <; where 8 is fixed so that U8 (s-l) t- < e. It is clear that 8 may be chosen
so small that mn exceeds an arbitrarily prescribed integer n', for all the values
of h such that ] h < 8.
We have lastly to prove that the sum of the first m terms of the series of
which the general term is
sin Jtw (1 — F h)] - (p (sin ne1r-7  ' (sin nq  )
l  6[ I ('1 \q ] /J \ t  'I  ')  v   cos ibLq  o,
is numerically less than e.
)il1 — 1  l
This series may be divided into two portions E  and E, where mnl is a
1       i
fixed number independent of' i, so chosen that the sum  -I is less than an
1 l 'Vq
arbitrarily chosen number '). The sum of the first 'ml terms of the series
under consideration can be made arbitrarily small, by taking 8 sufficiently
small; for the number of terms is independent of h. We have then only to
consider the sum E.
Since  qP differs from  an integer by at least, it follows that
q 
b' (sin nqer) is numerically less than some fixed positive number U', for all
values of nq. We therefore see that
7rof sin nqT)       I
E   \-,cosnP  <7r U.
Jul    n,            q
Further, m has been so chosen that tn- I h  < 1; from which we have
m hI-= h1-t~-+01 h,  where 0<0<1.     If' 8 be now   so chosen that
81-t + 8 < 1/2q, the two numbers nq-, nq (I + hb) differ from one another by
less than 2; moreover they are never integers, and contain no integer
2q'
between them, and they differ frnom the nearest integer by more than 2'
It follows that, for all values of y between sinllTrp and sin nr7r (P+  ),
where tnq has the values belonging to it in the series X, +(y) has a differential
coefficient numerically less than some fixed number U".




616


Functions defined by sequences


[CH. VI


sinHn fr - + h) -    (sin n      n r t f
WritinLg           q           (      q     in the fbrm
Writing                hitq
rsin n. -n    + h        s -  7r (sin.             sin1-nh
)                           (                      1    S  2,qY(  rh 
flq  Sil fq                                    2  1i
sin nq7rk - +  - sin qW 7r- )                 /    N q7rh
we see that this term is numerically less than  7  U". It now follows that
I  <,&"  ';
and this is numerically as small as we please, if we choose v and 8 sufficiently
small. It is therefore possible to choose 8 so small that the last of the
requisite conditions is satisfied, for all values of I h < 8.
It has now been proved that
f(+ h)-f (')            ' (sinnq  ) 7r
V     r _.   Y' — -  cos nqp r- +
lh           1.nq.8-1         q 
1+   i  (1 mP sinmqrh)
qS m=i hmn
where a and h converge together to zero.
The second series on the right-hand side of this equation can be written
in the form
1 =n  p (- 1l np sin nq7rh)  1        I (-'  sin qrh)
qs n=1       hs           q8 n=m+1      Ahn
where m is fixed as before, for each value of h. The second sum is arbitrarily
small for a sufficiently small value of 8. We have then to consider the first
sum, which may be written in the form
zr n=   n    11 fP (_ 1  p sin nq7rh) sin nq7rh
qS- n=  n-1 l-_ 1    sin nq7rh   nqwrh
and we now consider this sum in the different cases which arise when various
assumptions are made as to the nature of the derivatives of + (y) at the
point y = 0.
Case I. Let ( (y) have the derivative + oo, at y = 0 on the right, and the
derivative - o, at y = 0 on the left. It is clear that, for positive values of h,
all the terms of the series have one and the same sign, 8 having been chosen
so small that mh is also sufficiently small; also it is clear that the first term
of the series becomes numerically arbitrarily great for sufficiently small values
of h. It therefore appears that the sum of the series becomes indefinitely




423]            The condensation of singularities                617
great, as h approaches the limit zero from the right-hand side. If h be
negative, the terms of the series all have the same sign, the opposite one
fiom that which they have when h is positive, and as before, the sum of the
series is indefinitely great as h converges to zero.
It has therefore been shewn that,+ h) )-(s)
h
has the limit + oo on one side of the point p, and - oo on the other side.
q
The singularities of the derivative of f(x) at the rational points have the
same peculiarity as that of 4 (y) at the point y = 0; i.e. derivatives on the
right and on the left exist, which are infinite, but of opposite signs.
Case II.  Let 4 (y) have a differential coefficient at y = 0, which is either
+ oo, or - o.
It is then clear that, in case p be even,
( + h) -f()
h
has the same limit + oo, or - o, as ith)  If p be odd, the terms of the
series under examination have alternate signs, and no conclusion can in
general be drawn as to the nature of the derivatives of f(x) at the point
P =-.
q
Case III.  Let ) (y) have a finite differential coefficient at y = 0.
In this case, as is easily seen, f(x) has, at the point P, a definite
differential coefficient of which the value is
cc  sin n77r -
Vr:              cos nr.
m =1   ~s-1            q
Case IV. Let 4 (y) have finite derivatives at y = 0 on the right, and on
the left, which differ from one another. In this casef(x) has, at each rational
point, finite derivatives on the right, and on the left, which differ from one
another.
Case V. Let D+ + (0), D+ + (0), D- ) (0), D_L (0) be all finite and
different from one another. The function f () has then at -, at least when
p is even, the same peculiarity as 4 (y) at y = 0.




618


Functions defined by sequences


[CH. VI


EXAMPLES.
1
1. Let q( (y)==y sinl, kq (0) =0. The corresponding functionf (x) is given by
Sill 9-rX Sin
f(X) =2    ---- -,- where > 2.
1       8
This function is continuous, but has no definite derivatives at the rational points. No
assertion can be made as to the derivatives at the irrational points, because the differential
coefficient j' (y) has indefinitely great values in every neighbourhood of y = 0.
2. Let ( (y) = (y)f3, where a, 3 are positive integers such that 2a <t3, and the real
positive values of the root are taken. We then have
1    7)"  a
/(x') = 2>(   /, wherc s>2.
This function is continuous, and has, at all rational points, indefinitely great derivatives
on the right, and on the left, of opposite signs. No assertion can be made as to the
derivatives at the irrational points.
CANTOR'S METHOD OF CONDENSATION OF SINGULARITIES.
424.  A method of constructing a function which exhibits at an everywhere-dense set of points, some singularity, either in relation to continuity,
or to its derivatives, has been given by Cantor *. Let ) (y) denote a function
which is continuous for all values of y in the interval (- 1, 1), except y = 0;
and let 0 (0)= 0.   Let G denote an enumerable set of points eo1, w, o),...,
which may be everywhere-dense.      The method of condensation consists of
the construction of the function
j(x) =     2 c/ (X - ),
nt=l
io'~~~~~~~~~~~~= 1~~a
where c1, C2,..., c,,n... are positive numbers, so chosen that the series  c,,l is
1
convergent, and that S cb (x - wo,) converges absolutely for each value of
5= 1
x, and uniformly in every interval.
This method has two advantages over that of' Hankel.    In the first place,
the points wo, o.,... do not necessarily consist of the rational points of the
interval (- 1, + 1), but may form any enumerable aggregate.    In the second
place, for a value o,, of x, the singularity in question is exhibited by the one
term  cn(x - w,) only, of the series which represents f(x); whereas in
Hankel's method, the singularity of p (y) at y = 0 is exhibited, for x = pl/, by
an indefinitely great number of terms of the series which represents the
function formed by condensation.
* lMath. Annalen, vol. xix. See also Dini's Grundlayen, p. 188.




423, 424]


The condensation of singulalrities


619


Let now 0 (y) be discontinuous at y = 0; then, for any value of x0 of x,
which is not one of the values of G, the terms of the series xc,  - (x-o,)
are all continuous; hence, since the series converges uniformly in any interval
containing x0, it follows that f (x) is continuous at x0. Again, in order to
consider the continuity of f(x) at the point a,,, we may separate the term
c,&+ (x- O,,) from the rest of the series. As before, the series which consists
of all the terms except the one c,,o (x - co,) represents a function which is
continuous at x = Wo,, but c,4)o (x- w,,) has at 6,, a discontinuity of the same
character as that of 4q (y) at y = 0. It has therefore been shewn that f (x) is
continuous at every point which does not belong to G, but has at every point
of G a discontinuity of the same character as that of 4 (y) act the point y = 0.
If' ( (y) have a finite saltus k at y = O, the saltus of c,,^ ( — wo,) at Wn is kein.
Hence, on account of the convergence of Sc,, there are only a finite number
of points o,, at which the saltus of f(x) exceeds any fixed positive number.
The functionf(x) is therefore an integrable function.
Let it next be assumed that 4) (y) is continuous throughout (- 1, 1), and
possesses a differential coefficient for every value of y except y = 0; and that
the differential coefficients are all numerically less than some fixed positive
number B. It then follows that the four derivatives of 4> (y) at y=O  are
all finite; it also follows that  (h) is less than some fixed number A,
for all values of' I which are numerically less than some fixed number 8.
We now see that for any pair of points y,, y, such that y - y, < 8, we
have               I (yl) - _7  (Y)  < the greater of the numbers A and B, which may be
'yi-  'y.,~  Iv
denoted by C.
If x be not a point of G, the sum
E  C,' I4(x-h-+,A) —   (x-o,( )I is <C     E C,,
n = m-+1 -                              'l = mn +l
provided h I < 8; hence the series is uniformly convergent for all values of
h such that 0 < Ih < 8, and therefore it represents the value of f'(x).
In case x be a point w,, of G, we separate from the series which represents
fj(x), the term c,,; (x wo,). It appears then that the remaining part of the
series represents a function which has a definite differential coefficient X (o,,)
at wo.
We have therefore
f((,0 + h) -f(@coj)    (hA)
=A — ---        +  x(  )+ 
where ' converges to zero when A does: so. It thus appears that f(x) has no
definite derivatives at x = co,, but that it has at that point the same kind of
singularity as 4b (y) has at the point y = 0.




620


Functions defined by sequences


[CI. VI


EXAMPLES.
1. Let ( ((y)y - _y sin ( log 12). This function has a differential coefficient 0' (y) for
every point except y=0; and q' (y) oscillates between the values 1 - 1//2, 1 + 1//2.
The corresponding function f (x)) = sc, (v - co,) has a differential coefficient at every
point not belonging to G,,. At the point x= 'o,,, its derivative oscillates between values
C+X (o%)) and ac,,+X (co).
2*. Let + (y) =y; then q'(0) + oo. The corresponding function s2c, (a- w,) has
differential coefficients which are finite at a set of points not belonging to G. At a point
o,, of G, we have f' (o,,)= + oo. This example does not fall under the case considered
above, because I (' (y) I, for y I>O, has no upper limit.
THE CONSTRUCTION OF CONTINUOUS NON-DIFFERENTIABLE FUNCTIONS.
425.   A  general method of constructing functions which, although they
are continuous, possess at no point a finite differential coefficient, has been
developed by Dini+.     It will be sufficient to discuss here a restricted class of
such functions, which exhibits all the features of the somewhat more general
case considered by Dini.
Let u, (x), cu (x),...,, (x),... be functions each of which is continuous in
00
an interval (a, b), and which are such that the series E u,,(x) converges everywhere in (a, b), and defines a continuous function.         It will be further
assumed that u,, (x), for each value of n, possesses maxima and minima such
that the interval between each maximum and the next minimum is a number
8,, which diminishes indefinitely as n is indefinitely increased; and also that
u, (x) = - ua (x +  n), so that all the maxima of u, (x) are equal to one
another, and also all the minima, the maxima and minima being equal in
absolute value, and opposite in sign. Let D, denote the excess of a maximum
over a minimum.     It will also be assumed that U1,,(x) possesses finite differential coefficients of the first and second orders Un,' (), u," (x) everywhere in
(a, b); and that the upper limits of u,, '(x) 1, u'," () in (a, b), have finite
values, i.,
We have, for any two points x, x + h, in (a, b),
f( + h) -f(x)     u   (x + h) - u  (x)   -l   n ( + h) -    (x)
h                   h          +=l            h
Rm (x + h) - R,; (x)
+h  - '         -
where R, (x) denotes the remainder after m terms of the series       u,, (x).
' This function Zc,, (x - w,,) has been studied by Brod6n, see his paper " Ueber das Weierstrass-Cantor'sche Condensationverfahren," Stockholm Ofv., 1896, p. 583; also Math. Annalen,
vol. LI. See further Pompeiu, Math. Annalen, vol. LXIII, p. 326, where it is shewn that, if the
series be denoted by t, the inverse function x =G (t) is a continuous function with a limited
differential coefficient which is zero at an everywhere-dense set of points, provided the series
Ic2 be convergent. This function is accordingly everywhere-oscillating.
t Annali di Mat. ser. 2, vol. vII; also Grandlagen, p. 206. See also Lerch, Crelle's Journal,
vol. cmII, and Darboux, Annales de i'ecole no7rmiale, ser. 2, vol. vIII.




424, 425] Construction of non-liferentiable functions


621


This equation may be written in either of the following forms:f(x + h) -f(          + ) _,,, (X + h) - Un (X)  " R  +n ()-  (   )
h                 h           r                 hT,,since un  x), \     + h)- t,, (x) have the same upper limits in (a, b); the
h
number ti lies between 1 and - 1. Also
f(x +h)-f(x) uan ( + h)-urn (X)      ' +          h in+? 6,,U' (x) + rn  Un
h                 h            =1            n=l
+ Rm (x + h)- Rm (        2)
h...(2
where Cm is a number between I and - 1.
Let a neighbourhood (x, x + e), or (x - e, x) on either side of x be taken;
m may then be chosen so great, that several oscillations of ut,, (x) are completed in the chosen neighbourhood. Let the point x +h be taken at a
maximum or minimum of ut (x) in (x, x + e), or in (x - e, x); and let it be
at the first maximum or minimum of utm(xC) on the right or on the left of x, of
which the distance from x is  1,,,. The condition
| tm (x + A) - u,n (x) \  z D,,
is then satisfied; also  ] h  3 | Ar.  We may write
u,,n (x + h) - un (x) = - anrnDrn,
where ym is positive and > 1, and a,,= + 1, its sign depending on x and m,
and possibly on the sign of h.
The equation (1) may now be written in the form
f( +   ) -f(X)
h
=am  nDm I,,   2'h \1       2,,,    R(     h (,,+ (x)]
2h        1      nI+     'Yi~        D.......
where Wn/ is between 1 and -1.
Next, let x + h, be the next following extreme point of tu, (x) after x + h,
so that h and h, have the same sign and jhil > h l.       The difference
U,, (x+ h)- u, (), when it is not zero, has the sign opposite to that of
un (w + h) - un (x), and therefore
',, (x + hl) - u,, (X) = -  e,,l aX/,,n Di,
where O     e,,' < 1.
In (1) and (2) we may write hA instead of h, provided the values of qm and,. are changed; we find therefore from (3),
f(x + h) -f(x)  f( + hl) -f(x)
h               h,
i7 Aril   M )      41j 1"n,1,       1A,,  (x + h) - R, (x)
- - + ~?, +  R     ~ ^+   -
2 h                     nL r Li=1 n27m      Dm
a2   h    R,,+ (4 + h )- R. ()] 
-  '......... (4)
7m h,       DM         J




622


2Functions defined by sequences


CCH. VI


where 7,// is between I and - 1. Also from (2), we find
f(. +/ h) -f(/I) /f(x +,1) -f(x)
h               h,. h R.%  (x + h) - R, ( x)
-     ' -,. a      I........ --
-2,........ D..             (5)
when ',,' is between 1 and -1, and t,,,,  are each between 0 and 1.
However small the neighbourhood (x, x + e), or (x - e, x), be chosen, mn can
be chosen so great that x + h, s + hi both lie in that one of these neighbourhoods which is under consideration. As e is indefinitely diminished, m must
be indefinitely increased; thus, on passing to the limit, the results (3), (4), (5)
may be applied to obtain conditions for the existence or the non-existence of
differential coefficients of the function f(x).
(a) Let it be assumed that, as m is increased, the distance between
successive extremes of %, (x) becomes continually smaller compared with D,,,,
and so that -  t, and therefore also, has the limit zero when m1 is indefinitely increased.
In this case the limit of f-x +-/- (:i) is certainly infinite, unless the
h       I etainly    infilite, unless t'e
expression in the bracket in (3) has the limit zero.
If then RI,,,(x + h)- R,, (x) has, for values of mq greater than an arbitrarily chosen integer in', the same sign as a,,, i.e. as u,,, (x + h) - u,,, (xi), and if
21h 7-1
further D-    i,,,2 remains in absolute value less than unity, by more than
h   M,,m=l
some fixed difference, f(x + h)-f(x) cannot have a finite limit. The last
38 n-1
condition may be replaced by the condition, that       i E,n must remain
Din, n=l
numerically less than unity, by more than some fixed difference.
If these conditions are satisfied, one at least of the derivatives D+f(x),
D+f(x) is infinite, and also one at least of D-f(x), D_f(x) is infinite.
Moreover, in the case of functions of the type here considered,
n, (x + h)- ti,, (X)
has the same sign for positive as for negative values of A, hence it is impossible that f(x) can have an infinite differential coefficient with a fixed sign;
and therefore f(x) has at no point a differential coefficient, either finite or
infinite,




425, 426] Construction of non-dfclerentiable functions


623


(b) If it be not known that the condition in (a) relating to the sign of
R  (x + h) - R,,, (x) is satisfied, then f(x) has no differential coefficients,
provided R.,, (x + h) - RA,, (x) I has a finite upper limit 2R,,' for all values of
x, and also
38,, n-. 1  4Rm,
in  I,       Dl ll
remains less than unity, by more than some fixed difference. The condition
as to the limit of D2- is. the same as in (a).
DiLm
(c) Let us next suppose that "m has not the limit zero, but remains
Dinl,
less than some finite number, for all values of m. In this case D-  has a
h
finite limit; and we see from (4) or (5), that
f( +h)-f (x) f(x+ h,) -f(x)
h               h,
has not the limit zero, provided the expression in the bracket, in either case,
does not converge to zero.
If then R,m (x + h)- R,,, (x) has the same sign as u,,, (x + h) - u.,, (x), and
R,,, (x + hA) - R,, (x) has the opposite sign, and if further one of the two
expressions
In  12I-1
6( 2 Mr-I
D'I,,%=1
remains less than unity, by more than some fixed difference, then j (x) can
nowhere have a finite differential coefficient, although it may have an infinite
one, at some points.
(d) The same condition relating to 8 holding, as in (c), it is sufficient
to ensure that f (x) has nowhere a finite differential coefficient, that
i R  (x + A)) -                   RR (X),  A R  (x + h,)- Rm (x) 0
should never exceed a fixed number 2R,,', and that one of the two expressions
683,, "21  32 R,'   68, 2 m -1  32 Rm'
-7   1, +- -- 7A_,   C _S  + 
D  i       5 DU,     Dm n=  i    5 Dm
Can m1          m trn  Dm n=l  in
should remain less than unity, by more than some fixed difference. These
expressions are obtained from equations (4) and (5), by taking account of
the fact that h/h, cannot exceed 3/5.
426. Let vn (x) be a continuous function which has maxima and minima
equal to 1 and - 1 respectively, at distances d, from one another; where d),
is, for every value of n, less than some fixed number. Also suppose that
V,1 (x) = - v,, (x + dn), and that v,' (x), v," (x) are, for all values of x, in absolute




624


Functions defined by sequences


[CH. VI


value not greater than some fixed positive number A. We then take the
function tu, (x) of ~ 425 to be avn (bx), where the an's are constants such
that 2 I a, I is a convergent series, and the b,'s are constants such that anb,
does not converge to zero as n is indefinitely increased. The convergency
condition of 2 a,, I ensures that the series  a,,v,, (b,,x) converges uniformly,
and thus that f(x) is a continuous function.
We have in this case
in= -2 1 ace m, ), =  -
also   | RA (x + h) - R, (x) _i 2Rnm,  R(n (x + h,) - R,, (x) < 2R,,
where R,' is the remainder of the series X an I after m terms.
We see from (b) in ~ 425, that if amb becomes indefinitely great with m,
then f(x)-=,anv,(b,,x) has nowhere a differential coefficient, in case the
condition
3 Adin, In-I        2R   <
2 bnm Ia,   b(I 1  1    j am
by more than some fixed difference, be satisfied.
In case amb, does not necessarily become indefinitely great with m, but
has not the limit zero, we see from (d) of ~ 425, that f(x) is not differentiable,
provided one of the conditions
3Adm      b  a i+ 16 R'        3Ad, 2 m-i         16 R '
bman) 1       an   5 a, <1, b          20)a< b
bm\am\ 1               n I       Bbnaa m | 1 <5 am|      l
is satisfied in the limit.
Let us now consider the special case in which d, is constant, and = d, and
let all the functions vn(x) have their maxima and minima at the points
0, ~ d, + 2d.... Further let us suppose that b,+sl/b, is, for every value of s,
an odd integer. Then, if b,,,, (x + h) corresponds to a maximum or minimum
of v,,, (x), bm+s (x + h) also corresponds to a maximum or minimum of v,,,+ (x).
If we suppose all the a,,s to be positive, then the difference
00
R,, (x + h) - R,, (x) = S a, v,, [bn, (x + A)] - v,, (bxiw)}
m+1
is either zero, or else it has the same sign as i,,, (x + h) - t,, (x); on the other
hand the difference R  (x + h,) - Rm (x) has the opposite sign. We can
consequently apply the criteria in (a) and (c) of ~ 425.  The sufficient
conditions that  a,,vv(bx) is not differentiable become in this case,
3 Ad ''~l
23 Ab-d 2ab, < 1, lim abm, =;
2 aLnbi, 1
and when anbm has not the limit zero, then one of the conditions
3Ad rn-            3Ad2 1,- 
b,  atEbn < 1,  -, b     nb2 < a,
ambm n=1l          a,,,2 -n=l
by less than a fixed number, must be satisfied.




426]        Construction    of non-differentiable functions              625
In particular, let a = an, a < 1, bit = b7, where b is a positive odd integer.
If ab > 1, the condition of non-differentiability is satisfied if
3 Ad    anb"-1
2 nb     ab-1 
for every value of n; and this reduces to ab > 1 + |Ad.
If ab _ 1, the condition of non-differentiability is ab2 > 1 + 3Ad2.
If we let Vn (x) = cos x, we obtain Weierstrass' theorem* that, if b is an odd
integer, the continuous function
00
Ct ancos bnx  (a<l)
1
is not differentiable for any value of x, provided either ab > 1 +   r, or else
provided ab2 > 1 + 37r2, and also ab _ 1.  In the first case ab > 1 +3 -r, there
can be no infinite differential coefficient with a fixed sign, although at some
points there may exist infinite derivatives on the two sides with opposite
signs.  This was the first example of a continuous function          nowhere
differentiable which was exhibited.
EXAMPLES.
1. The function represented by 2 an sin bx, (a < 1), where b is an integer of the form
i
4p+l1, is continuous, but nowhere has a finite differential coefficient, provided one of the
two conditions ab > 1 + 7r, or ab2> 1 + 31r2, is satisfied, and ab? 1.
2. The functionst represented by
1              1              1
- cos (31x), E 3   cos (33x),     sin (33n x),
311n           33n            33s0
do not possess finite differential coefficients.
3. The continuous functions represented by
00       n
~~      an
1 1. 9... (4n+1 ) sin {1 5 9. (4n+l) x},
I.             85.9... (4nf+l)
where a> 1 +  r, do not possess finite differential coefficients.
* Crelle's Journal, vol. LXXIX.
+ See Wiener, Crelle's Journal, vol. xc.


H.


40




626


Functions defined by sequences


[CH. VI


THE CONSTRUCTION OF A DIFFERENTIABLE EVERYWHERE-OSCILLATING
FUNCTION.
427. The first attempt to construct a function with maxima and minima
in every interval, which should have at every point a finite differential
coefficient, was made by Hankel*. The function which he constructed is
however not an everywhere-oscillating function. By Du Bois Reymondt
the view was expressed that no such function can exist, but Dini+i regarded
the existence of such functions as highly probable. The first actual construction of such a function is due to Kipcke, who having first~ constructed
an everywhere-oscillating function with derivatives on the right and on the
left at every point, in a subsequent memoirll obtained a function having the
required properties. Kopcke's construction has been simplified by Pereno'T,
and the account here given is based upon the work of the latter.
On a straight line AB measure off segments AA', B'B, each equal to
7g-AB. Let 0 be the middle point of AB, and draw through 0 straight lines
r1, r2, r3,... r2l,+1, making angles with OA of which the tangents are
1/2n, 2/2n, 3/2n,... (2 + 1)/2n
respectively. Through A' draw a straight line r0 making with A'O an angle
of tangent 1/2n. Through the intersection (r0, r2) of ro and r2, draw a straight
line rl' parallel to rj: through (r', r3) draw a straight line r2' parallel to r2,
and so on. The straight lines ro, r,', r2',... r2^_, r2,,+ form an unclosed polygon


* Math. Annalen, vol. xx.                       + Crelle's Journal, vol. LXXIX.
' Grundlagen, p. 383.                            ~ Math. Annalen, vol. xxix.
11 Math. Annalen, vols. xxxiv. and xxxv.        ~ Giorn. di Mat., vol. xxxv, 1897.




427]   A   differentiable everywhere-oscillating function


627


above A'O. On OB' describe a precisely similar polygon on the other side
of AB. The figure is drawn for the case n = 2, and shews the half of the
figure belonging to A'O. The two polygons form a single polygon joining
A'B', and crossing it at 0. On ro take A'A" = AA', and describe an arc of a
circle touching AB at A, and ro at A". At each vertex of the polygon which
has been constructed, mark off on the sides adjacent to that vertex lengths
equal to -2 of the shorter side, and construct an arc of a circle touching the
two sides at the extremities of these segments so marked off. We have now
a figure joining A and B, and composed of arcs of circles and of straight lines.
This figure, by means of its ordinates perpendicular to AB, defines a continuous differentiable function, with a continuous differential coefficient
which is zero at A and B, and is -(2 + 1)/2n at 0. This function may
be denoted by (A/B)n.
Let x, y be a system of coordinate axes in a plane, and draw a quadrant of
a circle passing through the points (0, 0) and (1, 0), in the positive quadrant.
Let Fo (x) be the function represented by this quadrant, for the interval (0, 1)
of x.  The function Fo (x) has a maximum at x =; also Fo(0) = 1,
Fo' (1) =- 1. If a0 denote the value of Fo'(X) at x= 4, describe the curve
of which the ordinates are a, (O I4), from x = 0 to x =, and - a (i  1), from
x=~- to = 1. This curve represents a continuous function fi (x); and we
have
f' (O) =/f (i) =/() = 0,
and'                 f' ()= -4ao, fl () = 3ao.
The function         F (x) = Fo (x) + J, (x)
is such that
F/'(I) = - 1 ao, ') =   a     '(0) = 1, F' (1)  -1,   ' () O 0.
Thus F1 (x) has a maximum in the interval (0, 4), a minimum in (4, 4), a
maximum at x=, a minimum in (-, 3), and a maximum in (-, 1).
Let the interval (0, 1) be divided into sub-intervals, by means of the
points at which F,'(x) = 0; then in each of these sub-intervals F, (x) is
monotone. Then divide each of these sub-intervals into 2, 4, 8,... equal
parts, until the fluctuation of F1'(x) in each of these parts is - i: this is
always possible, since F ' (x) is a continuous function. Let cl(,'), c1(),...
denote all the points in which (0, 1) has been divided in this manner. In
any one part (cl,(S-, cl(8)), F1 (x) is monotone, and its differential coefficient
has a fluctuation  - 4. Let a,(l), a,(),... denote the values of F' (x) at the
middle points of the intervals (0, c(l)), (cl(l), c(2),... Describe the curves
a,  (   I cl () )2,  a   (c,  | c(2) ),,  a (   (c  () j  C,(3) )2...;
these form together a continuous curve which represents a function f2 (x).
Let
F2 (X) = Fo (x) + f + (x) + f, (x);
40-2




628


FuWnctions defined by sequences


[CH. VI


then F2(x) has, in every interval (c,(s-1), Ci(8)), a new maximum and a new
minimum. The length of each interval is < 1/22.
Proceeding in this manner, let us suppose that the function Fn(x) has
been formed. Take the points at which Fn' (x) vanishes, and, in case Fn (x)
has lines of invariability, the limiting points of those lines; these points
divide (0, 1) into sub-intervals in each of which F(x) is monotone. Then
divide each of these sub-intervals into 2, 4, 8,... parts, until the fluctuation
of Fn' (x) in each part is _ 1/2n; let c(1l), cn2, Cn(... be all the points of
division of (0, 1) thus formed. In any interval (cn(S-l), cn(s)), the function
Fn (x) is monotone, and the fluctuation of Fn'( ) is _ 1/2n. Let
an (1), an (2),... an (S),
be the values of Fn' (x) at the middle points of the intervals; and in the case
of a line of invariability, take as the corresponding value of the a., 1/2" or
-1/2'", according as the line of invariability is in the interval (0, i), or in
(2, 1). Let the curves a(,) (c,(8-') ln(8))n,+ be described, and let the function
represented by the totality of these curves be denoted by fn+, (x). Then the
function
Fn+1 (x) = Fn (x) + fn+, (x)
has a new maximum, and a new minimum, in every interval (c(,8-1), c,n()), and
the length of each of these intervals is less than 1/2n+1.
If this law of generation of the functions fn (x) be employed indefinitely,
we have a series
Fo (x) + f, () + f (x)   + J 2(+ (x) +...;
and it will be shewn that this series represents a continuous function which
is everywhere differentiable, and which has an everywhere-dense set of
maxima and minima.
428. Let
F,' (x) +  f (x) + / () +... + f,' (X) = S, ();
it will then be shewn that, for every value of n and x, Sn (x) is numerically
less than II 1 + -, which may be denoted by P. Let us assume that
U=1     21
i S, (x) is, for every value of x, less than II (1 + -, which may be denoted
I Sn                       i \.             may1 2nb d n
by Pn: it will then be shewn that Sn+ (x) I < Pn+i.
Let the point x be in the interval (cnS-l), c,n8)), where x < cn(8), the number
s depending upon the value of x; we have then, in accordance with the
construction of the functions,
an (,
n+i (x) = n (X) + an 2+,


where


1 > an _, - (2"+' + 1).




427, 428] A differentiable everywhere-oscillating function   629


In the interval (Cn(s-l), c,()), Sn (x) has a fixed sign, the same as that of
an(S), but this is not the case for Sn,+ (x). If an is positive, we have
|Sn+l (x)I<Pn      1 + < P.n+.
If an is negative, we have
I Sn, (x) < S (x) < Pn < Pn,+1;
it has thus been shewn that if ISn,(x) < Pn, then also ISn+ (x) (   < Pn+.
Now IF'(x)l is, everywhere in (0, 1), less than (1 + i), and therefore the
theorem  Sn (x)  P < P, follows by induction. A fortiori Sn (x) I is, for every
value of n and x, < P.
The numerically greatest value of fn+l (x) in the interval (cn(-l), cn(S)) is at
some point on the left of the middle point of the interval, and that value is
1   1   an S)
consequently <. 2 I+. 2n+l since the length of the interval is less than
1/2n+1. Also, as has been shewn above, an(s) < P, and therefore
I fn+l W | < 22+3;
and hence, since the terms of the series f (x) +f, (x) +... are numerically
less than the corresponding terms of the absolutely convergent series
P P           P
25    27   +  27+3 + * 
it follows that the series f, (x) +f (x) +... is uniformly convergent in the
interval (0, 1). It follows that the function F(x) defined as the sum-function
of the series F, (x) + f, (x) + f, (MU) +... is a continuous function.
In order to prove that the function F (x) is everywhere differentiable,
we shall shew that it satisfies the conditions stated in the theorem of ~ 398.
We have first of all to shew that the series f'(x) +f,'(x) +... is
convergent for all values of x in (0, 1). In case, for any value of x, all
the numbers So, (x), S+, (x),..., from and after some value of n, have all the
same sign, say the positive sign, we have
a(s)
m+ (X)- nt (X) +-2m+l
where m is the value of n in question. Also
(sq)
Sm+2 (x) < S.+ (x)+ 2m+2
with similar inequalities involving higher indices. From these inequalities,
we find
(sO)  (SO)       (Sp)
a     a~         a~       P
*  m     +.   m+p-1 < P
Sm+p(X> - Sm W   (-  m  +   + +       +~ = 
2     +1~  21A"t-  2m+p!::- 21a) 




630             Functions defined by sequences              [cIn. VI
and since m may be taken so great that P/2m is arbitrarily small, we -see
that n may be so chosen that S+p - SX (x) is arbitrarily small, whatever
positive integral value p may have. It has thus been shewn that, in the case
considered, the series is convergent.
It may happen that S,, (x) is zero, owing to x being at a point of division
a,,(); in this case all the functions fj /(x) with higher indices vanish, and
therefore all the functions S, (x) vanish, from and after the particular value
of in. It may happen that S,,(x) vanishes, owing to x being a point of
invariability of Fn (x); in this case Sn+ (X) may vanish if x is an extreme
of fn+i (x), and then x is a point of division as), and all the functions S, (x)
for indices m > n vanish. Thus if, for any value of x, S,, (x), S,+, (x) both
vanish, then So (x) vanishes for all values of in _ n. If Sn (x) vanishes, but
not S,,- (x) or Sn+l (x), x is a point of invariability of Fn (x), and
I       I     2P
S+1 W    2    2 i + 2n+1  2n+1 +
and the same reasoning is applicable as before. Let us next suppose that the
functions Sn (X) are never all of the same sign, from and after any value n,
and that for some values of n they vanish; let n,, 2,... be the values of n for
which Sn (x) has a change of sign, for example, let Smn (x) be negative or zero,
and  n,+i (x) be positive, and Sn2 () positive or zero, and Sn,+ (x) negative,
and so on. If S,, (x) is negative, we have
(s1)
-  a,
Sn+ (S) =   () +  n, a."
where                   1  a    - (2l+l + 1),
and since ca,, is negative, we have
I      P  P
Sn,+ (X) < 2n + 21 + < 2n,
account being taken of the fact that the fluctuation of F,,' (x) in the interval
in which x lies is -< 21. If S,, (c) is zero, so that x is a point of invariability
of Fn, (x), we have
Snl+l (X) - 2nl+1 (  2n~+1) <2n
In any case we find that
P    P     P
Sn,+p (x) < Snl+ (X) + 2  <  - +  2+  '
where                    p= 1,2,... n2-n.
Similarly, we find that
l ()        if   ()=0;
[ &2+1 (x) <, if Sn2(x)>= 0;




428, 429] A differentiable everywhere-oscillating function   631


and if Sn2 (x) > 0, we have
P     P    P
Sn2+p (X) < Sn2+1 (X) +22+1  2n2  2,+
for                     p = 1, 2, 3,... n - n2.
It is seen from these results that Sn (x) I becomes arbitrarily small for all
sufficiently great values of n, and thus lim S, (x) =. It has now been
2n=00
shewn that in every case the series
F0o (x) + i (x) + f2 () +...
converges for each value of x in the interval (0, 1).
429. It must next be proved that, if e be an arbitrarily chosen positive
number, then, for a given x, a number 8 > 0 can be found, such that, for each
value of h numerically less than 8, and for which x + h is in the interval
(0, 1), there exists an integer m, variable with h, and not less than a prescribed
integer in', such that the three numbers
F. (x + h) - F. (x) _  (   R, (x + h)   R, (x)
h                       h    '     h
are all numerically less than c; R. (x) denoting the remainder of the series
which represents F (x), that is, F (x)- Fm-, (x).
The case may be left out of account in which x coincides with one of the
points of division of (0, 1); for the function F(x) is then represented by a
finite series, and is differentiable, since f'n+p (c())- 0, for p > 1.
Let e, m' be fixed, and let us consider a point x in (0, 1); then a
number n _m' can be so determined that
P    1                            1
2n-2 <, and IS+p (x) - S'.+ (x) <,
where p, q are any positive integers. For any value of h, such that x + h
falls within the interval (c^-1), c(), the number m can be determined. Let
h be positive, and determine n1 so that x < c)  x + h  c(") < c(); then it
can be shewn that n, + 2 is a suitable value for m. We have
fnl+l+p \Cn>+l] --.n+l+ ((s)) P)
and             I fn1+l+p (c,  ) l< 2l+p, for p    1.
The point c(S) is in general between x and x + h, and therefore it determines
n1+1
two segments, kI, k2, where
(s = Cx )  -     + h = c(l) + k2 (
We have therefore
P                     P
I i+1,+i (x) I <2 +-1+1, I{L++2 (x ) | < 2nh+- l+2




632


Functions defined by sequences


[CH. VI


and so on; and from these inequalities we find that
7   '1      1          Pke1
Rnl+2 () I <  Pk {2n+2 + 2n+3 +**)< 2nl+1 
and similarly that
R,,1+2 ( + A) < 2L,+
Since ki, k2 are less than h, we have
Rnj+2 (x) <__ Rn, (x + h)  P
n+1+ < e, and,<               < e
h       2h                    -<2n- <,+  E
It has thus been shewn that m = ni + 2 is a value of m which satisfies the
required condition. The case in which h is negative can be treated in the
same manner.
We have now to prove that
F,+l (x + h) -  ni+ (x   ))    < e
--------- T   --------  -~ Sn+2 (X)  <   6.
We see that
Fn,+, (x + ah) -F,+,+ (-) -  + (x)
h
7    ---------  -~^Sn+2 ($)
{F  (x + h)-Fn,(x) -  1+ () Sn+l ( +) h))-f (x)       + (x
and if x, x + h are points in (cl), c(^) the absolute value of the first term on
the right-hand side is not greater than 1/2%n.
We consider therefore
fn,+l (x + h) -fn,+l (x) f  + ().
h
From the construction for fn,+1 (x), we have
fnl+l (x + h) - f, (l)  a()
A         D 2n,+l'
since x, x + h are in the interval ((S-l) c(S)). Let us take the case in which
Fl (x) increases from c(s-1) to c(); then, for any point x in the interval
between these two points, we have
(s)
fn1+4 ( + h) - fn/+ (x)  a(
h          - 2n'+l'
We shall find also a lower limit for this incrementary ratio. The point x is
such that the ordinate of a() (c(S-1) C()n+l is below the x-axis, and if, for that
point, the differential coefficient is negative, we have
fn,+l ( + A) -fn,+    (     x). 
= / —l — -l (X).




429]   A differentiable everywhere-oscillating function


633


Let the sides of the rectilinear polygon which was employed in the
construction of a((s- (c  (s)f)l+l be denoted by
ni n] nro', rl',... r 2n+l - 1, r7'2l, l  +1  - 2 1  * * * 2'  S i, So,
where ra' is equal and parallel to s,'. On r2', produced beyond (r,', r2'), take a
segment equal to r2'; then this segment is equal and parallel to s', and the
line joining the end of this segment with (s2, S3/) is parallel to r3, and will cut
r,' in a point p,. But s' is parallel to r2, and passes through (s2', s'); therefore
this segment is the prolongation of s3', and is consequently inclined to the xa(s)
axis at an angle whose tangent is -3 2. Hence, for a point between
(S) and p,, for which the ordinate is positive, we have
fnl+l (x + h) - fnl+l (xc) > _3 
h               2n,+li'
a(s)
But the greatest value off'n1+1 (x), in this case, is   and therefore
fn,+l (x + h) - fn1~, (x),     _( h          >f 2nn+l (-)-  n 
If a point p2 on r2' be determined, by making a similar construction with
r3' instead of r', then, for every point on the arc pA, p,, except p2,
~nl~l ( +      ~) - n,+ (x)> _ 4 as,
h               2n+i'
(S)
But the maximum value of the differential coefficient is, in this case, - 2 
therefore also in this case,
fn,+ (x + h) - fn,1+ () f  ()4 
+.        2n,/1' ~
This condition holds for every point on the curve which has a positive
ordinate. It holds also for points with a negative ordinate; because for such
points with a negative differential coefficient the relation
f,+l (x + hA)-/fl,+, (x))
h --       ' = n+l (W)
holds; and for points where the differential coefficient is positive, the
expression on the left-hand is positive, and that on the right-hand is
negative.
It has now been established that
2(S)                        a(s)
r  n.  4an,   f'1+, (x + h) - f,,,+ (x) <  %
n+lW 2n h,+1..          - 2n,+i '




634


Functions defined by sequences


[CH. VI


and it has already been proved that
Fn, (x + h) - F,, (x)       0
F  (  t-F    = F',F (x) +  w, where 1 _ 0 _ -1.
h                  2
We now see that
0     ans)  F1,+1 (x + h) - Fn, (x) _  _ 0nl (x) + 2 a' + 2,+s n
F+(x) +   0-4 as) < -x                             0    as,F!l+l (x 4- h) - Fnll (x)   ( ()<, 
and henceSn+2                                 < 
h                   j
since     a^ < P, and P/2nl-2 < 1e, and 10/2n < P/2n < e.
It has now been established that the function F(x) has at every point a
finite differential coefficient which is the sum of the convergent series
F0' (x) + f' () ++f ( +....
Lastly, it must be proved that F(x) has an everywhere-dense set of
maxima and minima.
It has been shewn that, in every interval (c-), cI)), the function FI(x)
has a new maximum and a new minimum, and that the length of the interval
is less than 1/28. If x, is a maximum of Fn (x), we have
F,, (Xo)= F.+i (xo), and Fn' (O) = Fn+1 (xo) = 0.
Moreover fn+ (x) is negative in the neighbourhood of the point x0, and
therefore F,,+ (Ox + h) - F,+, (xo) is negative or zero, provided  h is less
than some number kl. It thus appears that Fi+, (x) has also a maximum at
x0. If x0 is a point of invariability of Fn (x), it is no longer one for Fn+i (x),
and cannot be a point of invariability of all the functions with higher indices.
If X0 is a limiting point of a line of invariability, F,,+ (x) will have a maximum
or a minimum, or else a point of inflexion at x0. In every case Fn+l (x) will
have a maximum and a minimum in every line of invariability of Fn (x). For
any given interval, as small as we please, n can be determined so great that
the interval contains one of the intervals (c( -) c1) ) in its interior, and all
-1 Un-1
the functions Fn (x), Fn+, (x),... have maxima in this interval; and it follows
that F(x) also has maxima therein.
It may be remarked that F'(x), although definite at every point, has
discontinuities of the second kind at an everywhere-dense set of points. At
every point of continuity, this differential coefficient must vanish (see ~ 223).
The function F' (x) is not integrable in accordance with Riemann's definition.




CHAPTER VII.


TRIGONOMETRICAL SERIES.
430. THE theory of the representation of functions of a real variable
by means of series of cosines and sines of multiples of the variable is of
the highest importance, not only on account of the fact that such mode
of representation is at present an indispensable tool in the various branches
of Mathematical Physics, but also because this theory has exercised the
most far-reaching influence upon the development of modern Mathematical
Analysis. Historically, the questions which have arisen in connection with
this theory have influenced the development of the theory of functions of a
real variable to an extent which is comparable with the degree in which
the theory of functions in general has been affected by the theory of
power series. The theory of sets of points, which led later to the abstract
theory of aggregates, arose directly from questions connected with trigonometrical series. The precise formulation by Riemann of the conception of
the definite integral, the gradual development of the modern notion of a
function as existent independently of any special mode of representation
by an analytical expression, are further examples of the results of the
study of the properties of these series upon Mathematical Analysis.
It is a significant fact that the theory of this mode of representation
of a function had its origin in the attempt to investigate the form of a
stretched string in a state of vibration. The problem of the expansion of
the reciprocal of the distance between two planets in a series of cosines
of multiples of the angle between their radii vectores led to an independent
development'* of the theory of trigonometrical series.   The discussions
which arose in connection with the first of these problems were, however,
of much greater importance in the history of the development of the theory
of functions; they form the first stage in the development of what is known
as the theory of Fourier's series, in intimate connection with which the
modern theory of functions of real variables had its origin.
* The importance of this fact has been emphasized by H. Burkhardt in his work "Entwickelungen nach oscillirenden Functionen," published as a Jahresbericht der deutschen
Mathematiker-Vereinigung, vol. x, 1901 and later.




636


Trigonometrical series


[OH. VII


THE PROBLEM OF VIBRATING STRINGS.?2y = 2a2ly
431. The first general solution of the differential equation  a~= ~
at2   ax2
which determines the form of a string vibrating transversely, was given by
d'Alembert* in the form y =f (x + at) + b (x - at). He further shewed that,
if x= 0, x = 1, represent the fixed ends of the string, the form of the string
at any time t is representable by y =f(at + x) -f(at - x), where the function
f(z) is subject to the condition f(z) =f(21 + z). D'Alembert was thus led
to the search for analytical expressions which remain unaltered when 21 is
added to the argument.  In a second memoir, d'Alembert observed that
the motion is determinate if the values of y and  Y be assigned at some
3at
fixed time. Thus, in modern notation, if y=fi(x), a=f2(x), for t=0,
at
then for all values of x between 0 and 1,
f () -/(- ) =fi (),
/(^) V+(- ) =   fA2(x) dx;
a.
it follows that (x) is determined for all values of x between 1 and -1, and
thence, by means of the condition f(z) =f(21 + z), for all values of x.
The treatment of the same problem which was shortly afterwards given
by Eulert was in form of a similar character to that of d'Alembert, but
the difference of meaning assigned by these writers to the word "function"
was of fundamental importance in the controversy which afterwards arose
between the two mathematicians in relation to this problem. D'Alembert
understood by a function y=f(x), a single analytical expression, whereas
Euler employed the same expression and notation to denote an arbitrarily
given graph. Both, however, held the view that two analytical expressions
which are equal for values of the variable in a given interval must also be
equal for values of the variable outside that interval. D'Alembert argued
that Euler's mode of determination of the function in the solution of the
problem presupposes that y can be expressed in terms of x and t by means
of a single analytical expression, and that thus an undue restriction is
imposed upon the modes of vibration of the string. For example, in the
case in which the initial figure of the string is polygonal, d'Alembert
regarded the solution of the problem as impossible. The general effect of
the controversy is to exhibit on the one hand the narrowness of the restriction of the conception of a function as held by d'Alembert, to functions


* Memoirs of the Berlin Academy, 1747, p. 214.
t Memoirs of the Berlin Academy, 1748, p. 69.




431, 432]


The problem of vibrating strings


637


possessing at every point differential coefficients of all orders, and on the
other hand the looseness of the conception of Euler that the ordinary
methods of the Calculus are applicable without restriction to quite arbitrary
functions.
432. The formal solution of the problem by means of trigonometrical
series was given by Daniel Bernoulli* in a memoir in which he shewed that
the differential equation and also the boundary conditions of the problem of
the vibrating string, for the case in which there are no initial velocities, are
formally satisfied by assuming.rx   7rat. 2rx     27rat. 37rx    3rat
y = a sin   cos -   + a sin    cos      + a3 sin    cos g   +....
He asserted that this represents the most general solution of the problem,
and that the solutions of d'Alenbert and Euler must therefore be contained
in it. In a later memoir, he considered the case of a massless string loaded
with n masses vibrating transversely, and indicated an indefinite increase in
the number n.   A criticism  of Bernoulli's theory was published immediately
afterwards by Euler, who pointed out that a consequence of Bernoulli's
formula was that every arbitrarily assigned function of a variable x could be
represented by a series of sines a, sin x + a2 sin 2x + as sin 3x +.... This
appeared to Euler to be a reductio ad absurdum, since such a series could
represent only a function which is odd and periodic; the notion that a
function could be capable of representation by a certain analytical expression
only in a limited interval being contrary to established opinion at that time.
Bernoulli's solution was consequently regarded by Euler as lacking in
generality.  A considerable controversyt took place on the subject between
Bernoulli and d'Alembert.
This problem, together with the related problem of the propagation of
plane waves in air, was next taken up by Lagrange+, who obtained Euler's
results by the method of starting with a finite number of masses fixed at
intervals on a massless string, and then proceeding to the limit when the
number of masses becomes indefinitely great. In the course of his analysis
Lagrange came near to the determination of the form of the coefficients
in the expansion of a function in a series of sines of multiples of the
*Memoirs of the Berlin Academy, 1753.
+ For a detailed history of these controversies, see Burkhardt's Bericht, vol. i. The early
history of the theory of trigonometrical series is given by Riemann in his memoir "Ueber
die Darstellbarkeit einer Function durch eine trigonometrische Reihe," Math. Werke, p. 227. For
the general history of the theory of these series see Sachs, "Versuch einer Geschichte der
Darstellung willkiirlicher Functionen einer Variabeln durch trigonometrische Reihen," Schlbmilch's Zeitschrift, vol. xxv, supplement, and Pulletin des sc. math., ser. 2, vol. iv, 1880; also
Gibson "On the History of the Fourier Series," Proceedings of the Edinburgh Math. Soc.,
vol. xi, p. 137.. Miscellanea Taurinensia, vols. I, I, III.




638


Trigonometrical series


[CH. VII


argument. The defect of Lagrange's method lies in the lack of any investigation of the validity of the process of passing to the limit; no restrictions
upon the nature of the arbitrary functions were recognized by him as
necessary. The remarks made by Euler, d'Alembert and Bernoulli in the
course of the discussion of Lagrange's work failed to elucidate the difficulties
connected with this point, and no generally accepted theoretical views
emerged from the lengthy controversies, the general course of which has
been indicated.
The difficulties felt by the mathematicians of this period in regard to the
generality of the representation of a function by a trigonometrical series arose
in large measure from their restricted conception of the nature of a function.
To them it was conceivable that a function given by a continuous curve
might be so representable, but since they regarded a function obtained by
piecing two or more such curves together, not as one function, but as several
different functions, it seemed to them impossible that such a broken curve
could be represented by one trigonometrical series; a separate series seemed
to be required for each separate portion of the given composite curve.
Moreover, the idea was unfamiliar that a particular mode of representation
of a function need only be valid for some restricted range of values of the
abscissa; and thus only a periodic curve was regarded as capable of being
represented by means of a periodic series.
SPECIAL CASES OF TRIGONOMETRICAL SERIES.
433. Independently of the discussions of the problem of vibrating
strings and of other physical problems, a number of trigonometrical series
representing special functions of a simple character were obtained by Euler
d'Alembert and Bernoulli. The methods employed by these writers for this
purpose are of a character which fails to satisfy the requirements now
regarded as necessary for the establishment of such results; moreover, in
many cases the ranges of values of the variable for which the representations
of the functions by the series are valid were not assigned.
For example, the series
sin x - 1 sin 2x + ~  sin 3x -  sin 4x +....
cosx-   cos2x +  cos 3- - cos 4x-...
were obtained by Euler*, as representing ~x, wr72 i2 respectively; the
range of values of x (-7r, r) for which these representations are valid was
however not given by Euler, who appeared to regard them as valid for all
values of x.  These series were obtained by integration of the series


* Petrop. N. Comm. 1754-55, and Petrop. N. Acta, 1789.




432-434]      Special cases of trigonometrical series             639
cosx +cos2x+ cos 3+..., the sum    of which was maintained by Euler
to be - ~.
0 1
By D. Bernoulli* the series $ - sin nx was obtained as a representation of
n=1 1,
- (r - x), and the range of values of x (0, 27) for which this representation is
valid was assigned.  It was also observed that the sum  of the series is
discontinuous for x = 0, 27r, 4r,.... The following series were also obtained
by Bernoulli, and the ranges of the validity of the equations were assigned:I0           1 1     1  I
Cos nx=- 7r2 -7rx+x2
-=1 ns2      6     2     4
100     1       1    
= -s1i nx   = r2X -  7rcos n+ =  x lo
co           1      1       1       11
4n~l AB? d   490W  12  2 12    48
col         I        1
- Cosnx= log
n=ln         2   2(1- cos x)
It was remarked by Bernoulli that the sums of these series have discontinuities at x = 0, 27r, 47r,....
The following results among others obtained by Euler may here be
mentioned:
1          )  cos (2r + 1) x
2=o         2r+1I
1      ~       sin (2r + 1) x
rrX = Z (- 1)I2
r=O           (2r  1)2 
1     i87r 2 -  )X2=   cos((2r+1)x
8   47        r=O        (2r+l) +
The true range of validity of these equations will be given later.
LATER HISTORY OF THE THEORY.
434. No further advance was made in the subject until 1807, when
Fourier, in a memoir on the Theory of Heat presentedt to the French
Academy, laid down the proposition that an arbitrary function given
graphically by means of a curve, which may be broken by (ordinary) discontinuities, is capable of representation by means of a single trigonometrical
series. This theorem  is said to have been received by Lagrange with
astonishment and incredulity.
* Petrop. N. Comm. 1772.
t Bulletin des sciences de la soc. philomathique, vol. I, p. 122.




640


Trigonometrical series


[CH. VII


Fourier shewed, in a variety of special cases, that a function f(x) is
representable for values of x between - r and,r, by the series
- aO + (a, cos x + b 1sin x) + (a2 cos 2x + b2 sin 2x) +...
where     a=       f(x) cos nx dx, bn= -    f(x) sin nr dx,
7  - 7r                  Vj - 7
ao=J-    f(x) dx.
Fourier's results in connection with this subject are best studied in the
collected form in which they appear in his Theorie de la Chaleur, published
in 1822. Trigonometrical series of the above form, in which the coefficients
are determined as above, are known as Fourier's series. It should, however,
be remarked that Fourier also studied other trigonometrical series, in which
the cosines and sines do not proceed by integral multiples of the argument.
These latter series will not be considered in this work.
Although Fourier attained to correct views as to thie nature of the
convergence of the infinite series he employed, he did not give any complete
general proof that the series in the general case actually converges to the
value of the function; he indicates* however on general lines a process of
verification of such convergence which was not actually carried out until
Dirichlet took up the subject.
435. An attempt to prove Fourier's theorem was made by Poisson, who
started with the formulat
1 - h2
_ _('1) - 2h cos (x - x') + h2
-=  1   ff(X') dx'+ -  hn   f(') cos n (x- x') dx',
-IT           fT/' n l -- T r
which holds provided -1 < h < 1.
Poisson proceeded to shew that as h approaches the limit 1, the integral
on the left-hand side of the equation approaches the limit f(x), and
argued that f(x) is represented by the series obtained by putting
h = 1, on the right-hand side.   Apart from   the questions connected
with the limit of the integral on the left-hand side, the conclusion is invalid
unless it is shewn that the series obtained by putting h=1, is convergent. In
accordance with a known theorem, given by Abel, for power series (see ~ 356),
in case the power series is convergent for h= 1, it converges to the limit of
the sum of the series for values of h which are < 1, as h approaches the value
1; but no conclusion can be made as to whether the series is really convergent,
* See the Theorie de la~chaleur, chap. ix, especially ~ 423.
t Journ. de l'cole polyt. cah. 19, 1823, p. 404. See also his Theorie analytique de la chaleur.




434-436]     Formal expression of Fourier's series


641


or not, when h=1. A direct investigation of its convergence would be
required to make the proof a valid one.
Two proofs of the validity of the representation were given by Cauchy;
one at least of these is certainly invalid in its original form. Both of them
depend upon the theory of functions of a complex variable, and will consequently not be discussed here.  An example of an invalid proof of a
similar character to one of Cauchy's and also to Poisson's is the proof given in
Thomson and Tait's Natural Philosophy.
In 1829, Dirichlet* gave a proof that, in an extensive class of cases,
Fourier's series actually converges to the value of the function. His proof,
the first rigid one, was based upon a recognition of the distinction between
absolutely convergent, and conditionally convergent, series. Since a Fourier's
series, when convergent, is not necessarily absolutely convergent, it is imnpossible to obtain a proof of the convergence from the law according to which
the terms diminish, as Cauchy had attempted to do. As Dirichlet's proof,
apart from its historical interest, still repays a careful study on account of the
light it throws upon the mode of convergence of the series, it will be given
below, with some modifications and extensions which arise from later advances
in the Theory of Functions.
THE FORMAL EXPRESSION OF FOURIER'S SERIES.
436. Let f(x) denote a limited function, defined for the interval (0, 1) of
the variable x. A finite trigonometrical series of the form
rx. 2w7r. Sx                (n- 1) x
a1 sin -+ a sin.    +    a. sin --  +... +   an_, sin 
can be so determined that its value is equal to that of the function f(x) at
1 21 31     n- 11 
each of the points X =, -    -,.... It must be shewn that the
-     n'  n    n
coefficients a,, a2... an- can be determined by means of the linear equations
1{l. 7r. 27r. (n-1)7
f - =     a sin n -     +... + an_-, sin - 
n\n        n        n                   n
/21\. 27.r      2.27r. 2(n-1)7r
f ( -  = a, sin - + a sin -— +... + an-i sin 
\n I       n          n                    n
* Crelle's Journal, vol. iv, "Sur la convergence des series trigonometriques, qui servent a
representer une fonction arbitraire entre des limites donnees." See also his memoir in Dove and
Moser's Repertorium fir Physik, vol. I, 1837. Papers by Dircksen, Crelle's Journal, vol. iv, and
by Bessel, Astron. Nachrichten, vol. xvi, are on similar lines to those of Dirichlet, but of inferior
importance.


H.


41




642                    Trigonometrical series                  [CH. VII......'.................  e,,.....,,......~...................,
rl\. rr. 2r7r. r(n-l)7r
f ( -= a, sin - + a sin-   +... +   an_ sin
n fl/        n          n                   11
__n-ll                        2(')      +..a
f  fl11        (n - 1)7r. 2_(n-_) _ r             (n-1)(n-1)7r.al sin       +a2 sin          +   + a,,-, sin
Multiply the expressions on the two sides of these equations by
s.  r. 2s7r    (n- 1)S7r
sin    s sin  s... sin
n'      n'           n
respectively, and add the expressions on each side together. It can easily be
verified that
r7r. sin.2r7r. 2s'r            (n - )r7r   (n- l)s7r
sin- sin- + sin     sm    - +... + sin       sin          = 0,
n     n n     n      n               n           n
provided r and s are unequal integers not greater than n- 1; and also it can
be shewn that
sW 7     STT.        (n2-l)S7   1
sin2 -r + sin2 -  +. +  sin2 ( — 1) s 
nf       n               n       2
Using these two identities, we have at once
2[f /i. S7T    f21)i 2s7r           /n~I- Ir. s(n-1)7r
as =   -    sin m   +f    s. sin          sin     -.
n           n      n      n              n           n 
and thus the coefficients in the series have been determined so that the
series satisfies the prescribed condition.  Let us now assume that the
function f(x) is integrable in accordance with Riemann's definition, and let
the number n be indefinitely increased. The limit of the expression for as is
2 fl.S7/
then seen to be,j f( ') sin   d'. This process suggests the possibility
that the function f(x) may be represented by the infinite series. 7rX     27rx. S7rx
a, sin  +   -      + a. sin- +... +  +...
where the coefficients ca are given by
2 jf(-') s    Si7X d,
=  f(x ) sin -- dx,
for points x within the interval (0, 1). It will be observed that the series
cannot possibly represent the function at the point x = 0, unless f(0)= 0; nor
at the point x=l, unless f(l)=0. This limiting process is entirely insufficient
to shew either that the infinite series converges at all, or that, when it does
converge, its limiting sum is at any point equal to the value of the function
J (x) at that point.




436, 437]     Formal expression of Fourier's series              643
It will later be shewn by various methods that, for extensive classes of
functions, the series
2        S7rX at /. synXc,
1 =1       J0 
actually converges to the value f(x), for values of x within the interval (0, 1),
at which f(x) is continuous. This series is known as Fourier's sine series.
Let us now assume that the function f(x) sin 7  is represented within
the interval (0, I) by the Fourier's sine series.
This series is, in the present case, of the form
-2w. S7TrO   sx,. l  rx''  7rx'
7  sin  - I f(x) sin -  sin  dx',
I1=1           - -0 
which is equivalent to
1 X. S7rX s,'  s    -1 7rX'     S+l 7rX',  sin     f() cos ---      -cos          dx',
ls=1  l                               L
or to
1        X. 7rfl,I * s + s     7i   1       V-}x l        s7rx
sin-, jf(x') dx'+      sin        -sin -) cos -              dx
7rX
and this by hypothesis represents the function f (x) sin I-.
It thus appears that, on the assumptions made, the function f(x) is
represented by the series
l (l l   7/,2    s  8 f           S7TX,
1I f(') dax'+ - S cos-    ( ') cos 7' dx............(2).
0 o         s l =1    o
This series (2) is of the form
7rX       2v7rx SxTrX
0 +, c    +os + 2cos  +... + A cos  +...
and is known as Fourier's cosine series.
The cosine series, unlike the sine series, may possibly converge to the
values f(O), f(l), for x = 0,  respectively, when these functional values are
not necessarily zero.
437. Assuming for the present that the function f(x) may be represented
for the points of the interval (0, 1) by 'either of these series-(1) and (2), we
proceed to consider some obvious properties of the series themselves. The
sum of the sine series (1) has, for the point - x, the same value, with the
opposite sign, as for the point x. If then we. suppose that the function
f(x) is defined not only for the interval (0, 1), but for the interval (- 1, 1),
it appears that the series can represent the function for the whole interval
(-1, 1), only in case f(- x) = -f(x); that is, in case the function f(x) be odd.
41-2




644


Trigonometrical series


[CH. VII


Further, the series (1) is unaltered by adding to x any multiple of 21, and
thus the series, considered as existent for all values of x, defines a periodic
function, of period 21. If f(x) be defined for all values of x, it can only be
represented by the series, for all such values of x, provided f(x) is periodic
and of period 21, and also f(x) = -  (- x); otherwise the representation of
the function by the series is valid only for the interval (0, 1).
The cosine series (2) is unaltered by changing x into -x; therefore the
series represents the function f(x) for the interval (-, 1), only when
f(- x) =f(x), i.e. when f(x) is an even function. The cosine series, like the
sine series, considered as existent for all values of x, is periodic, and of
period 21; therefore the series can represent a function f(x), defined for all
values of x, only when f () is periodic with period 21, and alsof (x) = f(- x).
It is thus seen that, if the function f(x) be defined for the interval
(-, 1), it is in general not represented by either the sine or the cosine
series for the whole of that interval, although it may be represented by both
the series for the interval (0, 1). For the part of the function f(x) in the
interval (- 1, 0) is in general independent of the part in the interval (0, 1);
neither of the relations f(- x)= -f(x), f(- x)=f(x) being in general
satisfied. In fact there is in general no relation between the values of a
function, defined for the interval (- I, 1), at the two points - x, x.
It is however possible to obtain, from the series (1) and (2), a series containing both sines and cosines, such as to represent the function f(x) for the
whole interval (-1, 1). The function I {f (x) +f (- x)} is an even function,
defined for the whole interval (- 1, 1), and in accordance with the assumptions, representable for that interval by the series
87rX f~rX
21  { ((x') +f(- X') dx'  E cos   j  [f (') +f(- x')] cos -   dx.
Again, the function ~ {f(x) -f(- x)} is an odd function, defined for the
whole interval (-, 1), and is accordingly representable by
I" ~).S7TX [, Sin87T',rX
2    sinr   [f (x')-f(-x')] sin S  dx'.
l8=1     1  0
By addition of the two series, we find the series
I CI       1 f I     S7r
f (x') d' + Y         cos  (-   ')f(')dx'............ (3),
which is of the form
1CIO    at Cos  7TX\ rX  (       27rx       27rx\
0 + ai cos -- + 3 sin  ) + a2 cos -- + f2 sin -  +...
as representing the function f(x) for the interval (-1, 1). This series (3) is
known as Fourier's series, the sine and cosine series being regarded as the




437, 438]    Formal expression of Fourier's series


645


particular cases of it which arise when f(-x) = - f (x), or f(- x) =f(x)
respectively.
438. With certain assumptions, the form of the series (3) may be
obtained directly. Let it be assumed that a function f(x), defined for the
interval (-, 1), can be represented by the series
ao+ a( cos -+      1  sin  +... +  n cos — +   in sin  +...,
in the sense that this series converges to f(x) for each value of x in the interval. If it be further assumed that the convergence of the series is uniform
in the interval, and thus thatf(x) is continuous and consequently integrable
in the interval (- 1, 1), we may submit the series to a term by term integration,
n~r x        f7rx
even when it is multiplied by cos —, or by sin —. It would be sufficient
for our purpose to assume that the series, without being necessarily uniformly
convergent, is still such that a term by term integration is admissible, in
accordance with the criteria investigated in ~ 383, and that f(x) is integrable
in (-1, 1).
Making use of the fundamental property of circular functions, represented
ft cos norX cos nf'lrX
by the formula   COS rx  COS n'TIXr x
l    sin  1 sin  I dx=O, where n and n' are any unequal
f.  1Z r'x        2'n7rx
integers, and observing that   cosa   dx=     sin2    dx = 1, we thus
find that
a o=-C   /         1  f. /   nrrx  1'                   n17 7x
= i f() dx', an=       f(x') cos    dx',,8n=     f(x') sin   dx'.
Therefore we have, for the interval (-, 1), as the series representing f (x),
2 If (x') dx' + i    cos 7     f (x') cos   dxa
21 J" -l   n)+=l -            1 -l n1       7 qrX fn1 >x!      }
+ 2- sin -j-   /(-')s in ds',
nrr
or              f    2 (') dx' +  1l f (x') cos 7 (x - ') dx'.
2        1 -    =    - 1 -rrQX
If we replace 1 by x, no essential difference will be made in the formula;
thus there is no loss of generality in taking the interval (- rw, 7r) to be the
interval in which f (x) is defined, and for which it is represented by
1i,,  f r  () dx' + E,  f (') Cos n (f-d') d+......... (4).
Ts  epre n () wl  n=l ta T  - 
This expression (4) will be taken as the standard form of Fourier's series.




646


Trigonometrical series


[cH. VII


439. The form of the series (4) having now been obtained by purely
tentative processes, the reverse course will be adopted of taking the series
itself as the starting-point, and subjecting it to an examination with a view
of discovering under what circumstances it is convergent for all or some of
the values of x in the interval (- r, 7r), and of determining the value to
which it converges in case it is convergent.
In order that the series (4) may exist, whether it converge or not, it
is necessary that the coefficients  1    f(x') dx', -    f (') cos nx'dx',
- |  f(x') sin nx'dx' should have definite meanings. Until quite recently it
has consequently been assumed that f(x) is either limited in the interval
(- r, 7r) and integrable in accordance with the definition of Riemann, or else
that f(x) is unlimited in that interval, but possesses an improper integral in
accordance with one of the definitions which have been given of improper
integrals. The recent extension of the definition of integration, by Lebesgue,
to the case of functions which are not necessarily integrable in accordance
with Riemann's definition leads to a corresponding extension of the domain
of Fourier's series. It has been proposed by Lebesgue* to assign to the
series (4) the name Fourier's series, in every case in which f(x) is a summable
function in the interval (- r, 7r), whether the summable function be limited
in the interval or not, provided that, when the function is unlimited, it be
still integrable. Since only those improper integrals which are absolutely
convergent are included in Lebesgue's definition, there remains the case in
which the coefficients exist only as non-absolutely convergent integrals;
in this case Lebesgue has proposed to name the series generalized Fourier's
series. This terminology will be here adopted.
440. It being assumed that f(x), as defined for the interval (- r, 7r), is
such that the coefficients in the series (4) have definite meanings, it is easy
to express the sum of a finite number of terms of the series as a definite
integral.
Since I +cos + cos 20 +... + cos n= s     2Y       we see that S,)+,,
2 sin I 0
the sum of the first 2n + 1 terms of the series
2    f (x ) dc' +   ~ cos nx  f(') cos nsxdx'
1 -r-r        n=l (7       -r
+ - sin nx   f (/') sin nx'dx',
Vr       7T
* Lebesgue's treatment of the series is contained in a memoir " Sur les s6ries trigonometriques,"
Annales sc. de l'cole normale, superieure, ser. 3, vol. xx, 1903; in a memoir " Sur la convergence
des S6ries de Fourier," Math. Annalen, vol. LXIV, 1905; and in the Leqons sur les series trigonometriques, 1906.




439, 440]    Formal expression of Fourier's series             647
1 of     sin (2n +1) 2
is given by    S    =,,+l -            - f(    x'
7r r_            X- X
2 sin —
If we change the independent variable in this integral from x' to z, where
x'= x + 2z, and write 2n + 1  m, the expression becomes
sin mz
Sm  f- |      f(fx + 2z).  d2.
77- -(jr+^ -  ( x tz  sin z
In order that the series may converge at a point x, it is necessary that
Sm should converge to a definite limit, as the odd integer mn is indefinitely
increased.
It was first shewn by Dirichlet that, for an important class of functions
f(x), S, converges to the value f(x) at every point x in the interior of the
interval (- r, 7r) at which f(x) is continuous; that, at a point of ordinary
discontinuity of f(x), it converges to the value [{f (x + 0) +f(x- 0)}, which
is not of course necessarily equal to f(x); and that at the points x=7r or
- r, it converges to the value - [f(r - 0) +f (- 7 + 0)}.
An account will be given in the present Chapter of the investigations,
by various writers, which have as their object the determination of sufficient
conditions to be satisfied by the function f (x) in order that the series may
converge either throughout the whole interval, or at particular points of that
interval. It will appear that the convergence or non-convergence of the
series, at a particular point x, really depends only upon the nature of the
function in an arbitrarily small neighbourhood of that point, and is independent of' the general character of the function throughout the interval;
this general character being limited only by the necessity for the existence
of the coefficients of the series. These investigations have resulted in the
discovery of sufficient conditions of considerable width, which suffice for the
convergence of the series either at particular points, or generally throughout
the interval for which the function is defined. The necessary and sufficient
conditions for the convergence of the series at a point of the interval, or
throughout any portion of the interval, have not been obtained. This is not
surprising, in view of the very general character of the problem; and indeed
it is not improbable that no such necessary and sufficient conditions may be
obtainable. It is possible that the mere fact of the convergence of the
series at a point characterizes the nature of the function in the neighbourhood of that point in a manner incapable of reduction to any other form; so
that although the characteristics of various sub-classes of the functions
satisfying this condition may be obtained, as has in fact been done, yet
the whole class of such functions has no property capable of being stated in
any form different from the mere statement of the fact of the convergence of




- 0


C


648                   Trigonometrical series                [CH. VII
the series. It will appear that there exist functions, even continuous
functions, for which the series fails to converge at every point of the interval
belonging to an everywhere-dense set of points.
Recent investigations, an account of which will be given, shew that the
coefficients of Fourier's series have important properties which are related to
the functional values, independently of whether the series converges or not;
so that a divergent Fourier's series may be employed, in accordance with
recent ideas, for the representation of the function in a certain sense.
PARTICULAR CASES OF FOURIER'S SERIES.
441. Before proceeding to the theoretical investigations relating to the
convergence and the properties of Fourier's series, it will be instructive to
consider some simple cases of the use of the series. It will be assumed
that, for the functions employed, the series corresponding to a function f(x)
converges at every point to theve value  f(x+)+f(x-0)}.
If we employ the sine series to represent the function defined, for the
interval (0, 7r), by y= I (r - x), we find on evaluation that
2 
-   I (r - x) sin nxdx -
and thus the series is of the form
sin x + 1 sin 2x + 3 sin 3x +... + - sin nx +....
0n
The function defined for all values of x by
y = sin x +  sin 2x +... +- sinnx +...
is represented graphically in the figure. The function is discontinuous at
the points 0, 27r, 47r,... -27r, - 4r,...; the functional value being zero at all
those points. It is seen that the series represents the function  (r - x), not
only for the interval (0, 7r), but for the interval (0, 27r), except at the points
x =, x = 27r, where the sum of the series is zero. For the interval (- 27r 0)




440, 441]     Particular cases of Fourier's series


649


the function represented by the series is - 1 (wr + x), except at the ends of the
interval.
This series may be employed to illustrate some important points connected with the convergence of the series in the neighbourhood of the point
x = 0, at which the function represented by the series is discontinuous. To
this end we shall examine the series by a method employed by Fourier*, and
further developed by Knesert.
1.
Denoting sin x + - sin 2x +... +  sin nx, by s, (x), we have
ds.(x)                            sin (n +- 1) x
(     x) = cos x + cos 2x +... + cosnx =sn (  12
dx                                2 sinx 2x
therefore     (x)=        n       dx- 
n  sin (    ) +    i )     * r       ( -22 sin I  X:)
x      i= si ni (i + 1) x  x  _ x     2sin     )
-     x                o             2xs
_" f+r)X sin zx       r 
==( I 2   -ndz-4-+I (X).
On integrating by parts, we find that
i(  x -2sinl x cos(n+ )- )   fcos(n+ )x 4sin2      - x 2cos x
2x sin -x    n +-      J     n +    '    4x sin2.
- 2 sin1 x   4 sin2 X -x2 Cos -x
The expressions X- 2 sin.       -     x   — x  both become indefi2x sin x '      X2 sin2 ~x
nitely great, as x increases up to 27r; but if x be confined to the interval (0, b)
where 0 < b < 27r, they are both limited functions. It follows, since
cos (n +    <x)  1,
that a positive number A can be determined, independent of n and x, such
that [ I (x) < A/(n + -), provided x is in the interval (0, b). Hence it
appears that I (x) has the limit zero, when n is indefinitely increased,
whether x varies with n or not; in fact II (x)l is arbitrarily small for
sufficiently great values of n.
We have now
/(n+) x sin z        OA
Sn () -  (x) =           d —    +,
provide d 0  x  b; where 0 is  siuch that - 1  < 0 1.
d  snl( )-S(X)}2     sin4x ( dcif 0<x <
* Theorie de la chaleur, chap. in, ~ 3.
t Grunert's Archiv, ser. 3, vol. vII, 1904. See also Bocher's "Introduction to the theory of
Fourier's series," Annals of Mathematics, ser. 2, vol. vII, where numerical details are worked out.




650                    Trigonometrical series                 [CH. VII
X'r
and therefore sn (x) - s (x) has maxima and minima at the points x = 
t+ 2
where X = 1, 2, 3,....
It can now be shewn that, for sufficiently large values of n, at least,
27r \       27r \      47r        47r          r /       67r
Sn                      s n  --  +~)'
<2n + i      2n + 1 'n 2n       + I-  2n+1) '  nSo l + 1 )s \2n  ) I
are alternately positive and negative, the first of these differences being
positive.
f sin z
We have     s   dz=    sin    -       +    -... +  --  )    dz
o  z      o     \z   z +7    -+ 27r      z+ X-   7r
= u1   3-.. (- 1)+ +,
where u1,    2,... u  are all positive, and u1 > U2 > u3... > UA. Also
1    fV.         2
A <   -       sin zdz < -;
X-I7ro            X-l7r
hence lim ux = 0.
)A=oo
Further, it is well known that limlr sn   dz, which is the improper
sin z 
integral J      dz, is equal to r'; it follows that ul, u, - Ztu, u1- U2 + U3,..
20A
are alternately greater and less than - 7r. Since 2n  1 is arbitrarily small,
for sufficiently great values of n, it thus appears that the differences
/2X7        ( 2X7r 
s8 T2n +- 1 I  2n + 1)
are alternately positive and negative for X = 1, 2, 3,...; and that for X= 1,
the difference is positive.
O0
It thus appears that, for large values of n, the form of the curve y = sn (x)
in the neighbourhood of the origin is as in the figure; consisting of a wave-form
passing above and below the straight lines which represent y=s (x). The






441]          Particular cases of Fourier's series              651
27r
first maximum on the right of the point x  0, has as its abscissa x =  + 
2n+l '
and its height above the point whose coordinates are  n+1, s (2n + 
is nearly| sin dz -  r, which is nearly independent of the value of n.
The first minimum on the right of the point x =0, has for its abscissa x  2 
2n+ 1'
sn d below the corresponding
and is at a depth approximately 7rt-  -    dz below the corresponding
point of the locus y=s (x).
As n is continually increased, the abscissae of the maxima and minima
of Sn (x)- s () become indefinitely small, the magnitudes of these maxima
and minima remaining however nearly unaltered. If a particular value of x
be chosen, n can be so determined that I s (x) - s (x) I is arbitrarily small, for
such value of n, and for all greater values; but if a particular value of n be
27rchosen, there is always a value of x, viz. 2n+   for which s(x) - s () is
2n 1+
nearly equal to  si     dz -r.
o.
M
The graphs y = so (x), as n becomes indefinitely great, tend to the form
given in the figure, which consists of the continuous curve formed by the
straight lines of length 2  -  dz (> rr), through the points x = 0, 27r,
- 2r,..., and of the series of oblique straight lines which belong to the
curve y=s(x). The graph of the curve y=s (x)= lim sn (x) has been
n= 00
already given. The limit of the graphs of the curves y= s,, (x), and the
graph of the limit of Sn (x) differ in the respect that, for the abscissae




652


Trigonometrical series


[CH. VII


x=0, 27r, - 27r,..., the former contains the continuous straight lines of
si 7T
length 2     n z dz, whereas the latter contains only the single points on the
x-axis. Corresponding to any point P on the straight line LM through the
origin, it is possible to determine an indefinite number of pairs of values of
x and n, such that the distance of P from the point whose coordinates are
x, s, (x), is less than an arbitrarily prescribed positive number e. Thus the
double limit   lim  Sn (x) is indeterminate between the limits of inden= oo, X=O
termiv sin Z        sin z
terminacy                 sin  dz, -    sin z.
o0  z         oJ  z
By letting n increase indefinitely, and x at the same time diminish to
zero, in such a manner that nx has a as its limit, where a is any fixed positive
number not exceeding wr, we have as the particular value of  lim  sn (x),
n=oo, X=0
or lim s, (), the number fsin z dz. It will be observed that the repeated
n=oo   W                  Z
limit lim lim sn (x) has the value ~7r, or - -Tr, according as x approaches its
x=O n=oo
limit from  the positive, or from  the negative side.  The repeated limit
lim lim sn (x) has the value zero.
n=oo x=O
The distinction between the graph y = s (x), which represents the series,
and the limit to which the graphs y = Sn (x) tend, is clear, if it be borne in
mind that the limit y = s (x) is obtained by the special mode of first fixing a
value of x, and then letting n increase indefinitely; thus, for example,
s (0)= lim  n (0)= 0; whereas, as we have seen,   lim  sn (x) is indetern==00                                    n=co, x=0
minate between limits which have been found above. The difficulty which
has been frequently felt in understanding how a series, of which the terms are
continuous, such as the series here considered, can represent a function which is
not continuous, will be removed if the point just explained be fully grasped*,
that the sum  of the series at a point x is always taken to mean the limit
obtained by first fixing the abscissa x, and then afterwards making the number
of terms increase indefinitely.
It has already been shewn in ~ 343, that the points x = 0, 27r, - 2r,...,
must be points of non-uniform continuity of the series; moreover, other
examples have been already given, in which the peaks of the approximation
curves y-= s (x) remain of finite height above the curve y = s (x), however
great n may be. That the portions of the limit of the graphs y = s (x), in
* Some criticisms of Dirichlet's determination of the sum of a Fourier's series at a point of
discontinuity, made by Schlafli, Crelle's Journal, vol. LXXII, and by Du Bois Reymond, Math.
Annalen, vol. vII, where it is maintained that the sum of the series is indeterminate, are due to
a lack of appreciation of this point.




441, 442]     Particular cases of Fourier's series


653


the present case, have a length greater than 7r, the measure of discontinuity
of the function, was pointed out by Willard Gibbs*.
The expression  sin z2A
The expression   +    in  dz -  Tr + 2 +  which has been found above,
J0 o   z         22n + 1
for s, (x) - s (x), provided 0 < x < b < 27r, may be employed to shew that the
series converges uniformly in any interval (a, b), such that 0 < a < b < 27r.
(n+i)x sin z
For, by choosing n so great that     s    dz, for x?_ a, differs from Or by
less than a prescribed number ~e, which is possible on account of the con2A
vergence of the integral, and further choosing n so. great that 2+1 < ge, it
is seen that n can be chosen so great that, for the chosen value of n, and for
all greater values, I sn (X) - s (x) < e, for all values of x in the interval (a, b).
This expresses the fact that the series converges uniformly in the interval
(a, b). It is clear that the smaller a is taken, the greater must be the value
of n, so that (n +  ) a may be sufficiently large to satisfy the requirement
that f |         dz - -Ir < 2e; and that this value of n increases indefinitely
as a is indefinitely diminished. This is a verification of the fact that the
convergence of the series is non-uniform at the point x = 0.
442. Let f(x) be defined for the interval (0, 7r), by the specifications
f(x) = c, for 0 O   x < 7rT,
and                   f(x) =- c, for 17r  x s  r.
To find the sine series for this function, we have
ff(x) sin nxd     c  sin nxdx - c   sin nxdx
*dO,            or
= - (cos nr - 2 cos 1n7r + 1).
n               2
This integral vanishes if n is odd, and also if n is a multiple of 4, but i
n = 4m + 2, it has the value 4c/n. The series is therefore
8c
- ( sin 2x + I sin 6x + 1o sin 10x +...).
For unrestricted values of x, this series represents the ordinates of the
series of straight lines in the next figure, except that it vanishes at the points
O   T, 2,  17,...
It will be observed that, if the meaning off(x) be extended, so that it denotes
* See an interesting discussion on this subject in Nature, vol. LVIII, 1898, pp. 544, 569,
vol. LIX, 1899, pp. 200, 271, 319, 606, vol. Lx, pp. 52, 100, in which Michelson, Love, Gibbs, Baker
and Poincare took part.




654


Trigonometrical series


[ClI. VII


the sum of the sine series for every value of x for which that sum is continuous, then at the point rr, for example,
f(r +0) =, f(  - 0)= -c,


and the series represents at the point rT, the arithmetic mean
values.


of these two


-27. -T -.2


7-.


A  A
7


27r


In a similar manner, we find that the function defined for the interval
(0, rr) as before, is represented, for the interval (0, rr), by the cosine series
4c
(cos x - ~ cos 3x +   cos 5x-...).
7r


-27


-_               -r1


1  7r
I.I


For unrestricted values of x, the series represents the ordinates of the
straight lines in the figure, except that its sum vanishes at the points
-r', -7r, 7r..
443. Let         f(x) x, for 0 r x  Lrr,
and                 f(x) = rr - x, for 7r _7 x  7rT.




442, 443]     Particular cases of Fourier's series


655


In this case we find that
fo              fo *
jf(x) sin nxdx =J  x sin nxdx   (r - x) sin nxdx
2.  l
sin 1jr.
n2
Hence the sine series is
-(sin  - 2 sin 3x +  sin 5-...).
-'                52
- \            / 
For general values of x, the series represents the ordinates of the line in
the figure. The broken line in the interval (- r-, 7r) is repeated indefinitely
in both directions.
The cosine series which represents the same function for the interval
(0, 7r), will be found to be
r- -(cos 2x+ -  cos 6x+  cos 10+....
4    7\ V      32        52           /
This series represents for general values of x, the ordinates of the-line in
the following figure. As before, the-broken line in the interval (- 7, 7r) is
to be repeated indefinitely in both directions.
-7r            0               7r




656


Trigonometrical series


[CH. VII


EXAMPLES.
1. Prove that the series
sin x -  sin 2x + ~ sin 3x -  sin 4x +...
represents, for the interior of the interval (- r, 7r), the function ig.
For any value of x which is not a multiple of r, the series represents 2 (x- 2k7r), where
k is a positive or negative integer so chosen that x- 2k7r lies between wr and - r. The
sum of the series vanishes for all values of x which are multiples of 7.
2. Prove that the series
cos -   cos 2x +  cos 3x -  cos 4 +...
represents the function -1sr2 - x2, for the interval (- r, 7r).
3. Prove that


r- =       sin x  +  sin 3x + 1 sin 5x +...,
r = cos x- I cos 3x +  Ccos 5x-...,


for O<x<7r;
for - 7Ir<x<~7r.
2    2 ~


4. Prove that
1          1        1                 1       1
- r = sin x - -2 sin 3x + - sin 5x -...,  for -  r -T< X. - r.
4          32      52                 2       -2
5. Prove that
e- = 2  2n (1 - 5krcos nwr) sin nx, fo


2      n       ky
ekx =- 2    + 2 (9  - ekrl cos n7r) sin nx,  foi
ek —    2k ~ e cos n7rekx =     -                   cos nS,   for
k]7r     n=2l k2 +n2
6. Prove that
wr sin kx  sin x  2 sin 2x  3 sin 3x
2  sin k   12-k2   22-k2 + 32-k2 -         where
7r cos kx  1   k cosx  k cos 2x
2 sin k7r  2k -     2 + k2-22 -           where
not being integral.
7. Prove that
7r sinhkx  sin x  2 sin 2x  3 sin 3x
2 sinhn k7r12+k2   22+k2 +32 + k2    *
wr coshk(7r -x)  1    cosx   cos 2.  cos3x
2k   sinh kr     2k   12+k2  2 +k2 + 3 +k2 


0<x <r,
r O::Sr.
0 - x< 7r,, O _ X _T;
where 0 x <r;
where 0   x  r.


k i


DIRICHLET'S INTEGRAL.
444. It has been shewn in ~ 440, that the sum S2n+, of the first 2n + 1
terms of Fourier's series, is of the form
1 ' (r(-x)        sin mz
82n+1 =-        f (x + 2z). —  dz,
w he  -(7   + x)q    sin z
where                       m = 2n + 1.




443, 444]               Dirichlet's integral                     657
The function f(x) has hitherto been defined for the interval (- 7r, wr)
only; we may now, as a matter of convenience, extend the definition of f(x)
to values of x which do not lie in this interval.
We shall assume that, for all values of x which are not multiples of wr,
the functional values are defined so that, for all such values of x, the
condition f(x + 27r)==f(x) is satisfied. In case f(7r) =f(-7r), we may, if
we please, suppose that f/(r) ==f(~ kqr) for all integral values of k; but in
any case it is indifferent what values are assigned to the function at the
points + k7r. Whenever f(7r) and f(-7r) are unequal, one at least of the
points wr,- 7r is certainly a point of discontinuity off(x),
for        f (r + 0) =f(-7r + 0), and f(r - O) =f(-r - 0),
in accordance with our extended definition off(x); it is accordingly impossible
that both of the sets of conditions
f(7r) =f (7r + 0) =f(/r - 0),
f(- 7r)=f(- r   0) =f(- 7r -0)
can be satisfied, if f (7r) and f (- r) be unequal.
Let us now write f(x + 2z) = F (z), then the function F(z) is periodic in
z, with period nr, except possibly for those values of z for which x + 2z is a
multiple of 7r; this exception is, however, immaterial, since the values of
integrals are unaffected by alteration of the functional values at single
points.
We may now write Sm in the form
1ft' P, sin mz
Sm =       F(z) s-in  dz,
r -j,, r     sin z
which may also be written in the form
1 f~n    sin mz    I   rl     SiF l M
7r j o    sin z       Jr oT    sin z
It thus appears that the investigation of the limiting value of Sm turns upon
the existence and value of the limit, when m is indefinitely increased, of an
integral of the form
[  sin rmz
f F    (z) sin z dz;
Joy      sin z
the second integral in the expression for Sm being essentially of the same
form.
This integral is known as Dirichlet's integral, the term being, however,
generally applied to the more general form,
sin mz d,
F(wh) i 
eJo su0in z
where a is such that          0 < a ^ 'Mr.


H.


42




658                   Trigonometrical series                [OH. VII
The term Dirichlet's integral is also frequently applied to the closely
allied expression
sin z
| F(z)-    dz, where 0<a <   t7r.
It will be shewn that, with certain assumptions as to the nature of the
function f(x), the integrals
^rcl, o  sin zrn
fsin mz
|F (-z)-__- dz,
J 1r(o  ()sin z
converge to the values
F (+ 0),   F   0),
2rF     )' 2-F(-o),
respectively, as m is increased indefinitely; so that S, converges to the value
{/( + 0) +f( - 0)}.
At a point of continuity of f(x), this agrees with f(x).  When x = -7rr,
S, converges, under certain assumptions, to the value
i {/f( - o) +f(- 7r +  )}.
DIRICHLET'S INVESTIGATION OF FOURIER'S SERIES.
445. As a preliminary to the consideration of Dirichlet's integral, some
properties of the integral
or
2 sin mz z
dz
J   sin z
are required.
We have
2 sin mz d    2 
2 sin   dz =do [ + 2 cos 2z + 2 cos 4z +... + 2 cos 2nz] dz =
sin z                                               2
If we divide the interval (0, ), of integration, into the portions
(   T\  /r, 27r\       /rr  r+ 17r),        r    )
0)  m,                              ' '" -   -m  '  ~.  )'
sin mz
we see that, in these portions, the integrand sinz has alternately positive and
negative signs; thus if we write




444, 445]   Dirichlet's investigation of Fourier's series?:29


7rrTr...
p2 sin mz
pn-= (- I) -i   dz,
Tr  '
m"  -  -, 
= po - pl + p2 +.. + (- 1)-1 p.-i +... + (- l )np,


we have


where all the p's are positive.
In p,.-, sin nz is always of the same sign, and.    is monotone! and
sin z
decreases as z increases, hence. 1.        2       r.- 1 r
p,.-l < (-1~ —   __ ).I      sin mz. dza < m cosec r ---;
sin -         -1r  m
sin
m


and similarly
It follows that
For pn, we have
hence


2   r7r
r-i > -cosec.
m   m
2   rrr
pr-1 >- cosec- > pr.
m   m
1   n7r  1
- cosec -> pn > -
m   m 'm
2   n7r
pn-: >- cosec -  > pn
r'  ~ 9


It follows that, if 2p < n,
'<Po-p.+p,
2-< po- pI + P2-...+ p2p,


and


'iT
- ' PO -plS-p -~2  pzp-l

Let us suppose that the function F(z) has a finite upper limit, for the
values of z such that 0 < z 5 jrr, and further, that it is in the whole interval
positive and monotone, and such that it never increases as z increases;
it is consequently an integrable function.
In the integral
F (z) sin  z dz,
o o i   sinmrs z
where a < l7r, we proceed to divide the interval of integration as in the case of
gr
f2 sin mz.  d.dz
smin z
42-2




660                   Trigonometrical series                 [OH. VII
into alternately positive and negative portions; thus if
rTi
Sq (-   l y-     F. -   dz,
sin mnz dz
= (-1)       F (z) sn   dz,
Sq  vin /tsin Jq   m
where q is a positive integer such that
q7r    aq+ 17r
< a_
m         m
we have
in F  z) dz = So- sl + S2-... + (- 1)r-l,_l +... + (- 1)q Sq,
Jf       ) sinl z
where so, s, s,... sq are all positive. On account of the supposition made as
regards F(z), we have
pr-lF (r   -) > s._l > pr-_lF   ), and Sq- pqF ()
From these inequalities it follows that
Sr-p, _1 - F   > prF () > Sr;
and this holds for all values of r from 1 to q.
We have consequently the result, that
f[a -  \sin mz
Uj F(z) sin      dz,
sin z
is less than          So - si + 2-... - sp_ + sp,
and greater than  So - s + s2 -... — s2_, where 2p  q.
From these inequalities, with the help of those obtained above, we have
U>(p (-pl) F-      (p2 -3)F ((pi2p-2- p2p-1) F (2p 
>F    Y^r) (PO - Pl + p.-p3 +  + p~p-2 -p2p-l);
also          U< poF (+ 0)-F (2    )(p,-p,+p-... -p2).
On using the theorems which have been proved relating to the p's,
we obtain
{(          )m   2 pp)
and           U<p mi.-              )}F(2 +  PP)      m)




445]        Dirichlet's investigation of Fourier's series          661
where, in accordance with the supposition made, p is any integer such that
qno
2p <q < _
Now let m and p both increase indefinitely, but in such a way that
2p has the limit zero. Since
m
2p7r
2       1     m
P2p <         < 
2 sinp 2pr  p7r sin 2p 7r 
m            m
we see that 2p has zero for its limit; and hence
rrfprrnzrr7
F (P) (- p2P)
has 2 F (+ 0) for its limit. Again
7r
7r     7r   2   m
P<+ p<       - p < + -—;
'2 J7r.rm
sin -
m
in
r   2
and hence po has a limiting value not greater than - + -. It follows that
7r
po {F (+ O)-.F (a w)} +    + p2p) F (Pm2)
has for its limit the value  -F (+ 0).
It has been proved that U lies between two numbers, each of which
has 7rF(+ 0) for limit, when m and p are indefinitely increased in such a way
2
that 2p has the limit zero; hence the limit of
U7  JF(z) in     dz
0      sin z
is                               F (+0),
where a is such that           0 < a - 27r.
It follows, as a corollary from this theorem, that
sin mzd
fF(z)       dz
sin z
has the limit zero, when m is indefinitely increased; where a,. / are two
fixed numbers, such that 0 < <   <a  7r.




662


Trigonometrical series


[CH.! VIi


446. — We have now seen that, if F(z) be a limited and positive function
which never increases as z increases from 0 to 7r, the integral
7r   
F (z) sin mz 
LF(z). —dz
Jo      sin z
converges to the value - F (+ 0), as m is increased indefinitely. The function
F(z) may be freed from the condition that it must be positive in the
whole interval. For if F(7i) is negative, we may apply the theorem   to
the function C +F(z), where the constant C is chosen so that
2( 2)
is positive; thus
'r
oC +F(z) sin ~mz dz
Jo           51fsin z
converges to the limit       - {C + F (+ 0).
2
7T. 2 sin mz
Now                         C   sn.-   dz
0 s    in z
7r
converges to the limit -C; hence   F(z).-n   dz converges to    (+0),;2        hsin z                        2
where F(z) is not restricted to be positive.
Again, the theorem holds for a function F (z) which is monotone and
never diminishes; for we can apply the theorem to the monotone function
-F(z) which never increases.
The theorem has now been established, that if F (z) be any limited,
monotone function, defined for the interval (0, Opr), then
2(z) sin mz 
F(z)       d
Jo      sin z
converges, as the odd integer m is increased indefinitely, to the value  F (+ 0).
The theorem also holds if the upper limit of the integral be any fixed
number a, such that 0< a <   r.
It has been shewn in ~ 195, that any function with limited total
fluctuation is expressible as the difference of two monotone functions. Hence
the results.which have been established can be immediately extended to.the
case of functions of this class.  We have, therefore, the theorem that, if




446, 447]  Dirichlet's investigation of Fourier's series


663


F (z) be a function defined for the interval (0, -or), and with limited total
fluctuation, then the integrals
[aC^),sin mz d  P      sin mz
F (z).n    dz,  F (z).    dz,
Jo    sill z            sin 
where                 O < a  17rr, 0 < a < /3 < 'r,
converge, as the odd integer m is increased indefinitely, to the values - F (+ 0), 0
respectively.
If we apply this result to the two integrals contained in the expression
for Sn, the sum of the first 2n + 1 terms in Fourier's series, we obtain the
theorem that, if f(x) be a function with limited total fluctuation, defined
for the interval (- r, 7r), the sum of 2n + 1 terms of the series
2 f _f(x') dx' +   -  cos nx ff(') cos nx'dx'
27  r    /          7        -—,
+   sin nx   f (x) sin nx' dx
converges, as n is indefinitely increased, to the value, {f(   0 + o) +f(x.- 0)}.
It will be remembered that a function with limited total fluctuation is
essentially integrable, in accordance with Riemann's definition; and that it
can have discontinuities of the first kind only, so that at every point the
functional limits f(x + 0), f (x- 0) exist.
In the case x = + 7r, the limit to which the sum of the series converges is
{f(7r - 0) +/(- 7r + 0)}.
At a point x of continuity of the function f(x), the limiting sum of the
series is f(x); at a point of discontinuity of f (x), the limiting sum of the
series agrees with the value of the function at the point only if
f() =-2 I{/(X + 0) +f(x - 0)}.
At the points r, - r, the limiting sum of the series agrees with the value of
the function only if f(7r), or f(- r), is equal to
1 {/(7 -0) +/(- 7 + 0)}.
447. It is now clear in what sense the given function f (x) is represented
by the corresponding Fourier's series.  The representation is necessarily
complete for all points at which the function is continuous, with the possible
exception of the end-points + 7r, which cannot both be points of continuity
of the extended function, unless f (7r) =f(- r). At a point of discontinuity,
or at an end-point + 7r, the series represents the function only if the functional
value is properly chosen in relation to the functional limits at the point; in




664


Trigonometrical series


[CH. VII


the case of the end-points these functional limits are those of the periodic
function obtained by extension of the given function beyond the domain for
which it was at first defined, this extension being such thatf(x) =f(x + 2vr),
as explained in ~ 444.
The functions with limited total fluctuation include, as a particular case,
functions which satisfy the following conditions:(1) The function is continuous in its domain at every point, with the
exception of a finite number of points at which it may have ordinary discontinuities, (2) the domain may be divided into a finite number of parts, such
that in any one of them the function is monotone; or in accordance with the
more usual but less exact expression, the function has only a finite number of
maxima and minima in its domain.
These conditions are known as Dirichlet's conditions, and his proof, in
its original form, applied to the case only of functions which satisfy these
conditions.
448. Dirichlet extended his results to the case in which there are a
finite number of points in the domain (- r, vr) in the neighbourhood of
which If(x) I has no upper limit. In this case the Fourier's series must be
so interpreted that the integrals in the coefficients are the improper integrals
/  (x)dx,    s. nxf() dx,
-7r          - Vs Si
the function being such that these improper integrals exist.  From  our
somewhat more general point of view, we shall suppose that the function
f(x) is such that, when arbitrarily small neighbourhoods of these infinite
singularities are excluded from  the interval (- r,?r), in the remaining
part of the interval f(x) is of limited total fluctuation; and further it
will be assumed that the improper integral
f /(x) dx
-7r
exists, and is absolutely convergent.  Under these conditions, it can be
shewn that the theorems still hold, that the integrals
sinMZ mz
F (z) s      d.,  for 0 < a </   r,
sin z
and                   F (z) sinm  d, for 0 < a < 17r.
J   sin  z   o 
converge to zero, and to  F(+ 0), respectively, as m is increased indefinitely.
If, between a and f, there is a point c in whose neighbourhood I F(z) has
no upper limit,
F (z) sin mz d
sin z




447, 448]  Dirichlet's investigation of Fourier's series


665


is interpreted as the limit of
sin mz              sin mz 
F (z)          sindz +  F(z)   d,
where 8, e have, independently of one another, the limit zero; assuming that
such limit exists.
Let 8' < 8, then
[If6-S'c-5' B() rsin mz             d c —6
i   -J  \ F(z) -sn  d: < cosec a   i F (z) dz;
where the expression on the right-hand side is arbitrarily small, on account of
the absolute convergence of the integral of F (z), and is independent of the
value of m.
Now, if F(z) dz converges absolutely at the point c, we can choose 8 so
small that, for every 8' < 8,
rc-S'
cosec a i:  F (z) I dz
Jc-'
is arbitrarily small; hence the integral
F (-z) sinmz dz
J a       sin z
for a fixed m, converges to a definite value, as 8 converges to zero. Similarly
it can be shewn that
7r F ( sin mz 
JF(z)+ sin z
converges to a definite value, as e converges to zero. It has thus been
shewn that
sin mz          -      sin m  mz
F (z) -    dz = lim    F (z) sin   dz
Ja     snz          Jnin Z  8=0  sin z
sin nmz
+ lim    F ().i   dz=    ()+,(m);
e=0 Jsin z
and we have now to shew that, (m), 2^(m) converge to zero as m is
increased indefinitely. It has been already seen that 8 may be so chosen
that, for all values of m,
(z) sinmz
Fz).s     dz- f'(m) <n,
fz a      bSill Z
where v is a fixed arbitrarily small positive number. Now, for a fixed value
of 8, ml may be chosen so great that, if m > mi,
n (z) si   dz <,
Ja        sin z




666


Trigonometrical series


[CH. VII


where g is arbitrarily small; hence, if m _- m,,
I -1 (m) I <?q + C,
and therefore k, (m) converges to the limit zero; similarly #2 (m) converges
to the limit zero.
If, between a and /, there are any finite number of points such as c, we
may divide the domain (a, 3) into a finite number of parts, such that each
part contains only one such point as c, and apply the above result to each of
the integrals which are taken through one such part.
The integral f F(z)sn m. dz can be divided into two parts
() smin 
F(z)s      dz +    F(z) sin    dz,
o    sin z       a       sin z
where a, is so chosen that all the points of infinite discontinuity of F(z) are in
f   sin mz xr
(ac, a); we thus see that j F(z)   x   dz converges to 2 F(+0), when m is
sin z                 2
indefinitely increased.
It has now been shewn that:-if f(x) be such that, when the arbitrarily
small neighbourhoods of a finite number of points at which f (x) I has no upper
limit have been excluded, f(x) becomes a function with limited total fluctuation,
then the Fourier's series
2 |   f(x') dx' + 2     f(x') cos n (x - x') dx'
converges to the value  [f (x + 0) +f (x - 0)}, at every point in (- 7r, 7r), except
at the points of infinite discontinuity of the function, provided the improper
o r
integral   f(x) dx exists, and is absolutely convergent.
APPLICATION OF THE SECOND MEAN VALUE THEOREM.
449. An alternative method of investigation of the limit to which the
sum  of Fourier's series, corresponding to a function with limited total
fluctuation, converges is obtained by employing the second mean value
theorem*. As before, we employ the known result that any function with
limited total fluctuation is expressible as the difference of two functions
each of which is monotone and does not decrease as the variable increases;
and it is therefore sufficient to consider the case of such a monotone function.
* This method was first employed by Bonnet, who used his form of the mean value theorem,
see the Memoires des Savants etrangers of the Belgian Academy, vol. xxIII. The method is
also used by C. Neumann, see his work Ueber die nach fKreis- iKgel- und Cylindesfunctionen
fortschreitenden Reihen; also by Jordan, Cours d'Analyse, vol. n, where it is applied to the case
of functions with limited total fluctuation. The method is employed, and discussed in great
detail, in Dini's work Sopra le Serie di Fourier.




448, 449]   Application of second mean value theorem               667
(1)  We have, as before,
7r
" sin  mz     2
(2) If                   0<a< 3<1,
I, in a e  dzn  i  t  s n  mz dz + t   m.  sin nzndz
which lies between a and            J 
|       P3sin  z  2 m4
s m. dz <- (cosec a + cosec ) < - cosec a;
J asin z     m                    mn
here, in accordance with the second mean value theorem, 7 is some number
which lies between a and 8.
It follows that Jsin    dz converges to zero, as m  is increased inSln 
definitely.
f sin mzdr is, in absolute
In a precisely similar manner, it appears that  s   dz is, in absolute
4
value, < -, and thus the value of the integral converges to zero, as m is
indefinitely increased.
(3) If aO                       id  - 0.
For                       f        sin  d  <  -.


and therefore
'hence, if a -  r,
Now
hence   do
h   J since
since


IJa    w I 'o' h'
lih sin 0      2
lim   d    O _ -<-;
h=ao a 0        C
fsin       2  7r
ia -  dO -- < 2
d f" sin O     sina
d-Ja     do =-, when a>O;
diminishes as aincreases, provided  r; therefore
diminishes as a increases, provided 0 < a < 'n-; therefore


f sin 0 
-- dO = 9,
we have              f: sin^, i
2
if a < 7r, and < -, if a > r.
7,




668                   Trigonometrical series               [CH. VII
From this result, it follows that
(f sin 0   dO  r,  where 0  a< f?.
Ia  0
450. After having established these preliminary theorems, we proceed to
consider
7r
pF2^) sin mz dz,
F(z) sin   dz,
where F(z) is monotone, and does not diminish as z increases.  We have
rIT                     T
~~~~~~~~7r~ ~
C26      m s -in m z    '2 sinmz    Pn/     -Rmz.sin mz
F (z) s.   d dz=F(+0)      -dzj     {F(z)-F(+O)J i      dz
Sill z   sin          z o       sin             Z
C2nz      r     sin mz
+   {F(z) F(+O)} s n     dz,
where, is fixed. On applying the second mean value theorem, we have
7r
fF(iz)5.   dz  - F(+ 0)
sin z
f2 -r.,              ' sin  sin m                      C  srri  m\
=       () sin mz dz + F( + 0)-F (+ 0)}j sfn mdz
Jo       z                            sin z
+^(- o)~j (   lf2 sin m - d,
7r
+       -0 -F(+O... dz,
{, sin z
7r                           z
where, is between /, and, and G(z)= {F() - F(+ 0)} sin  Also G(z)
increases as z increases from 0 to,a.
A' *sin mz              si sin (n             0
Again        G (z)  -- dz =                = G (I) --     dO,
J      z            J    z            m    u
where 0 c S  < /,.
Although it is unnecessary for the purpose of the present investigation,
it may be remarked that we cannot have   = 0, which would involve the
equality
{ G (a) - G (z)n   dz = 0.
in m
This is impossible unless G (z) = G (,p), for all values of z in the interval (0, u).
For let z=-, then
sin mz dz
Jz
-o     {     (")}sinz'     [           Az'++ r\) sin z'
-Jo {     m)    z            ()              z' + 7r
(f([  \ (27w"          sin z' 
~+   1 ^ ^)-^ ---- r  o -z'.......~
m   /2 +27r




449-451]     Uniform  convergence of Fourier's series


669


and on the right-hand side, each integral is positive and less than the
preceding one, the signs being alternate; and such a series cannot vanish.
The number: depends on m, and on the function G (z); it may happen
that as m is increased indefinitely, ~ diminishes indefinitely in such a way
that mn  has a finite limit a. Whether this happens or not, we see from
(3), that J G (z) in  dz does not numerically exceed 7rG (p), and,/ may
z
be so chosen that this is less than the arbitrarily chosen positive number 2e.
Let a fixed value of /u be taken so that this condition is satisfied.
7r.     e( sin,z  i 2 sin rizz
Since    sin   dz,   s     dz both converge to zero, as m is indefinitely
sinsmz    f, sin z
increased, even though, in general depends on m, since, by (2), each
integral is numerically less than - cosec/u, we now see that ml can be so
fixed that
7r|F) sin mz dz(+O)
sin z         20
is numerically < e, if m - ml; and therefore the expression converges to zero,
as m is indefinitely increased. Thus the theorem has been established upon
which the proof of Dirichlet's theorem depends, in the generalized form for
the case of a function with limited total fluctuation.
UNIFORM CONVERGENCE OF FOURIER'S SERIES.
451. It is known that the limiting sum of a series of continuous functions
of a variable is non-uniformly convergent in the neighbourhood of a point of
discontinuity of the sum-function, but that the sum-function is not necessarily
uniformly convergent in an interval in which it is continuous. In the case
of Fourier's series it can be shewn that, if the function f(x) be of limited total
fluctuation in its domain (- 7r, 7r), then the series converges uniformly in its
whole domain, provided f(x) is continuous in its domain, and provided also
(-7r +0) =/(7r-0), so that the function obtained by extending f(x)
beyond the range (- r, 7r), in accordance with the periodic lawf(x) =f(x + 2vr),
is also continuous at the points - r, + r. It can be shewn more generally
that, provided f(x) is of limited total fluctuation in (-7r, 7r), and is continuous in an interval (a, b) contained in (- r, 7r), so that (a, b) contains no
discontinuity of the function in its interior or at its ends, then the series
converges uniformly in (a, b). If f(-7r+ 0), f(r - 0) are unequal, the
points - r, 7r must be included among the points of discontinuity, and




670


Trigonometrical series


I [CH- VII


therefore cannot be end-points of (a, b). It has been shewn in ~ 450, that
sin mz dz                        4
F(z) sin    d-      (+  ) < 2r G (~ )+     [~(U  + O) [  F(+  )]
mn sin ILF \2  o) F(+     ]
<  G ()           [F + m   [I+0o) - F(+ o)] + F --  F(+ O)
Using this inequality, and the corresponding one for the function F(- z), we
have, at any point of (a, b),. 4).
S2+1-f( ) I< (/ cosec +-      cosec J) [If(x + 2). -/(x)
+ f(x - 2~) -f(x) I]
4
+ 4   cosec, [ /f(x + 7r) -f()I +   (- )f(-  )  () |]
-<, cosec, [ jf(x + 2/x) -f(x) if - +-            cosec,,
where A is some fixed number, independent of m, and depending on the
upper limit of If(x)) in the whole interval (- r,; r).
Since f () is continuous in (a, b), and is, by the theorem of ~ 175, therefore also uniformly continuous, a value li of p can be chosen such that, for
every value of x in (a, b), If(x + 2/L) -f(x) |, If(x - 2/) -f(x) I are less than
an arbitrarily prescribed positive number e, provided i /a1. Also a value
/I2 of p/ may be so chosen that ef,2 cosec /2 <  T, where  ' is an anrbitrarily
assigned positive number. Take for, the lesser of the values /A,, /L; then
we see that
A
I+n+ — f() I < X + - cosec t, 
m
for every value of x in (a, b). It follows that, since W7 and m are independent
of x, for all values of n greater than some fixed value n,, S2n+, -f(x) < < 8,
for the whole interval (a, b), where 8 is an arbitrarily chosen number, and n1
depends only on 8. It has now been shewn that S2,n+ converges uniformly to
f(x) in the interval (a, b). The function F(z) has been assumed to be
monotone; for the general case we have only to consider the difference of
two such functions.
The following theorem has now been established:In the case of a function f (x) with limited total fluctuation in (- r", 7r),
the corresponding Fourier's series converges to the value f(x) uniformly in any
interval which contains, neither in its interior nor at an end-point, any point of
discontinuity of the function.
It must be remembered that the point wr is to be reckoned as a point of
discontinuity, unless the conditions f(7r - 0) =f(r)=f (- r + 0) are satisfied.
A similar remark applies to the point - 7r.




451, 452]      Limiting values of the coefficients               671
THE LIMITING VALUES OF THE COEFFICIENTS IN FOURIER'S SERIES.
452. If the function f(x) be limited, and satisfy Dirichlet's conditions,
or in the more general case in which f(x) is a function with limited total
fluctuation in the interval (- r, 7r), an estimate may be found for the upper
limit of the general coefficient in the Fourier's series, as n increases indefinitely.
Let f(x) =f (x) -f2 (x), where f, (x), f2 (x) are monotone functions; then
/r:                         ra.            /:
Ii ) cos        = i (-7r + )   cos ax dx +fi (t -0)  cos nxdx
-7                   a
= - { (- T + 0) sin a -f, (7r - 0) sin na};
therefore   ff (x) cos nxdx   -          +f0)-f (-  r -   0)}
where a is some number between -rr and rt. Since a similar result holds
for f f(x) cos nxdx, we see that the coefficient -    f(x) cos x dx is
numerically less than A, where A is some fixed number.   In a similar
n'
manner it can be shewn that the coefficient of sin nx is numerically less than
B
-, where B is some fixed number.
~n'~~~   ~n
Since a series, of which - is the general term, is divergent, it is seen
that the convergence of Fourier's series is in general not absolute, but
depends upon the variation of sign in the terms.
Next, let us suppose that near the point x =/3, where - wr _/3 g  7r, the
function f(x) has indefinitely great values, and is such that, near the point /,
f(x) =0 (x), where v < 1, and 0 (x) is limited, and has only a finite number
(x - P)
of oscillations, in the neighbourhood of the point /. In this case the integral
itr
/ (x) cos nx dx is an improper integral which is represented by the sum of
i'3-e ^(X),           (X)
I_, (x- ) cos nx dx,      (x/)^ cos nx dx,
i- (X -1Y            j   (X^/ - 
where e has the limit zero. We may consider the portion
+-  cos nx dx




672                    Trigonometrical series                  [CH. VII
of one of these integrals, where a is so chosen that b (x) is monotone in the
interval (/ + e, a). This integral is equivalent to
cos nx,.    ~ / \ l"-3 cos n (y + 3).
( )  (    edXo or to    (a) — dy)
where a' lies between 3 + e and a, and x = y +/. Let ny = z, then
f- cos n (y +)            t (a-J ) Icos z    sin 
dy = n"-         — )   cos n/3 - ~ 7 sin nsi  dz.
e    Y"               n f   n             zv 
Now the two integrals
rnna- 3) COS Z  r f(a-13) sin z
"    d,         27 d
X cos z d    0 in z,
- dz,     -    dz
converge to the values
2 cos vl. r (v)'  2sin1r r (.) respectively.
It may be proved in a similar manner that the integral
I   (Xz )(V cos n z dz
contains a portion of the same character as that already obtained. It thus
appears* that the coefficient of cos nx in the Fourier's series is less in absolute
value than A/nl-1, where A is some fixed number; and a corresponding
result holds as regards the coefficient of sin ix.
453. The following more general theorem will now be established —:If f(x) be a function which is integrable in a  interval (, b), the integral
being either a Riemann integral, or an absolutely convergent improper integral,
in accordance with the definition of de la Vallee-Poussin, or of Harnack, then
f (x)sin nxdx, and ff (x)cos nxdx both converge to the limit zero, as the
positive number n (either integral or not) is indefinitely increased.
All those poi    f  nt of discontinuity of f(x) at which the saltus is n a fixed
positive number k may be enclosed in the interiors of the intervals of a finite
set {3}, of which the sum is arbitrarily small. Let (a, /) denote one of the
intervals of the finite set which is complementary to the set {8}. Sincef(x)
* See Heine's Kugelfunctionen, vol. i, p. 62.
t This theorem is a generalization of a theorem given by Stackel, nearly equivalent in form to
that case of the above which arises when f (x) is continuous in the interval. See Leipzig. Ber.,
vol. LIII, 1901, also Nouvelles Annales, ser. 4, vol. I, 1902.




452, 453]      Limiting values of the coefficients              673
is absolutely integrable in (a, b), the set {8} may be so chosen that the sum
of the integrals of If(x) I taken through the set {8} may be less than an
arbitrarily chosen number. It follows that f(x)sinnxdx taken through
rb
the set {8} is numerically less than C, and therefore f f(x)sinnx dx differs,
Ja
for each value of n, from 2 ff(x) sin nx dx, by less than r; the summation
referring to all the intervals such as (a, /). If e be an arbitrarily chosen
positive number, the interval (a, 3) can, in accordance with the theorem of
~ 185, be divided into a number r, of equal parts, such that the fluctuation of
f(x) is, in each part, less than k + e.
We have now
s+l
ps             s=r-i     s/3-    Ca+-(13 —a)
If(x) sin nx dx  S     a +    -  ) —  r   sin nx dx
as=O        r   ja+_3-a)
s=r-1 ra+ -(3-a)            s+  2;      r |  s y(x~)-  (a+   r   )] sin nx dx,
r
r  f(x) - f (a  s/- aIr
where                      r f(})-   (a+/- ~  J < k +e, for each sub-interval.
From the above identity, we now deduce that
f (x) sin nx dx <- U + (k + e) (3-a),
where U may be taken to denote the upper limit of If(x)l in all the
intervals such as (a, 3).
By addition of this result as applied to all the intervals (a, /3), we have
2t
f f(x) sin nx dx <- U + (k + e) (b - a)
where t denotes the sum of the values of r which correspond to the different
intervals.
It now follows that
r6                  2t
f(x)sinnxdx <    + - U +(k+ -e)(b-a);
Jao,            n
and therefore  lim     f (x) sin nx dx _ ' + (k + e) (b - a).
Since [, e, k are all arbitrarily small, it has therefore been proved that
rb
lim   f (x) sin nx dx = 0.
n=o0 a
The proof is precisely similar, in case cos nx be substituted for sin nx.
H.                                                        43




674


Trigonometrical series


[CH. VII


454. The above theorem is a particular case of the following theorem
due to Lebesgue*:If f (x) be any function, either limited or unlimited which has a Lebesgue
rb                     rb
integral in the interval (a, b), then  f f() sin nx dx, and   f (x)cos nx dx,
converge to the limit zero, as the positive number n (not necessarily integral)
is indefinitely increased.  The theorem  also holds if the interval (a, b) be
replaced by any measurable set of points.
First, let f(x) be a limited function.  The interval (a, b) can be divided
into a finite number of measurable sets, in each of which the fluctuation of
f(x) is less than an arbitrarily chosen positive number e. Let f, (x) be a
function which is constant in each of the sets, and is equal to one of the
values off(x) in that set.  We have, since Ifi(x)-f(x) I < e,
f (x) sin nx dx-     fi (x) sin nx dx < e (b - a).
Let e,, e2,.. ep denote the sets of points in (a, b), for which f, (x) has the
values cl, 2,... p,; then
Pj fi (xC) sin nx dx =  c  s in nnx dx.
Ja                 q=l  J eq
It has been shewn in ~ 83, that the set eq is contained in a finite, or
infinite, set of intervals of measure m (eq) + 7, where ] is an arbitrarily chosen
positive number; it follows that sin nx dx taken through eq, differs from the
value of fsinnxdx taken through the set of intervals which includes eq, by
less than A. Moreover fsin nx dx taken through this finite, or infinite, set of
intervals, converges to zero, as n is indefinitely increased. For, if the intervals
be arranged in descending order of magnitude, the integral is numerically
less than  ' + 2r/n, where the integer r is such that the sum of all intervals
after the first r is less than the arbitrarily chosen positive number. '; therefore, for a sufficiently large value of n, the integral is numerically < 2g, which
is arbitrarily small. It follows that
lim    f(c)sin nx dx < e (b - a) +-    cq;
n=    a                               1
keeping e, and therefore p, fixed, V may be diminished indefinitely, and
therefore
rb
limr   f() sin nx dx    e (b - a).
* Annales sc. de iecole normale, superieure, ser. 3, vol. xx, 1903. The theorem is stated by
Lebesgue for the case a=0, b=27r, of the Fourier's coefficients corresponding to f (x).




454]           Limiting values of the coefficients               675
rb
Since e is arbitrarily small, it follows that  f()sin ndx converges to
zero.
Precisely the same argument would apply if the interval (a, b) were
replaced by any measurable set of points.
Next, let the function f(x) be unlimited; then a set E exists, containing
all those points of (a, b) for which
If(x) > N;
and the number N can be chosen so great that
f,l(x) dx<e,
where e is arbitrarily small. If E1 be the set of points complementary to E,
we have
( f(x) sin nxdx -     )sin f ( d)   ( sin  n xdx;
and it has been proved above that
j f () sin nxdx
converges to zero, as n is indefinitely increased. It follows that
lim   ff(x) sin nxdx <e,,=.=  a
and, since e is arbitrarily small, that
lim  f (x) sin nxdx = 0.
n-= - cO
The substitution of cosnx for sin nx makes no essential difference in the
proof. As before, the substitution of any measurable set of points for the
interval (a, b) makes no essential difference in the proof.
It will be observed that the theorem does not apply to the case in which
f(x) has only a non-absolutely convergent improper integral in (a, b).
If we let a = - r, b = + r, we obtain the following theorem:The coefficients-of cos nx, sin nx in any Fourier's series whatever converge to
zero, when n is indefinitely increased.
This is a property of the coefficients which is entirely independent of the
convergence or non-convergence of the Fourier's series.
It does not necessarily hold for the generalized Fourier's series, as defined
in ~ 439..
43-2




676


Trigonometrical series


[CH. VII


455. The following method for estimating an upper limit to the value
of the integral
AJ    sin z


where


O<a<f/3<,
2'


is due to Schwarz*, and furnishes an alternative proof that the integral
converges to zero, when m is indefinitely increased. It is applicable to the
case in which p (z) is limited, and has a Riemann integral in the interval
(a, /).
Let the interval (a, /3) be divided into the parts
(/ pr7\.  /Pr   p+l  Pr    q71
(a    ' \m ') ' ( \           ' a '
where p, q are positive integers so chosen that
(p- 1) 7       pr q7r q<       + 1l r
m            n m mb         m
In the first interval let
(Z)=    () +      (Z);
in the second interval let
b(z) =     ) +   2+ (2);
in the third interval let
b(z)=    (P +.2      (2.);
in the fourth interval let
(z) =   (P  2) + qr4 (z),
and so on. The given integral I may be written in the form


p7r         pl+ir
I =  p\ r nM sin mz dz+   sin z 
\m  [ja  sin z         sin Jz
m
_p+27r_ r_ _   sinm Tz
+qb [P+ ---  [L sinvZdz +
\m   - i+pr sin 
+.........
f, sin mz
+   +(z) sin  dz.
f   (zsin z


p+-37r
n sin mz 
+f m  sn   dz
mp-+2r Sll Z
n


* See Sachs' "Versuch zur Geschichte der Darstellung willkiirlicher Functionen."  SchlWmilch's Zeitschrift, supplementary vol. xxv, 1880.




455]           Limiting values of the coefficients            677
If we use the notation of ~ 445, this may be written in the form
pfr
/=     - \)      --- s. dz + (-l )P pp}
+ + (P +2) (- 13)1 (p+l -  pp+2) + * 
fJ~ a  in mxsin z
~ =. (z)-    dz,
p7r          pr
sin tz      -     sin  a 
+         )(pp+,- pp+2) +.... +  (z) sin  Z
pTr          pIT
rf sinmz,     m  sin mz
Now                   s             s.-n dz -I  a dz
Ja  smZ      J p-lzr smaln
is numerically less than
si  fI sin mzdz,
sin a J  ln
2                        7r           2      7r
or than - cosec a, which is less than -. Also pp<  < -  and p > pri;
m                       ma           m sin a  ma
hence we have,. p,,. sin mz,    7 rC  /                        2r. 7rc
dz <     +0 + c    p    plp~       +<,
1- \ + (z) sin  d  <   -- + C - - pp + pp+ - pp+2 +...< -- 
I ~JaYj)sin z   ma      ma                       ma
where c is the greatest of the numbers  (pr), -  (  ).... 
Again,               # r (z) sin mz I ' ar (z),
where -r (z) denotes the fluctuation of b (z) in the rth interval; hence
|   sin mz,    a (z)     r   af (z)d
~  z)-. —dz              = --- z^^\  z- dz,
sin z        sin z    2     z
where a (z), in any interval
(t7r  r + 17 r
is equal to the fluctuation of b (z) in that interval.




678                   Trigonometrical series                [CH. II
We have now: obtained Schwarz's theorem, that the value of the integral!f  z) sin m  dz,for 0 <a< /3  r,
sin z
is numerically less than
2c7r 7r    (Z).
mna  2-a   Z
c denoting the upper limit of l (z) I in the interval (a, 9), and c (z) the
fluctuation of + (z) in the interval
(r7r  r+17r)
and having this constant value throughout that interval.
The function  b (z) being finite and integrable, in accordance with
Riemann's definition, we have
f| -'k dz < - j  (z) cz <  a- S   (Z) A,
where A,. is the length of the rth interval. The sum Ero (z) A. becomes, in
accordance with Riemann's condition of integrability, arbitrarily small, by
making m large enough; and thus we see that
if    sin mz
(z)-.-   dz
eJ ()  sin z
has the limit zero, when rn is indefinitely increased.
SUFFICIENT CONDITIONS OF CONVERGENCE OF FOURIER'S SERIES
AT A POINT.
456. The function f(x) will be throughout assumed to be integrable in
accordance with Lebesgue's definition, through every finite interval, so that
the corresponding Fourier's series exists, whether it be convergent or not. It
has been shewn in ~ 444, that, at a point x, the sum of the first 2n + 1 terms
of the series is expressed by
2n+i - 1   If(x+ 2z) +f(x - 2)] sin (2n + 1) z dz.
rJo                          sin z
The function f(x + 2z) +f(x - 2z) being integrable in the interval (0, 7r),
of z,    +f (      ~ + ~ 2z) +f(x - 2z)] is certainly integrable in any interval, consin z
tained in (0, 7), which does not contain the point z 'O,:at which sin z
vanishes. It follows therefore, by applying the general theorem of ~ 454, that
71  r     +.  \ / _      sin (2n + 1) dz
[f( + 2z) +f (x - 2z)]     sin      dzn + 1)
wr,                          sin z




455-457] Sufficient conditions of convergence at a point 679


converges to zero, as n is indefinitely increased; where /t is any fixed number
such that 0 <  ~ -tr. The investigation of the limiting value of S2X,+ is
therefore reduced to that of
I  1~~~~s1 in (2n +l)z
fr [f (x + 2z) +f(x - 2z)] sin 
sin z
where,u is arbitrarily small, and subject to the condition 0 < u <  Otr.  For a
given value of x, this integral involves only the functional values off(x) in the
neighbourhood (x-2/u, x + 2/)) of the point x; and consequently the convergence of the series, at this given point x, depends only on the nature
of the function in the arbitrarily small neighbourhood of that point. The
following theorem has thus been established:If f(x) be any function, limited or not, which has a Lebesgue integral in
the interval (- r, 7r), the convergence of the corresponding Fourier's series at
any particular point depends only on the nature of the function f(x) in the
arbitrarily small neighbourhood of that point.
This theorem was first established* by Riemann, for the case of a function
integrable in accordance with his definition. The theorem of Dirichlet and
its extension, which have been investigated above, contain sufficient conditions of convergence of the series in the whole interval (- r, Vr). It
appears however from the preceding theorem that the convergence or nonconvergence of the series at a particular point depends only on the nature of
the function in an arbitrarily small neighbourhood of that particular point,
and is independent of the general character of the function in the whole interval
(-7r, 7r), this character being limited only by the condition that f(x) must
be integrable through the interval, in accordance with Lebesgue's definition,
so that the Fourier's coefficients exist as absolutely convergent integrals.
Various conditions will accordingly be found, the fulfilment of any one of
which is sufficient to ensure the convergence of the series at a particular
point.
457. The theorem that    F(z)   n   dz, where 0< O < /3 -, consin z
verges to the limit zero, when m is indefinitely increased, was deduced
I  sin mz
in ~ 446, as a corollary from the theorem that  F(z)-  dz converges to
the limit 7rF(+ 0), in the case in which F(z) is a function with limited
total fluctuation. It has now however been shewn that the convergence of
sin mZ
I F(z).sz dz is really independent of the condition that the function
J)a   sin z
should be one of limited total fluctuation. Referring to the investigation in
~ 451, it is now clear that it is sufficient for the convergence of
F s (z).z    dz, to the limit ' -rF(+ 0),
S   sin r d D                   p 
* See his memoir " Ueber die Darstellbarkeit," 1Math. Werke, p. 227.




680


Trigonometrical series


[OH. VII


that F(z) should be of limited total fluctuation in the arbitrarily small
interval (0, ip), of z. For the condition of convergence at xi is unaffected by
alteration of the values of f(x) outside the interval (xi - 2/, xi + 2Pu). We
thus obtain the following sufficient condition of convergence of the Fourier's
series at a point x:I. That the Fourier's series may converge at a point x, to the value
I {f(x + 0) +f(x - 0)}, it is sufficient that a neighbourhood of x can be determined, so small that the function f(x) is of limited total fluctuation in that
neighbourhood.
As a particular case of this theorem we have the following:Ia. That the Fourier's series may converge at a point x, to the value
~ {f(x + 0) +f(x - 0)}, it is sufficient that a neighbourhood (x - p, x + /) can
be determined, such that the function f (x) is monotone, both in (x - p, x) and in
(x, x + f).
1 F  sin (2n ~ 1) z1                                  4 
Since -   i            dz converges to the value 1, the condition of
7r Jo     sn z                             2
convergence of the series at a point x, is satisfied if
sin mz
fj  {f(x +  2z)-f (x + 0)} sinz dz
sin mz
and               J   {f(x - 2z) -f(x - 0)} smsi dz
sin z
each converge to the limit zero, as m is indefinitely increased. It is however
not necessary that these conditions be satisfied; all that is necessary is that
the sum
sin mz
[f (x + 2z) +f(x- 2z) - lim f (x + t)-f (x - t)] sin  d
t=o                   sin z
should converge to zero. This remark enables us to obtain conditions of
convergence of greater generality than those which would be obtained by
assuming that each of the two separate parts of this integral converges
to zero.
At a point x, of continuity of f(x), the necessary condition is that
sin mz
{f (a + 2z) +f (x - 2) - 2f(x)} sin  dz
should converge to the limit zero.
Writing, for convenience
2z=t, f(x+t)+f(x-t)=       (t),
the condition of convergence of the series is that
~ (t) - 0 (+ 0)  t
f t ~)sin Isin st dt
should converge to zero, as m is  initely increased.
should converge to zero, as m is indefinitely increased.




457]     Sufficient conditions of convergence at a point         681
Now                   (t)-      ) (+ O)  t
t       sin 2 t
is certainly integrable in any interval, contained in (0, 7r), which does not
contain the point t = O. In case
) (t) - + (+ 0)
t
is integrable in the interval (0, /.) of t, so also is
(t)- 0 (0)  t
] t   sin  t;
and then the theorem of ~ 454, suffices to shew that
(t) -_   (+- -   -- sin I mt dt
sin It 
converges to the limit zero.
We have accordingly obtained the following theorem:II. If | (t) - f (+ O) be integrable in (0, At), where 0 < i~ _ r, and / (t)
t
denotes f(x + t) +f ( - t), then the Fourier's series is convergent at the point x.
This condition is satisfied when f(x + 0), f(x- 0) are both definite, and
f(+ t) -f(x0)      f   -t) -f(-) are both integrable in (0, p.); or else
t,t
when f (x + 0), f((x- 0) are not necessarily definite, but f(+ 0) is so, and
t
q (t)- tb (+ 0) t is inteirable in (0, 4).
The series in either case converges to 2  0 (+ 0).
II a. If x be a point of continuity of f(x), the Fourier's series converges
to the value f (x), at the point, if f (x + t) + (x - t) - 2f (x) be integrable
in the interval (0, /u); and in particular if
f(x + t)-f(x)    f(x - t) -f (x)
t       '        t
are both integrable in (0, a).
The condition that   (t) -   - (~ 0) should be integrable in (0, p) is
t
satisfied if lim  (t) -   (+ 0)  lim  b (t)-  (+ 0) are both finite. In case
t=o      t         t=o      t
x be a point of continuity of the function, this condition is satisfied if the
four derivatives D+f(x), D+f(x), D-f(x), Df(x), at the point x, are all




682


Trigonometrical series


[CH. VII


finite, and in particular iff(x) have a finite differential coefficient. We thus
obtain the following theorems:III.  The Fourier's series converges at a point x, if
lim   (t) - (+ )   lim   (t)- (+)
t=o      t          t=       t
be both finite, 4 (t) denoting f(x + t) +f(x - t). In particular if x be a
point at which f (x) has an ordinary discontinuity, and
li f(x + t)-f ( + )     lim f(x- t)-f (-O)
t=+o        t           t=+O        -t
be both either definite, or indefinite between finite limits of indeterminacy, then
the series is convergent at the point x.
IIIa. If f(x) be continuous at x, the Fourier's series converges, at the
point x, to the value f(x), if the four derivatives of f(x) at the point be finite,
and in particular if f(x) have a finite differential coefficient at the point.
It is however not necessary for the integrability of ' (t)- b (+ 0) in the
t
neighbourhood of t = 0, that it should be limited in that neighbourhood. It
is sufficient (see ~ 281, Ex. 1) that
P(t)-sb(~0) <A
(t)-  (+0),_ A    or that I (t) -  (+ 0)   At-,
t '
where 1- a is some positive number, and A is some fixed positive number,
for all values of t which are not greater than some fixed value /. We thus
obtain the following sufficient conditions of convergence of the series:IV. The Fourier's series converges, at a point x, if, for all values of t
not greater than some fixed positive number pA, the condition
i (t) -  (+  ) - Atk,
be satisfied, A and k denoting fixed positive numbers.
IV a. The Fourier's series converges to f(x), at a point x of continuity of
the function, if If(x + t)-f(x)l < Atk, where A  and k are fixed positive
numbers, for all values of t numerically less than some fixed positive number iA.
At a point of ordinary discontinuity, it is sufficient that both
If(x+t)-f(x+ O) and If(x-t)-f(x-O)
should satisfy this condition.
A more general sufficient condition of integrability of  (t) -   (+ 0) in
t
the neighbourhood of t= 0, is that, in a sufficiently small interval (0, /s), of t,
|I (t)-  (+o )+0)  A —                           1a
log - log log.   log log...
t       t... 




457, 458]    Conditions of uniform convergence


683


where A and a are positive numbers (see ~ 299). We therefore obtain the
following condition of convergence of the series:V. The Fourier's series converges, at a point x, to the value
lim  f (x + t) +f(x - t)},
t=o
if, for all positive values of t not exceeding some fixed value /, the condition
A
I V (t)- q (+ 0) o_  1     1 -               1a
log t   log log t  ogo..... g  
be satisfied, where A and a are fixed positive numbers. In particular it is
sufficient that both
If(x + t)-/f( + O) ], If(x - t) -f(x - 0) 
should satisfy this condition.
CONDITIONS OF UNIFORM CONVERGENCE OF FOURIER'S SERIES.
458. Conditions will now be investigated that the Fourier's series,
corresponding to a summable function f(x), either limited in (-7r, 7r), or
unlimited but integrable, may be uniformly convergent in a given interval
contained in (- wr, 7r). With a view to this investigation, the following
general theorem* will be established:The function f(x) being summable and integrable in (- r, wr), each of the
four integrals Sf (x + 2z) x () siZs dz, taken through any interval (a, 13),
such that 0 _ a < 1 3  - 7r, converges to the limit zero, as the positive number m
is indefinitely increased, uniformly for all values of x contained in the
interval (- 7r, 7r); the function x (z) being any function with limited total
fluctuation in (a, /3).  The function f (x) is assumed to be such that
f(x + 27) =f(x), for -  < x < 7r.
More generally, sin mz or cos mz may be replaced by 0 (mz), where b (z)
is any limited summable function, of which the integral, taken through any
finite interval whatever, is less, in absolute magnitude, than some fixed
positive number, independent of the particular interval. It is unnecessary
that m be restricted to be integral.
First, it will be assumed that f(x) is a limited function. It is sufficient
to consider the case of the integral f f(x + 2z) X (z) sin mz dz; the proof in
the case of the other three integrals being precisely similar.  Also, the
substitution of qc (mz) for sin mz makes no essential difference in the proof.
* Hobson, "On the uniform convergence of Fourier's series," Proc. Lond. Math. Soc., ser. 2, vol. v.




684


Trigonometrical series


[CH. VII


Let U and L denote the upper, and the lower, limit of f(x + 2z), for all
values of x in the interval (- r, wr), and for all values of z in the interval
(a, /3). Let the interval (L, U) be divided into p portions (Co, cl), (cl, 2)...
(Cq_, Cq)... (Cp_1, Cp), where Co = L, cp = U, and where Cq- Cq-i < e, for every
value of q. Let the function f, (x + 2z) be defined as follows:-For those
values of x + 2z for which Co, f(x + 2z) < cl, let f (x + 2z) = C; for those
values of x + 2z for which c, f (x + 2z) < C2, let f, (x + 2z) = c,; and generally,
let fi (x + 2z) = c_, when cq_- f (x+ 2z) <cq; when f(x + 2z) = cp, let
fi (x + 2z) = c. For any particular value of x, it may, for example, happen
that there are no values of z such that C  f (x + 2z) < c,; in that case there
are no values of x + 2z, with the given value of x, for which f, (x + 2z) = o.
We have
A r   
| f (x + 2z) X (z) sin mz dz -  fi (x + 2z) X (z) sin mz dz <e (/3- a) X,
where X is the upper limit of | (z) I in the interval (a, /); and this holds for
all values of m, and of x in (- r, 7r).
We have also
I  (x + 2z) X (z) sin mz dz =  cq  % (z) sin mz dz,
J~a ~q=o                 eq
where eq is that set of points z, at which Cq -f(x + 2z) < Cqi; this set eq
depending upon the value of x.
In the interval (- 27r, 27r) of the variable x, let Eq denote that set of
points at each of which cq f(x) < cq+,. Let the set Eq be enclosed in a
finite, or enumerably infinite, set Hq of non-overlapping intervals, such that
m (Hq)- m (Eq)= 7r. For any fixed value of x, the set eq consists of that part
of Eq which lies in the interval (x + 2a, x + 2/), contained in (- 2vr, 27r). Let
that part of the set of intervals Eq which lies in (x + 2a, x + 2/3) be denoted
by Fq; then it can be shewn that m(Fq)-m (eq) - A. For, if possible, let
m (Fq) - m(eq) =  +, where 7 is a positive number. Let the set eq be
enclosed in the interiors of non-overlapping intervals of a set Lq, all in the
interval (x + 2a, x + 2/3), such that m (Lq) < m (eq) + y; and let Hq denote
that set of intervals which consists of Lq together with that part of Hq which
is not in (x + 2a, x + 2/3). We have then
m (Hq) = m(Hq) +J m (Lq)- m (7Fq)
< m (Hq)- 
< m (Eq).
As Eq cannot be enclosed in a set of intervals Hq, of measure less than m (Eq),
it is impossible that the positive number y can exist; and therefore
m (Fq) - m (eq) - q7.




458]


Conditions of uniform convergence


685


It is to be observed that the number r is independent of the parameter x, in
the integral.
We have now
f|  (z) sin mzdz d-  (z) sin mzdz <X V.
Let the intervals of the set Hq, in descending order of length, be denoted
by 71, 2 73,....... In case the point x + 2a, or x + 2/, is interior to an
interval of Hq, we divide that interval into two parts, and assign separate
indices to those parts. We may choose r so that
gm (Hq) - (y + 7y +... + fr) < 7.
Of the intervals 71, 72,... 7r......  let those which fall in (x + 2a, x + 2/3)
be 7s1, 7S2  *......, where sI< s< S...; and let st be the greatest of these
indices which does not exceed r. We have then
Yst+i + 7t+i + *. < 7, and  m (Fq)-(+7s + S2 +.. + 7S) < V;
or denoting by Dq the finite set of intervals 7,S, 7y2,... 7'st, we have
m (Fq) - m (Dq) <.
The number t, of the intervals in the set Dq, cannot exceed the number r,
which is independent of the value of x.
We now have
Fq X (z) sin mz dz -  (z) sin mz dz <?X.
The function x (z) having limited total fluctuation in the interval (a, i3),
it may be expressed as the difference %X (z)- % (z), of two functions % (z),
X2 (z), each of which is monotone in (a,.3).
The integral J    (z) sin mz dz may, by means of the second mean value
theorem, be expressed as
xi (p)f sin mz dz + % (X)  sin mz dz - X (i) j sin mz dz
rA
- %2 ()     sin mz dz,
where i, /' are two numbers in the interval (/L, X). We thus see that
AX               4
X (z) sin mz dz < <   (X  +  2),
where,I, 2 are the upper limits of | xi (z) and of X, (z)\ in (a, i/); the interval
(/i, X) being supposed to be contained in (a, 8/). We have now
JX (z) sin mz dz <  ( + 2) t <   (  + 2) r,
since t cannot exceed r.




686


Trigonometrical series


[CH. VII


By combining the inequalities which have been obtained, we find that
f(x + 2z) (z) sin mz dz <e(fi-a) + (2Z + -m%      2 cq,
where -' denotes  i + X; and this holds for every value of x in (- 7, 7r).
Let a positive number ' be now arbitrarily chosen; we can then choose e so
that e (,3 - a) % < I. Having fixed e accordingly, and consequently also the
numbers co, c1,... Cq being capable of being fixed, we next choose v so that
q=_p
2%Vr 2 Cq < 3-; the number r is then fixed. We can now choose a value ml
q=O
4r _clq=p
of m, such that -      cq < c, form  im,n.
9    q=O
It has now been shewn that, having given a positive number ', arbitrarily
small, a number nil can be so determined that
f(x + 2z) (z) sin m d  <,
for nm min, and for all values of x in the interval (- r, r). It has therefore
been shewn that, when f(x) is a limited summable function,
J f (x + 2z) X (z) sin mz dz
converges to the limit zero, as m is indefinitely increased, uniformly for all
values of x in the interval (- vr, 7r); and consequently also for all values of x
in any interval (a, b) contained in (- 7, 7r).
Next, let f(x) be no longer limited, but still integrable in accordance
with Lebesgue's definition.
If ' be an arbitrarily fixed positive number, a positive number N can be
so determined that
the integral being taken over that set of points KN- in the interval (- 27r, 27r),
for each of which If(x) I > N. If kN be that part of KN which lies in the
interval (x + 2a, x + 2,f), for any fixed value of x belonging to the interval
(-r, 7r), we have, a fortiori
If (x)i dx < 2 /X
Let the function f,(x + 2z) be defined by the specifications
f(x + 2z)=f(x+ 2z), if If(x+ 2z)    N;
and              f2( +2z)=0, if    f(x+ 2z) > N.
Thus f, (x + 2z) vanishes at all the points of the set KN; and it is a limited
summable function.




458, 459]      Conditions of uniform     convergence             687
We have now
f (x + 2z) % (z) sin nz dz
=ff (x + 2z) x (z) sin mz dz +  f2( + 2z) x (z)sin m2z dz.
By the first part of the theorem, we see that a value ml of m can be
determined so that
f    (x 2+ ( 2) X(z)sinmz dz <
for rm _ mi, and for all values of x in (- 7r, 7r).
Also               f (x + 2z) X (z) sin mz dz <2 S;
hence we have shewn that
f (x + 2z) x (z) sin mz dz < g,
provided m _- m, for all values of x in (-7r, 7r). The theorem has therefore
been completely established.
459. Some particular cases of the general theorem established in ~ 458
will now be considered.
(1) Let a = 0, /3 = 7r, and let x have the single value 0; also let
x(z)=1, and m = 2n. Then, changing z into -x, we see that
/(x) sin nx dx,  Jf () cos nx dx
both converge to zero, as n is indefinitely increased. Taking the integral
which involves f(x - 2), we see that
~                  0
-f (x) sin nx dx,    f(x) cos nx dx
also converge to zero. By addition, we obtain the theorem already established in ~ 454, that the Fourier's coefficients
ff (x) sin  d
f f(Wc) nx dx
converge to zero, as n is indefinitely increased.
(2) Let 0 < a, / = ~ r, and X (z) = cosec z, which is of limited fluctuation
in the interval (a, 7r). We see then that, if (a, b) be an interval for x,
in which f(x) is limited, and be also such that f(a - 0), f(b + 0) are finite,
then
|fi  [f(x + 2z)+f(x- 2z) -f(x+  O) -f(x -  0)] sin  dz
converges uniformly to zero in the interval (a, b), of x.
converges uniformly to zero in the interval (a, b), of x.




688


Trigonometrical series


[CH. VII


For, by the theorem, the two integrals
f f(x+ 2z).     dz,    f(x - 2z).  dz
jasinmz     a           sin z
converge uniformly to zero in (a, b); also If(x + 0) +f(x - 0) is less, for all
values of x in (a, b), than some fixed finite number, and
Jr sin mz dz
sinz dz
Ja sin z
converges to zero, as mn is indefinitely increased. It thus appears that, in
order that a Fourier's series may converge uniformly to zero, in an interval
(a, b) contained in (- r, 7r), it is sufficient that
facL                          sin mz
j [f (x + 2z) +f(x - 2z) - 2f(x)] s    dz
should converge to zero, as m is indefinitely increased, uniformly for all values
of x in (a, b). We know that it is a necessary condition for such uniform
convergence that f(x) should be continuous in (a, b), including the endpoints a and b.
In accordance with this result, it depends only upon the nature of the
function f(x) in the interval (a-2a, b + 2a), where a is arbitrarily small,
whether the Fourier's series converge uniformly in (a, b) or not; the nature
of f(x) in the remainder of (- nr, 7r) being irrelevant, subject only to the
restriction that f(x) must have a Lebesgue integral in (- r, vr), whether it
be limited or not. We have therefore obtained the following theorem:If (a, b) be any interval contained in (- r, 7r), such that f (x) is continuous
in (a, b), including the end-points a and b, then the answer to the question
whether the Fourier's series converges uniformly in (a, b), or not, depends only
upon the nature of f(x) in an interval (a', b') including (a, b) in its interior,
and exceeding it in length by an arbitrarily small amount. The function f(x)
may be of any character in the part of (- r, vr) outside (a', b'), so long as it
has a Lebesgue integral in (- r, vr).
This theorem contains, for the theory of uniform convergence, the parallel
to the theorem of ~ 456, that convergence or non-convergence of the series,
at a particular point x, depends only on the nature of the function in an
arbitrarily small neighbourhood of x. The latter theorem is that particular
case of the theorem here established, which arises when the interval (a, b) is
reduced to a particular point x.
(3) It has been shewn in ~ 451, that, if f(x) be a function with limited
total fluctuation in (- r, r), the Fourier's series corresponding to f(x) converges uniformly in any interval (a, b) which contains no point of discontinuity of the function, either in its interior or at its ends. By applying the
theorem obtained in (2), we now see that the following extension of this
result holds:




459, 460]      Conditions of uniform     convergence             689
The function f(x) being summable and integrable, whether it be limited
or not; if (a', b') be any interval, contained in (- r, 7r), and such that f (x) is of
limited total fluctuation in (a', b'), then the Fourier's series, corresponding
to f(x), converges uniformly in any interval (a, b) in the interior of (a', b'),
provided the function be continuous in (a, b), including the points a and b.
(4) Let the function X (z) be defined by X (0) = 0, and
1    1
(z) = -s, for z>0;
XAz) v  z  sin z
also let a = 0, /3 = / <  7r. We then see that
f [j(x + 2z) + f (x-2z-   2f (x)] (  sin   zsin   dz
converges uniformly to zero, as m is indefinitely increased, in any interval
(a, b) in which f(x) is limited.
It thus appears that, if
R [f(x + 2z) + f (- 2z)- 2f(x)] siz dz
converges uniformly in (a, b), then so also does
[f (x + 2z) +f(x - 2z) - 2f(x)] sin mz z.
l)'      z
Therefore, the condition of uniform convergence of the series in an interval
(a, b), in which f (x) is continuous, including the points a and b, is that
J [f (x + 2z) +f(x - 2z)- 2f(x)] sin   dz
should converge uniformly to zero in the interval, as m is indefinitely increased.
The number, may here be taken arbitrarily small; in fact, in accordance
with the theorem of ~ 458, if 0 < /u < Ct, the integral
J                                   dA,^ z
converges to zero, as m is indefinitely increased, uniformly in (a, b).
460. We proceed to apply the result in (4), of ~ 459, to obtain sufficient
conditions for the uniform convergence of the series in an interval (a, b).
Denoting       f(x + 2z) +f(x - 2z) - 2f(x), by F(z),
we have, if                   0 < u <,
I  F(z).              FIf ) sn   7       F(z)
(-z) sin mz dz     i F(zsinmz dz +      F) sin mz dz.
Z                   z
Let it now be assumed that, for every value of x in (a, b), the integral
|/i  (z) dz exists as a Lebesgue integral, and that 7' F(z) dz converges to
Hz                                            4z
H.                                                         44








690                   Trigonometrical series                 [OH. VII
the limit zero, as /Al is indefinitely diminished, uniformly for all values of x
in (a, b).
We have then
f^ F(z)               F (z)
F (z) sin mz dz <  F() dz.
The number,1 can now be chosen so small that, if ' be an arbitrarily fixed
positive number, the inequality
(z) dz<
is satisfied for this value of /, and for every value of x in (a, b). The
number au having been so fixed, we can fix a value m,, of m, such that
-(z) sinmz dz < <,
for m > nvh, and for every value of x in (a, b). We have then
7 F-( sin nz dz <go,
for m   r ml1, and for every value of x in (a, b).
The following theorem has therefore been established:It is a sufficient condition for the uniform convergence of the Fourier's
series, in an interval (a, b) in which f (x) is continuous, the points a and b
included, that
If(x+2z)+ f(x-2z)- 2f(x) dz
Jo             z
should exist for all values of x in (a, b), and should converge to zero, as t is
indefinitely diminished, uniformly for all values of x in (a, b). The condition
is satisfied if the two integrals
f(x - 2z) -f(x) z       f (x - 2z) -f(x) ] 
both exist, and are uniformly convergent.
In particular, the series is uniformly convergent in any interval (a, b) in
which one of the four derivatives, and therefore each of the other three
derivatives, is limited, the end-points a and b being included. A special case
is that in which f(x) has a limited differential coefficient in (a, b), including
a and b.
The condition is also satisfied if
f(x + /3) -f(x) | < Ck,
for all values of x in (a, b), and for all positive values of / not greater than
some fixed positive number; where C and k are positive numbers independent




460, 461]      Conditions of uniform     convergence             691
of x. This is a generalization of the sufficient condition, obtained in ~ 457,
for the convergence of the series at a particular point.
F (z)
461. Let the integral    -   sin mzdz be expressed in the form
2rr  3rr  47r
- rm  rm      rF (z)
f+f + +... +    -F- sin mzdz,
LJ O   J 27 r      J2r   Z
} '&    f1    m
where r is an integer such that
2r7r  27r
0! -- --- < -.
m     m
We assume that (a, b) is contained in an interval (a', b'), in which f(x) is
limited; if then we choose / to be less than the smaller of the numbers
(- a'),     (b -
we see that F(z) is limited, for all values of z in (0, /), and for allvalues
of x in (a, b). We have now
27r                  2rr
fm
sinmZ
mF(z) inZ dz <          I iF(z) dz
< 27r x upper limit of I F(z) in (o0,  ).
Since a continuous function is uniformly continuous, the two functions
f (x + 2z) -f(x),  f(x- 2z) -f(x)
converge to zero, as z converges to zero, uniformly for all values of x in (a, b).
It follows that
2Tr
[m     sin mz
JF(z) s     dz < m
where q7 converges to zero, as m is indefinitely increased, and is independent
of the value of x.
Next, we have
__\ sin __        4/m                             2r7r   \
F (z)    -    dz -<  - 2rm x upper limit of F(z) in -—,     )
4/m
<   -— /m  x upper limit of i F(z) I in (0, Ou)
< Vm)
where rm' converges to zero, as m is indefinitely increased, and is independent
of the value of x.
44-2




692             Trigonometrical series      [CH. VII
The remaining part of the integral may be written in the form
2?r 32r          47r
F ( +    ) F (Z+    F( )+
27r      37T     47T
z - -    z+- -   z+ -
2r-2 7r\     2r- 1 7r
F z+        F z++ - -- -- -  - -- -      sin mziz,
2r - 2     ( + 2r - 1I 
- 2 -z+- -- -- zz 2r-l,
m           m 
which is less, in absolute value, than
2rz + 27 z + 3w'      + 2r..  dz
Jo  z + 27r  z + 37r  * *'  z + 2r -lr
and this does not exceed
sr     ~z + 2s7r  z + 2s + 17r
f s=r-1 { F( -~28) F~  +    d
Js-l  z + 2s7r  z + 2s + 1 r
Now
(z+ 2s -7r  z + 2s + 7r2s
z + 2s7r  z + 2s —+ 1   Fz + 2,s7r
F        F (z~ 2s) 2 )    F(z2
_ m   _    m _ m               _+ m7
z + 2s + 17r     (z + 2s7r)(z + 2s + 17r)
hence
F(z + 2s7r) F( + 2s + r  I(z + 2s7r) F  2s + 17r
m     _    m        V   y     V   L
z+ 2s7r  z + 2s + 7r        (2s + 1)7
2s(2s+ 1)7rw  m)
We now see that the part of the integral to be estimated is, in absolute
value, less than
+. + 2r+-) A + 2.upper limit of \F(z) in (0,,);
where A is the greatest of the numbers
F( + 2s7r  F+ 2s + l r) 
for s = 1, 2, 3,... r-1, and for all values of z in (0, 7r).




461]


Conditions of uniform convergence


693


The upper limit of I F(z), or If(x + 2z) +f(x - 2z)- 2f(x), for the
interval (0, /u), of z, and for all values of x in (a, b), depends upon /u, and is
a number 2u (/), which may be made as small as we please by taking /4
sufficiently small.
Also
@ +  +          I+2+-+o-+...+ -  A<(C,-+log2r)A,
where C, is a number between 0 and I, which converges to Mascheroni's
constant. Let Dm denote the greatest value of the difference of the values
of f(x) at the ends of an interval of length vr/m, contained in the interval
(a - L, b +,/), for all possible positions of such sub-interval; thus A _ Di.
It has now been proved that, for 0 < / < n-7r,
[/(x + 2z) +f ( - 2z) -2f(x)] sin mz  dz <   n, +   +  (G)
Z
+ D     C, + log e + log (- m2 )}
2rvr     27r
where 0 is less than unity, and such that / -  =0. 
m       rm
We have now to find a sufficient condition that it be possible, with /,
fixed, to determine a value m of m, corresponding to an arbitrarily prescribed
positive number g, such that
sin mz 
F (z)   m  dz <s, for m -m,
IJo       m
and for all values of x in (a, b).
If /, be a number such that 0 <, < p, we can choose p~ so small that
U (  <1) < &.
The number,l having been so chosen, we can now choose a value ml, of m,
such that
/A..sin mz 
J F(z)    -Z dz <, for in _nm1,
Z        6
and for all values of x in (a, b); this follows from the uniform convergence
of the integral to the limit zero.
We then have
|1A   sinmx,     1m        / Z  I-      i  mi,        20ir\
F (z) i  d   <   +     m +  + D    Cm. + log  + log /(  -- 0,
the numbers '/m, Dm, 01 now having the values of the former?',, Dm, 0
which correspond to /1t instead of /. We can now choose m2 so that rn < 6,
for in? m2; also we can choose m3 so that q'm < 6, for in  m3. Again, since




694


Trigonometrical series


[CH. VII


Dm converges to zero, as m is indefinitely increased, the function f(x) being
assumed to be continuous in the interval (a - 2/b, b+2/,+), we can so
determine m4 that
D.n {  + log  1  2 --  )  <  for qn _ in,.
Let us now assume that it is possible so to choose mn, that
D, log - <    f, for n  n m5.
7r
Taking am to be the greatest of the numbers nm, nm2, M3, m4, m95, we now
have
F (z) sin dz d< <, for m   TI.
Therefore, with the assumption made, that Dmlog - converges to zero,
7r
as m is indefinitely increased, it has been shewn that the convergence of the
Fourier's series in (a, b) is uniform. The following theorem has now been
established:If (a', b') be an interval such that, for every pair of points x, x +/3,
contained in that interval, {f(x + /3) -f(x)i log/3 converges to zero, as /3 is
indefinitely diminished, uniformly for all values of x, then the Fourier's series
converges uniformly in any interval (a, b) contained in the interior of (a', b').
More generally, it is sufficient that {f( (x +,) +f (x -/) - 2f(x)) log/3 should
satisfy the similar condition.
For the case in which the interval (a, b) is reduced to a single point, the
condition of the theorem becomes a sufficient condition of convergence of
the Fourier's series at that point. Thus a sufficient condition of convergence
of the series at a point is, that a neighbourhood of the point can be
determined, such that, if e be any prescribed positive number, another
positive number /3 can be determined, such that
{f(x + /) -f/(W) log / < 6,
for every pair of points x, x + 3 in that neighbourhood. This condition was
given by Dini*.
The condition of the theorem is satisfied, in particular, if, for every pair
of points x, x +,/, in (a', b'), when /3 is sufficiently small,
If(x + /) -f(x) I < /3,
where C, k are positive numbers independent of x. In this form, the condition
was given by Lipschitzt, as a sufficient condition of the convergence of the


* Serie di Fourier, p. 49.
t Crelle's Journal, vol. LXIII, p. 308.




461, 462]  Further investigations of Dirichlet's integral  695


Fourier's series at a single point, the condition being applied to a neighbourhood of the point.
The condition of uniform convergence in the interval (a, b), stated in the
theorem, is satisfied, if / can be so determined that, for all pairs of points x,
x + 3f, in the interval (a', b'),
If(x + 0) -f( ) I <                              k
log. log log   {log log....+
where C, k are positive numbers, independent of x.
FURTHER INVESTIGATIONS OF DIRICHLET'S INTEGRAL.
462. Many investigations relating to Dirichlet's integral have had as
their object the determination of sufficient conditions for the convergence of
a Fourier's series at a particular point. These investigations have resulted
in the discovery of many special conditions sufficient for the convergence
of the series, some of them of great generality.  The most important of
these conditions have been already discussed in the present chapter.
An account will now be given of some investigations of Dirichlet's
integral, given* by Kronecker, Holder, and Brodent. The mode of procedure of the last writer will be here adopted.
It has been shewn that the question of the convergence of the series
at a point depends upon whether   F(z) sin  z dz(O < e    ) converges
Jo       z
to the limit zero, as m is indefinitely diminished; the function F(z) being
such that F (+ 0) = 0.
The integral   F(z)Si-Zdz may be expressed as the sum
Joeintgra ~Ba)sin m'x
Ix (m)   sin mz              sin m
F (z)      dZ +     F(z)  — dz,
Z          (i)Z
where x(m) is a function of m, which is positive and < e, for every value of m,
and is such that it converges to zero, as in is indefinitely increased; i.e.
lim X (in) =0.
Let us consider first the integral
X (m)F   sin mnz
Ix-      F(z) }     dz.
* Berliner Sitzungsber. 1885, "Ueber das Dirichlet'sche Integral," by Kronecker; and in the
same volume, " Ueber eine neue hinreichende Bedingung..." by Holder.
t Math. Annalen, vol. LII.




696


Trigonometrical series


[CH. VII


Since si     < m, we have I I  < mX(m)Ux, where Ux is the upper limit
of {F(z) in the interval (0, %(m)); and this upper limit Ux converges to the
limit zero, as m is indefinitely increased, because F(+ 0)= 0.
The condition that lim I I  =0 is that X(m) must be such that
lim mx(m)U = 0;
im2 = Qo
and this condition will be satisfied if %(m) be such that mX(m) remains
less than some fixed number, however great m may be.
The condition may also be satisfied if mX(m) becomes indefinitely great,
as m does so, since Ux may be such that mx(mn)Ux has the limit zero.
1
In particular, if  (m)=-, we see that
1
F(z) sin  z dz
z
has the limit zero; and therefore
fox (m)    sin =nz         (x )      sin mz
lim      F(z)      dz = lim       F(z)       dz < Ux log m(m) 
m =0 o          Z           = o   1      Z
On
We have now seen that
i x (  z) sin nzz
lim      F (Z)  -dz
m= co 0         Z
vanishes, provided X(m) be such that lim X(m)= 0, and also such that
nz= 00o
one at least of the numbers mx(m). Ux, log ({tm (m)} Ux has the limit zero;
where Ux is the upper limit of IF(z) in the interval (0, x(m)).
If we take X(m) =, where: is a positive constant, and change z into?TX, then write    =, we obtain Kronecker's* theorem
nm             m
lim f    F(x) s-  x dx = O, when F (+ O)= 0.
a=0           X
If we choose x(m) so that one of the above conditions is satisfied, we
have
imf F(s) sin mZ d        f        sin mz d.
wn==c *o             n=a Xoo  (in)   Z


* Loc. cit., p. 642.




462]      Further investigations of Dirichlet's integral          697
The integral on the right-hand side can be divided into three parts, by
dividing the interval ( (m), e) into the sum of the three intervals
(X(m), 2p7r/m),  (2p7r/nm, 2qr/m),  (2q7r/m, e),
where p, q are positive integers, such that
2p7 vr        27r            2q7r  27r
0 <     - X (m) (   27, and 0 < e -  -2qr 
m            Im              in mm
Of the three integrals, the third clearly has the limit zero; and the first is, in
absolute value, less than F, log  2pr, where Fi is the upper limit of F(z)l
mX(mn)
in the interval (0, 2p7r/m). Hence the absolute value of the first integral is
less than F, log (1 + m2  ), of which the limit is zero; unless mX(m) has
the limit zero, in which case the absolute value of the integral is less than
F in{2pr _      }
or than 27rFl; and the limit is therefore in this case also zero.
We have now left the integral
2qpr
m F() sin mz 
JF(z) - dz.
Z
m
27I
If we divide the interval into portions of length, and then change
the variable in each portion, we can reduce this integral to the form
(Z+   2L+    F (z+F 2t+ 1+r
f,s-in  [      z       +2-         dz,
=p     E z + 2t7r  z + 2 + 21r  J
which is less than
_     z+267r         +2     7     dz.
LO        L_ iz + 2+          + 17   J
It follows that, for a given e, the limit, when m is indefinitely increased, of
f[e n sin mz
F(z) in     dz
is zero, provided it is possible to choose x(m), consistently with the
conditions which have been laid down, so that, if 8 be an arbitrarily small
positive number, a value mn, of m can be found, such that
=q-1 F(     w   ) m F(Z  2,+    ) 
o  c=P   z+ 2L7r      z + 2t +1         < 




698


Trigonometrical series


[CH. VII


for all values of m, such that m  iml. It can easily be verified that the
limit of this integral is independent of the value of e, and also of the
particular choice of the function %(m).
If we take nm(mn)= 2n - 1, where n is a fixed positive integer, and
write zrw for z, and a for 7r/m, the sufficient condition takes the form
v =2a2       F E'(<^+< A) (c  n 
lim      2   (- 1)" — dz = O,
o-=oo   h7=2          z + -
where FE    denotes the integral part of 2; this is equivalent to a condition
Lo-i                          2-'
obtained by Kronecker*.
463. The sufficient condition which has been obtained above, that the
limit of oF(z) sn mdz may be zero, is of a very general character, and
includes, as special cases, various sufficient conditions which have been
obtained by special methods.
The condition will be satisfied if, corresponding to a number 8 as small as
we please, it is possible to choose x(m) so that a value ml of m can be found,
such that, for in > vi, all those values of z between 0 and 7r, for which
+( 2(    v        + ) (   + 17r
^~-1 \  \m       _      _m       \
{ Fz+ 2t^T         z+ J   2L+l7) J
numerically exceeds 8, form a set of points of zero content, provided also
the absolute value of this sum, for every value of z, has a finite upper limit.
For in this case the integral is less than r78 + f dz, where the integral is
taken over sub-intervals which include all the points at which I I > 8, and
this is less than
ra + 2i fdz,
where Zi is the upper limit of I. Since dz is, by hypothesis, arbitrarily
small, 7r8+  ifdz is arbitrarily small, and thus the limit of the integral
is zero.
An important deduction from the general theorem is that
lim   F(z) sin nz dz  0,
m=oo 0       Z


' Loc. cit., p. 651.




462, 463]    Further investigations of Dirichlet's integral          699
provided the function F(z) is such that
JfE F(z) dz
Jo I  z -
has a definite finite value.  This theorem has already been established in
~ 457.
For, we may write the sum
F (i  + 2T)   F (Z+ 2 + 1 7r
L=9    Z + 2t7r      z + 2t + 1r 
in the form
F Z 3+ 2'rr          Zz z+ 2t + 1 7r
i 1  _    _ m  J _. L=q-i 1 F  +     J+.
-=P m     Z+27r       =  m    Z+2t+ l7
*m2                    m
now if IF(z)/z I is integrable in the interval (0, e), each of the sums in this
last expression converges uniformly, as m is increased indefinitely, to the value
p F(z) dz,
J  Z
and hence the limit converges uniformly, for every value of z in (0, 7r),
to zero.
Since F(z) is an integrable function, we know that
2 F (z)
Z
where 62 > e1 >0, has a definite finite value, hence it is immaterial what
positive value less than u7r, e may have in
dz.
The theorem last proved may be used to shew that the two conditions
lim    F(z)- si    z = 0,
JO2?=   sin OZ
lim    f () ----  dz = 0,=o o        S0 Sin 2
are equivalent to one another.
We have
f (    sin s z     F 1   sin rzz                     sin?z,
F(z)     -dz      F(z)        dz - iF(z)z.      (z)      dz
o      sin z      o        z         Jo               z




700


Trigonometrical series


[CiH. VII


where   (z) denotes  (1 - -   ); which for a sufficiently small value of
Z     sinl z
z I is expansible in a convergent series of ascending powers of z. Since
lim zF (z)  (z) = 0,
z=o
and F(z) +(z) is an integrable function, we see, by the last theorem, that
lim   F(z).z (z) sin mz d= O
nm= ooJ             Z
therefore
f [e -rr/ \ sinml,,. f6 -. sin mz'
lim   F(z) sin   dz   lim   F(z) sin  dz.
m12Z= 0J      Z       M = Go     sin z
It has thus been shewn that any condition which is sufficient to ensure that
F(z) sin n dz
has the limit zero is also sufficient to ensure that
F (z) sn-dz
sin z
has the limit zero. This has already been established in ~ 459 (4), in a less
restricted class of cases.
464. If e and X (m) are such that
h=2cl-1     h(
E   (-1)-  F (h= 2p          m
is less than a fixed positive number N, for every value of m, it can be shewn
that
lim=   F() sin mz
n/F(z) —         dZ =0.
For, by a known arithmetical theorem due to Abel, we have
( AZ + 2t77^    Z4r27r7\) -1 7                  +h7I
1=q   (Z-    1)    (Z +     1 r)     h=2q-1     F(    r)       N
=-1  V   m  J_  \l-m=, (- ])h           <,=    z ++27r 2z+2t+-lr              h= 2p       z + hrr      + 2p7'
2prr          2rr
where 2p is determined by the condition 0 <  - X (m)   -.
Let mX (m) have the fixed positive integral value 2n - 1; then p = n, and
N
thus the absolute value of the above sum is less than, which may
z + 2n7r
be made arbitrarily small, for 0 < z < e, by choosing n large enough; and
thus the theorem is established.
If the interval (0, e) can be chosen, such that, in this interval, F (z) is




463-465]     The non-convergence of Fourier's series              701
monotone, the condition is satisfied. Suppose F(z) does not diminish as z
increases from 0 to e, then all the terms in the sum
l —1 OF(z + 2L) _     z + 2 + 17}\
are negative or zero, and the numerical value of the sum is less than ] F(e) |;
and thus the condition is satisfied. This is the case which was considered by
Dirichlet, and has been otherwise investigated in ~ 446. Again
F(z+2t7r _ F(z + 2 + I r)
is, for all the values of z, not greater than the fluctuation of the function F(z)
in the interval
2t7r  2t + 27r
hence the condition for the vanishing of the limit is satisfied if the total
fluctuation of F(z) in the interval (0, e) is less than a fixed finite number.
This is the case which was considered by Jordan, and was given in ~ 446.
The special conditions obtained in ~ 457 and ~461, due to Dini and Lipschitz,
may also be deduced from the general theorem of ~ 462.
THE NON-CONVERGENCE OF FOURIER'S SERIES.
465. Various sufficient conditions for the convergence of the Fourier's
series, corresponding to a given function f(x), have now been investigated.
The continuity of f(x) at a particular point x is neither necessary nor
sufficient to ensure that the Fourier's series, corresponding to f(x), converges
at the point x; it being assumed that the function f (x) is such that the
Fourier's series exists. Du Bois Reymond* gave the first example of a
Fourier's series, corresponding to a continuous function, which fails to
converge at points of a certain everywhere-dense set.
It is not definitely known whether a Fourier's series, corresponding to a
continuous function, can be such that the series fails to converge at every
point of an interval. It has however been proved t by Fatou, that, in case the
coefficients of the series be such that lim nan = 0, lim  b n, = 0, the series is
n=o       I n= 00
convergent at a set of points of which the measure is equal to that of the
whole interval (- 7r, r).
An example, due to Schwarz+, will be here given, of a function which is
everywhere continuous, but for which the Fourier's series fails to converge at
* Abhandlungen der bayerischen Akademie, vol. xnI, Abthg. 2.
t Acta Math. vol. xxx, p. 379.
+ See the history of the theory of Fourier's series, by Sachs, Schlimilch's Zeitschr. Supplement,
vol. xxv.




702


Trigonometrical series


[CH. VII


a certain point. It will here be shewn * that the series is, at that point, in
reality, oscillatory. It will then be shewn that the function may be employed
to construct another continuous function, for which the Fourier's series fails
to converge at each point of an everywhere-dense set.
Let the product 1.3. 5... (2X + 1) be denoted by [2X + 1], and let the function (b (z) be defined for the interval (0, a), where 0 < a < - r, in the following
manner:-In the interval (7r/[X - 1], 7r/[X]), let b (z) = cA sin [X]z, where cx is
a constant, depending upon the value of X; let X have all values Xi, XI + 1,
X, + 2,..., where X, is a fixed integer, and we may suppose a so chosen that
a=7r/[X- 1]; also let <b(0) = 0. If the sequence c,, Ci+1, CA,+2,... be so
chosen that it converges to the limit zero, the function / (z) is continuous at
the point z = 0, but it has an indefinitely great number of oscillations in an
arbitrarily small neighbourhood of that point. If the constants CA satisfy the
further condition, that cAlog(2X + 1) becomes indefinitely great, as X is
indefinitely increased, it will be shewn that the integral
f0  (z) sin (2n +1) z 
will increase indefinitely, as n has successively the values of integers in
a certain sequence. Thus the Fourier's series, corresponding to the continuous function defined by f(x) = 0, for - Tr < x  0, and f (x) =  (,I), for
0 _< x 2 2a, and f(x)= 0, for 2a ' x _ wr, does not converge at the point
X =0.
Let 2n+ 1 =.3.5...(2/i+ 1) = [/]; then
f (z) sin [] Z dz
may be written in the form
r/[p-1 sin []                ] si1n [r] z/-  sin  [/u] z dz
c^      sin   J- dz +  E c,.      -             dZ
J /[=]    Z        r=A Jt /[r]        Z
+  E  1 /[,-1] sin [r] z sin [/,] z
+   Y  Cr, j                  dz.
r= +l  J rr/[r] 
The first integral may be written in the form
7T/[Li — ] 1 - cos 2 [/] dz
J  /[X ]    Z
which is equivalent to
c, log (2, +    1) -  cM of- |  cos 2 [P] z dz,
where S3 is some number between 7r/[j] and 7r/[, - 1].
* See Hobson, "The failure of convergence of Fourier's series," Proc. Lond. Math. Soc., ser. 2,
vol. III.




465]


The non-convergence of Fourier's series


703


Now let c, log(2t + 1) increase indefinitely with,u. This is consistent
with c, having the limit zero; for we have only to take
c, = {log (2/u + 1)}where s is some fixed positive number, less than unity.
Since,L /f    cos 2 [u] z dz
7r  7T/[M]
is numerically not greater than c,/r, we see that, with the supposition made
as to c,, the expression
r/[l-1-] sin2 [zi] z
J Tr/[fk]  z
becomes indefinitely great, as, is increased indefinitely.
To evaluate        r/ f/[r-1] sin [r] z sin [/] dz,
To evaluate         el I                   dz,
r=Xl  J r/[r]      Z
we see, by writing sin [r] z. sin [,] z as half the difference of two cosines, and
applying the second mean value theorem to each integral, that the absolute
value of the expression is less than
r=,            1    1[ ]-[r] [   + [r]
r,^  T i  -I ),
or than  Y.
r=,r tn  r  [-      -       1] + 1-[]/[-1]  2 + 1 + [r]/[ - 1]
which is less than
CA,   [r]   1
7r  [, - 1] ' 
and this is less than
C;,       1            1           )
r4 {   -/ 2/-1  (2- 1)(2/-, 3) }'  j
Therefore the absolute value of the integral is less than 2cA/7rr,; and this
becomes indefinitely small, as, is indefinitely increased; and therefore the
limiting value of the expression is zero.
Lastly, we have to consider the expression
i /[r ' -1] sin [r] z sin [/] z d
r=S+nl  J 7r /[r]   Z
Since           sin   z
Since          jsin  _ < [~], and I sin [r]  _ 1,
the absolute value of the expression is less than 7rc,+,; and this has the limit
zero, when P is indefinitely increased.
It has now been shewn that
(z) sin. [1] z 
/o.




704


Trigonometrical series


[CH. VII


increases indefinitely with A, where [] = 1. 3. 5... (2/L + 1), provided cx has
the value {log (2X + 1)}-, where 0 < s < 1.
466. We proceed to consider the case in which 2n + 1 = (2p + 1) [/ - 1],
where p is an integer which varies with /L in such a manner that it always
lies between 0 and PL.
In this case, as before, we divide the integral
[f(z) sin (2n + 1)z z
Jo        z
into three parts
F1/[-l] i     sin (2p + 1)[ a- ] d
C,,      sin [P/] z
J  rl[]                z
U~d  z
-  1 1;r/[r-11 sin [r] z sin (2p + 1) [ - 1] z dz
r=  L      J7r/[]        z
+  E    r/[r -1] sin [r] z sin (2p + 1) [ - 1]z d.
+   Y, CE                                 dz.
r=x+l  J7r/[r]           z
The first part is equal to
C[f] fj   [cos {[ - 1] (2 - 2p)z} - cos {[ - 1] (2L + 2p + 2) z}] dz,
27r i[~1
where /3 is a number between 7r/[/t] and 7r/[A - 1]; and this expression is
less, in absolute value, than
cC4i]{ f   11                    1         )
r 1[ - 1] (2~ - p)     [I - 1] (2 + 2p + 2)>'
or than
c, I1 + 1/2P ^ 1 +  1/2/p  [
7r     PIP- p/  1 1 + P/2J
If, now, p increases with J in such a manner that p/l/ is always less than
some fixed number which is less than unity, then this expression diminishes
indefinitely, as / is indefinitely increased. It would also be sufficient that
p/l = 1 - ec {log (2p + 1)}-8',
where s' < s, and c, = {log (2/ + 1)}-s; the positive number Kc being fixed.
The second part of the above integral is less, in absolute value, than
Yl c}r]                 _ 1  +          1.=,, v7r 1(2p+ 1) [k-1]-[r]    (2p + 1) [-1] + [r]'
or than
e x            [[r _     ] {  1  r 
7r  [1-] {2p + 1 - [r]/[/ - 1]   p + 1 
and this is less than
+l      - 1      +(2-)(  - 
p7r     2/~- - I  (2- 1) (2/ - 3) + '"




465-467]    The non-convergence of Fourier's series


705


or than 2cA,/pr. Therefore the expression diminishes indefinitely, as p is
indefinitely increased.
That the third part of the above integral has the limit zero is seen from
the fact that its absolute value is less than c,+l (2p + 1) [/ - 1] 7r/[j/], or than
7rc,+l (2p + 1)/(24 + 1).
It has now been proved that
fe q, ( )sin(2n +1)z dz
Jo  Wdz
has the limit zero, if 2n + 1 increases indefinitely through a sequence of
the form
[1-1l](2p + 1), [H2,-1](2p2 +1), [/-l](2p3 +),...
where /,u, 2, A3,... is an increasing sequence of integers, and p1, P2, p,...
are such that p/v  1 - Kc {log (2, + 1)}-s', where s' < s.
It has now been shewn that the limit of the sum of the Fourier's series
oscillates; the limit being infinite, or zero, according as one or other of two
particular sequences of values of n is chosen.
467. In order to construct a continuous function which is such that the
corresponding Fourier's series fails to converge at all the points of an everywhere-dense set, we take the following definition off(x):If -7r _ x- c I, where: is a fixed point in the interval (- r, 7r), let
f(x) = 0; if 0 _-_ x - if 5 2a, let f(x) = (  2  where 0 (z) is the function
f(~)=O~;if O~-,letf)(x-'   ( 2), where b (z) is the function
that has been already discussed. In case  + + 2x < r, we take f(x) = 0, for
+ 2a< x    7r.
The limit of
dx',
I   f   (X) sin(2n + ) ('- x)'
at the point f, depends upon that of
1      (,,()sin (2n+ l) dz
It has been shewn that this limit is zero, or is indefinitely great, according
as 2n + 1 increases indefinitely through one or other of two sequences.
It follows that the Fourier's series for f(x) does not converge at the point f.
Let the function (x) be now denoted by +r (x, ); and let tf, 2,... r **.
be an enumerable set of values of I, everywhere-dense in the interval (- r, 7r).
Let us consider the function
F(W) =,       + (X, 1,)  +  C+ (, +  C  ( r) +...,
where c1, c2,... c,... are numbers so chosen that the series c, + c +... + Cr +...
is absolutely convergent.


H.


45




706


Trigonometrical series


[CH. VII


Since the upper limits of all of the functions I J (x,:) | have one and the
same finite value, it follows that the series which defines F (x) is uniformly
convergent in the interval (-7r, 7r); and thus that the function F(x) is
continuous.
The expression
1     f (') sin I (2n + 1) (x' - x)
2 Fj,,    sinx'(x'-x2  - dx'
27r      _ F (  sin I (x'- x)
is accordingly equal to the sum
1   C       (/,   sin2 (2n + 1) (' - x)
2 Y. C2 sind(x'-x)                     x,
which may be written in the form,iX, (x, n) + C2X2 (x, n) +.. + c, (x, n) +....
We have lim X, (x, n) = r (x,:), unless x = ~; at which point the limit
n=oo
may be either 0 or oo, according to the mode in which n is indefinitely increased, or may have other values between 0 and oo. A similar statement
holds as regards lim X2 (x, n) at the point 2; and generally lim Xr (x, n) is
n=-c.                                    — =o
r.(x, er), unless x=,., in which case the limit depends upon the mode in
which n becomes infinite.
At the point a,., the term  c,.X,(x, n) has its limit indefinitely great,
provided n is indefinitely increased in a proper manner; but it might happen
that the limit of
C,.+ir+l (r, n) + c,+2%.+2 (r, n) +.. 
is also infinite, although each term has a finite limit. In that case the limit
of the whole expression for the sum of the series might be finite, or zero, in
whatever manner n were made to become indefinitely great. If this happened
for a particular set of values of c,, c2,... c.,.., it would no longer happen if
these numbers were replaced by
c el, c2ee2, c3ele2e3,., *  ee2... e r,..,
where e,, e2, e3,... is a properly chosen sequence of diminishing positive
numbers. For, if
Cr,.xr(,, n) + {C,+, X,.+(r, n) + Cr+2Xr+2 (r, n) +..}
had a finite limit, when n is indefinitely increased, being dependent on the
form oo- o, the expression
ee2... erCrr(., n) + e e2... er+1 {Cr+iXr.+ (er, n) + C,.+2er2Xr+r+2 (,, n) +...}
would also have a finite limit, or become zero, only in case
Cr+lXr+i (a., n) + Cr+2er+2,x,+2 (r,, n) +...
cr+l  CTir+ (er, n) + Cr+2Xr+2 (, n) +..




467, 468]       The method of arithmetic means                   707
had unity as its limit, when n is indefinitely increased. But this limit can
be altered by changing er+i without altering e,.+2, e,+3,...; and thus er+i can
certainly be so chosen that this expression does not converge to unity, when
n is indefinitely increased.
It has therefore been shewn that, by altering the numbers c,, c2, C3,... in
a suitable manner, the infinite limit of crXr (r, n) will no longer be removed
by means of an infinite limit of the sum
Cr+lXr+l (4r, n) + Cr+2SX,+2 (r, n) +...
It has thus been shewn that it is possible so to choose the numbers
00
cl, c, Cs,..., that the continuous function F (x)=  c,.r (x,.r) is such that
its Fourier's series fails to converge at each point of the everywhere-dense
set {In.
THE SERIES OF ARITHMETIC MEANS RELATED TO FOURIER'S SERIES.
468. It has recently been shewn* that a divergent series may be utilized
in various ways for the representation of a function, and consequently for the
calculation of approximate values of the function. The simplest of these
methods is that due to Cesaro, of taking the arithmetic means of the partial
sums of the series.
If sn be the nth partial sum of the convergent series of numbers
a, + a... +   a +... + 
and S, denote the arithmetic mean - (s, + s +... + s,), then S, converges to
n
the sum of the given convergent series; or lim Sn = lim s, = s. To prove
n= o    n=i00
this theorem, let e be an arbitrarily chosen positive number, and let r be an
integer such that s -, s,+,  s -, I... are all less than e. We have now,
if n > r,
\1   {s*+ +sr+2 +..+s 
n      S= (S, + S2 +.. + Sr) _ + {' -+- Sr+2 -+ *- + S \ n
and keeping r fixed, n may be so chosen that (s, + s2 +... + s,.) is numerically
less than e; also the second term in the expression for Sn -  is equal to
s +, where      <, and by taking n suficiently great, this is numerically
less than 2e. It follows that S, - s < 3e, from and after some fixed value
of n. Therefore, since e is arbitrary, we have lim Sn = s.
n=oo
* For an account of these methods, see Borel's Legons sur les series divergentes.
45-2




708


Trigonometrical series


[OH. VII


It may however happen that, when the series EaC is not convergent, S,
still converges to a definite number 2; this number 2 may then be regarded
as the sum of the divergent series, in an extended sense of the term " sum."
For example, in the case of the oscillating series 1-1+1-+..., Sn
converges to zero, which may therefore be regarded, in the new sense, as
the sum of the series.
If, however, Ea, diverges to oo, or to - oo, and is not oscillatory, Sn must
also diverge. For Sn+m > S +S2 +  +Sn +       N, if n can be so fixed that
n+ m        m+n 
Sn+), s8+2,... all exceed the positive number N. Therefore lim Sn+m -N;
m=oo
and since N may be taken arbitrarily great, if Yan diverges to + o, it follows
that lim S, = + oo.
n=oo
From the point of view of the theory of sets of points, it may happen
that the points PI, P2,... Pn,... which represent s,, s,... Sn,... do not
converge to a single limiting point, but that the set of points PI, P,... P,...,
where Pn is the centroid of the points PI, P2,... P,, has a single limiting
point P, which represents the number,.
In the case of a series u (x) + u (x) +... + u (x)+... involving a
variable x with a given domain, it may happen that the series fails to
converge for some, or all, of the values of x, but that the function S (x), of
which the value for each value of x is the sum of the series, in the extended
sense, has a definite value in the whole domain of x.
469. This method of summation has been applied by Fejer* to the case of
Fourier's series. It can be shewn that, in a large class of cases, the Fourier's
series corresponding to a function f (x), when summed by the method
of arithmetic means, - converges to the value 2 lim tf(x + h) +\f(x -  )},
h=O
when this expression has a definite meaning; no assumption being made as
to the convergence of the Fourier's series when summed in the ordinary
manner.
Since Sn (x)= -    f (x') dx' + - S-  f (x) cos s (x - x') dx',
27jro          7I '=1 J -rr
~\  FIT  ( I  tl-1 n -  s
we have     Sn (x)=         + ~  --     cos s (x- x') f(x') dx';
from which it is easily found that
\ 1  ( JsinIn(X'-X))., -  2
Sn (x) = 2n1   S('in 21( <  /- (') dx.
* Math. Annalen, vol. LVIII; also Comptes Rendus, for December 1900, and for April 1902.
Fej6r considered only the case in which the function has a Riemann integral. Lebesgue extended
the result to the more general case; see the Legons sur les series de Fourier, p. 94.




468, 469]        The method of arithmetic means                   709
Writing x'= x + 2z, and remembering that f(x') is defined as a periodic
function, for values of x' not necessarily in the interval (- r, vr), we have
1 r^ /msin nz\2
Sn ()=         sin z  / f(x + 2z) +f(x - 2z) dz.
It is easily seen that
1     f/l sinnz2d  1
n7rJ    sin      2'
therefore we have
1      fb/Qsinn YZ)\ 2
Sn (x) - I lim [f (x + h) +f(x - h)] = n-  sin )F() dz,
where F(z) denotes
f(x + 2z) +f (x - 2z) - lim f (x + h) +f(x - h)},
h=-O
and it is assumed that x is a point at which lirn {f(x + h) +  (x - h)} has a
h=O
definite finite value.
_1     f1~ sin nz 2 d
The expression       -     (sin n  F (z) dz
n7ra \ sin z
will be first examined, in order to find whether it tends to a definite limit, as
n is indefinitely increased. The number a is fixed, and such that 0 < a 27tr.
We have
1 f^ /sinnzx\2               7 1  f  1 r,   1[cos nz
j     -sin-nz  F (z) dz= 12  si     (z) dz -  - 1Tr cos2nzF() dz 
n7rja  sin z           2n7rja sin2z          2n7rJa  sinz 
1 fr.l
and                      2nr     si   F (z) dz
2n7ra sin2 Z
is less, in absolute value, than
12n  in a  F(zI dz,
2n7r sin2 aJo  (z) dz,
or than         a      f () dx +      lim f(x+ h) +f( - h)} ]
othn2n7r sin2 a          2   h =Oj
It follows that, at any point x, at which lim {f(x + h) + f(x - h)} is definite,
h=O
t can be so chosen that
-2nrf sin z F (z) dz < E.
It also follows that, in any interval of x, contained in another interval in
which f(x) is limited, and such that lim {f(x + h)  f(x - h)} has everywhere
h=0
a definite value, this inequality is satisfied for all values of x in the interval,
provided n has a, sufficiently great value, independent of x. Again since




710


Trigonometrical series


[CH. VII


cosec2z has limited total fluctuation in the interval (a, - 7), it follows from
the theorem in ~ 458, that
cos 2nzf(x  2z)dz
sin2       - Z
converges to zero, uniformly for all values of x. It is thus seen that
1   If cos 2nz
2n-I   -    -F~z)dz
2nvrj   smin2 
converges to zero, at any point at which lim {f(x + h) +f(x - h)} exists and
h=0
is finite; and that the convergence is uniform in an interval contained in
another interval in which f(x) is limited, and in which the limit is everywhere
definite. The following theorem has now been established:If f(x) be either limited, or unlimited, but possess a Lebesgue integral in
the interval (-,  7r), then-  ) F(z) dz, where 0 < a < 2 1r, converges
nhe  J a \sin z                    2       rg s )
to zero, as n is indefinitely increased, at any point x at which
lin {f(x + h) rf(x - h)}
h=0
has a definite finite value. The convergence is uniform in any interval (a, b),
in which f(x) is limited, it being assumed that lim f (x + h) +f(x - h)} has
h=O
definite values everywhere in (a, b), including the end-points a and b.
We now have to investigate the limiting value of
n- f    s( -— ) F(z)dz,
n7r Jo  \ sin z /
which may be expressed as
1   /I(22+~1)  sinnz2      1                 nz2 
Bar I           sin nz                     sin 
-wj    F(z) sins    dz +    j      F(z). —   dz
n/n-J o     'r,- sin z / d + — 1(n+)      \ sin z 
ni f   2   \ sin z/
where 7r/(2n + 1) < a < -r.
The first part of this expression is numerically less than
(2l) F(z) dzj
7r(o
n                                                  7r
or than   -+  x the upper limit of IF(z) in the interval (, 2n- I+)  At
a point x at which lirn f(x + h) +f(x -- h)} is finite and definite, n may be
h=0
chosen so great that the ulpper limit of F(z) I in (O, 2n+  ) is less than the




469]


The method of arithmetic means


711


positive arbitrarily chosen number e. Moreover, in virtue of the theorem
established in ~ 185, in any interval (a, b), in which f(x) is limited, and in
which lim If(x 4 A) +f(x - h)} has everywhere definite values, including the
h=O
ends of the intervals, n may be so chosen that the upper limit of IF(z)l in
(, 2    + ) is less than e, for that value, and for all greater values, of n, and
for all values of x in the interval. Therefore nl can be determined, such that
1 K2n+1r  O"(/sin nz 2
-- F1 (/(     (z)     dz < e,
n7rJ   0      \sinz/    I
for n > ni, and for all values of x in an interval in which the specified
conditions are satisfied.
The second part of the expression is numerically less than
(2)"1
n]   /(2n+l) (2) 2 | F(z) l dz;
or than -     |/P  -2dz, which is less than iP; where F is the upper
4n ^j/(2n+l1) Z
limit of I F(z) in the interval (0, a). Now, for any value of x, at which
lim {f(x + h) +f(x - h)} has a finite value, a may be so chosen that 3F < e;
h=o
also one and the same value of a will satisfy this condition, for all values of x
in an interval in which lim {f(x + A) +f(x - h)} everywhere exists, including
h=O
the end-points, and in which f(x) is limited.
The number a having been so fixed, and n, then fixed as above,.and such
that 2     < a, n2 can be determined so that
2n~ + 1
1 fr\ /sin nz\2
F F(z)           dz < e,
n7T Ja     \ sinz /
for n > n2, and for all values of x in the specified interval.
If i be the greater of the two integers n1, n2, we now have
1    'sin nz  /
nr1   (sin nz) F(z) dz <3e,
for n > n, at a point x at which lim {f(x + A) +f(x - h)} has a definite finite
h=0
value. Moreover n will, if properly chosen, suffice for all points x, in an
interval in which f(x) is limited, and in which lim {f(x + h) +f(x - h)} has
h=0
everywhere a definite finite value, including the end-points of the interval.
The following theorem has now been established:If f(x) be any function, limited or unlimited in (- r, 7r), which has a
Lebesgue integral, and therefore a corresponding Fourier's series, the function




712


Trigonometrical series


[CH. VII


S& (x), which is the arithmetic mean of the first n terms of the Fourier's series,
converges to i lim f (x + h) +f (x - h)} at any point x, at which this limit has
h=O
a definite finite value, as n is indefinitely increased. Moreover, the convergence
is uniform in any interval (a, b) in which f(x) is limited, and in which
lim {f(x +h) +f(x - h)} has everywhere, including a and b, definite finite
h=O
values.
In particular, Sn (x) converges to f(x) at any point of continuity of f(x);
and it converges to f(x) uniformly in any interval in which f(x) is continuous,
the continuity existing at the end-points, on both sides of those points.
No assumption has been made as regards the convergence of the Fourier's
series when summed in accordance with the ordinary method.
470. Let the upper and the lower limits of indeterminacy of s (x), for a
Fourier's series at a point x, be denoted by s(x) and s(x); either of these
may have one of the improper values + oo, or -oo.  It is easily seen that
the points s, (x), s (x),... s, (x)... must be everywhere-dense in the interval
(s(x), s (x)); in fact this holds for any series for which the limit of the nth
term is zero, as n is indefinitely increased. For, if e be positive, and arbitrarily
small, n may be so chosen that I Sn (x) -- sn+i (x), I sn+i (x) - Sn+2 (x),... are
all < e; therefore in the interval (s(x), s(x)), there exists no sub-interval of
length e which contains no points of {S (x)}. Since e is arbitrarily small, it
follows that the points s, (x) are everywhere-dense in the interval (s (x), s (x)).
There can only be a finite number of values of n, for which s, (x) does
not lie in the interval (s(x)- e, s(x)+ e); let this number of values be r,
and let the sum of the corresponding values of sn (x) be S. We have then
Sit (x) - S (x) = -  1 (x) + S2 (x) +... + Sn (x)} - S (x),
nf
(n - r) \s (x) + el + 2  -          r --
<-r      (      }     - s (x) < - +  - -s (x) + e} < 2e,
n                   n      n
if n be taken sufficiently great. Therefore, from and after some fixed value
of n, Sn(x) is less than s(x) + 2e. Similarly, it can be shewn that S,(x) is
greater than s(x)- 2e, from and after some fixed value of n. Since e is
arbitrary, it follows that, when S,,(x) converges to a definite number S(x),
as n is indefinitely increased, S(x) lies in the interval (s(x), s(x)).
From this result the following theorem now follows:If f(x) be summable in (- r, +r), then, at any point x at which
lim {f(x + h) +f (x - h)} has a definite finite value, the upper and lower limits
h=O




469-471]


The method of arithmetic means


713


of indeterminacy of the sum of the Fourier's series, corresponding to f(x),
form   a limited, or an unlimited, interval which contains the number
-lim If (x + h) +f(x - h)} to which the arithmetic mean S(x) of the first
h=O
n partial sums of the series converges. In particular, at a point of continuity
of the function, f(x) is in the interval of which the ends are the limits of
indeterminacy of the series at the point x.
It follows, as a particular case of this theorem, that, at a point x at which
the Fourier's series converges, and at which 1lim f f(x + h) + f(x- h)} exists
h=O
as a finite number, the Fourier's series converges to that number. In particular,
at a point of continuity of f(x), if the series converges, then it must converge
to the value f(x).
It has been shewn above that, for a point x, the numbers s (x), s2 (x),...
sn (x)... are everywhere-dense in the limited, or unlimited, interval (s (x), s (x)),
of indeterminacy of the sum of the series. It follows that, if 4 be any chosen
point in this interval, a sequence s,, (x), s2 (x),... s, (x)... can be determined,
which converges to the number:. At a point x, at which {S, (x)} converges,
we may take: to coincide with the value to which the sequence converges;
and we thus obtain the following theorem:At a point x, at which lim {f (x + h) +f (x - h)} has a definite finite value,
h=0
the Fourier's series corresponding to f(x) can be replaced, by bracketing the
terms of the series in a suitable manner, and amalgamating the terms in each
bracket, by a convergent series of which the sum is lim ff (x + h) +f(x - h)}.
h=O
In particular, at a point of continuity of f (x), a suitable sequence of partial
sums of the Fourier's series converges to the value f(x).
In the particular case of a function f(x) which has either a Riemann
integral, or an absolutely convergent Harnack integral, in (- vr, 7r), the set
of points at which lim {f(x + h) +f(x - h)} has not a definite value, forms
h=0
a set of points of measure zero. It therefore follows that, for such a function,
the set of points at which it is impossible, by bracketing the terms of the series
suitably, to convert the series into a convergent series of which the sum is
1-lim  f (x + h) +f (x - h)}, has the measure zero. The requisite system  of
h=O
bracketing is in general dependent upon the particular point.
471. The method of ~ 469 can be applied to the case in which


* See Hobson, Proc. Lond. Math. Soc., ser. 2, vol. III, p. 55, where this is established for the
case of a limited function, by a different method.




714


Trigonometrical series


[CH. VII


lim {f(x + h) + f(x - h)} has no definite value at the point x. Let F, (z)
h =0
denote f(wx + 2z) +f(x - 2z); we then have
1 f(-sinn    Fz) dz
~nr 0               (sin Zn2i                      d
=_ 1   F (z).- nz dz + l-    F (z)0si    do.z.
77rJo      \sin z/     nqra ~s
Now a can be so chosen that F1 (z) - F, (+ 0) < e, and F (z) - F1 (+ 0) > e,
for all values of z such that 0 < z - a. We then have
I    [      2                                  2     -      sin
1rt^/sinnz\ 2           1             " sinnz 2      1 Psirn
-r 0    sin z. F (z) dz < -— e + F, (+ 0)}  dz+sin   -  dz.
j       s   ) Z         n7r)j              sin               s     dZ.
In accordance with the first result in ~ 469, the second term on the righthand side converges to zero, as n is indefinitely increased. Also the first
1 CP/sinz\2, 2 
term  converges to  {e +F1(+ 0)}, since j-    (n     dz = 7r. Hence we
5   2 f'e +   /' 01  Tr Jo \ sin z /
have lim Sn (x)  4L {e + F1 (+ 0)}, and, since e is arbitrarily small, we have
lim Sn (x) = IF, (+ 0); and in a similar manner, lim Sn (x) > -1 (+ 0).
= 00                                       n=o
It therefore appears that lim Sn (x), lim Sn (x), both lie in the interval
nC'oo      - on=o
(IFi (+ 0), - iF (+ 0)), which is certainly in the interval of which the ends are
2 {/(f( + 0) + f  (- O)} 2 -{f ( +  ) +  (x - o)}.
If S, (x) converge at the point x, the value to which it converges must lie in
the interval of which the ends are these two numbers. If the Fourier's series
converge at the point x, it necessarily converges to the same value to which
Sn (x) converges.
We have therefore the following theorem:If a Fourier's series be convergent at a point at which f(x) has a discontinuity of the second kind, its sum at the point lies between  {f(x + 0) +f(x-O)}
and - {f (x + 0) +f (x - 0)}, or may be equal to one of these numbers.
If, at a point x, the Fourier's series diverges to either + oo, or to - oo, but
does not oscillate, then S,(x) diverges to the same improper limit. The
above discussion shews that this can only happen when F1 (z) has its upper
limit, or its lower limit, for z = 0, indefinitely great.
It therefore follows that the Fourier's series can only diverge to + oo, or
to - o and be non-oscillatory, at a point at which the function has an infinite
discontinuity.
The series may oscillate between infinite limits of indeterminacy at a
point at which the functional limits are all finite.
The results here obtained include the theorem of ~ 469, as the special
case which arises when F (+ 0) = F1 (+ 0).




471, 472]           Properties of the coefficients                   715
PROPERTIES OF THE COEFFICIENTS OF FOURIER'S SERIES.
472. The coefficients
a = -I f(x) dx, an= -       f (x) cos nx dx, b = -   f (x) sin nx dx,
— rn        -        Tr J7 -                    r J -r
which occur in the Fourier's series which corresponds to a function f(x), may
be termed the Fourier's constants related to the function f(x). They possess
important properties which may be regarded as connected with a general
theory of the Fourier's constants related to a given function which possesses
a Lebesgue integral. These properties are independent of any assumptions
as to the convergence of the series; and thus the relation* of the constants
to the function may be denoted by
f(x)    a0 + (a, cos x + b,sin x)+... + (a, cos nx + b, sin nx)+....
The following important property of the Fourier's constants will be here
established, for the case of a limited function which is integrablet in
accordance with Riemann's definition:+ The function f(x) being limited, and integrable in accordance with
co
Riemann's definition, in the interval (- 7r, wr), the series IaO2 + X (a2 + bs2)
s=1
converges to the value       {f(x)}2 dx; where as, bs denote the Fourier's
constants corresponding to f(x).
The function Sn (x) denoting, as in ~ 469, the arithmetic mean of the first
n terms of the series, and S (x) denoting lim Sn (x), it has been seen in ~ 469,
n=co
that S (x) =f(x), at a point of continuity of f(x). The points of discontinuity
of f(x) form a set of points of measure zero. At one of these points, S(x)
has a definite value if lim [f(x + h) — f(x - h)} exists at the point; otherh=0
wise S (x) may be indeterminate between finite limits of indeterminacy, the
upper one of which does not exceed  i f(x + ) + f( - )}, and the lower one
of which is not less than 1 f (x + 0) +f(x - 0)}. The function S (x) is therefore limited, and determinate, except at points of a set of zero measure. It
follows, since
{f(x) - S (x)2 = limn {f() - S (x)}2,
B2=oo
* See Hurwitz, Math. Annalen, vol. LVII, 1903, p. 426, where this terminology was introduced
for the case of a function possessing a Riemann integral.
t The theorem has been extended by Lebesgue to the case of functions which are integrable
in accordance with his definition. See the Leqons sur les series trigonometriques, pp. 100, 101.
+ See de la Vallee-Poussin, Annales de la soc. scien. de Bruxelles, vol. xvii, p. 18. Also
Hurwitz, loc. cit.
~ This involves an extension of Lebesgue's theorem given in ~ 384, that, if s (x)=limsn, (x),
where s(x) is limited for all values of n and x, then  (x)dxto
where s.(x) is limited for all values of n and x, then  s (x) dx= lim | s, (x) dx, to the case in
a      n=0o a




716                      Trigonometrical series                     [CH. VII
that       lim    f {f(x)                        - S (x)}2  d = I {(x) - S (x)}2 dx= 0.
n=cOG -C T                    -7Tr
s=n n-s
Since            Sn (x) =    a+ 2      (a, cos sx + bs sin sx);
s=l   '/
we find that
If({ ) - Sn (x)}2 dx =   ff f(x)2 dx -   a 2 +   n-     (2 + b2)
- -7rrT~   */ —,r*~ I               s=l    '
We then have also,
— 71r
f   rf(x)  )                            sn +    s2dx
=    {f ()}2 dx - 7r ao2 +    (a2 + b2)} + 7    8=2 s(a2 +2)
— It                    8,9=1          n S=l
SX)        -    S2  2    2j).
=      (x) -  ao -   (a cossx + bs sin sx) d  +       (a2=  + 
rrr
Since lim  f   {f(x)- Sn ()}2dx = 0, we see, from    the last expression,
n=G J -7r
that lim  z     s2 (as2 + b2) = 0; and therefore, from  the first expression, it
n=o  n s=1
follows that
ff {f(x)}2 dx - T  ao2 +   (aS2 + bS2)
J — Ir                SS=l1
converges to zero, as n is indefinitely increased.     Therefore the series
1  r22  
-aO + E (as + b22) converges to the value -       {f(x)}2 dx.
8=1                                  7r -
From this theorem, Hurwitz has deduced the following more general
theorem  which was first discovered by Liapounoff* and by de la ValleePoussin:If f (x), ) (x) be two limited integrable functions, then
aoaoO' + 2 (aas' + bb')
s=l1
which, although s (x) is limited, it may be indeterminate at points in (a, b) belonging to a set E
of measure zero. To prove this extension, let Z. (x) be a function which =sn (x) at every point
of C(E), and =s (x) at each point of E. Then s (x)=lim Z (x), everywhere in (a, b). Since
neoo
Y (x)-s {(x) is zero, except at the points of a set of zero measure, it follows that. (x) is
summable, since s, (x) is so. Therefore  s (x) dx = lim  2Z (x) dx =lim  s (x) dx; and since
a      n=oo a        n=w a
b          bf_ rb                     
s(x) dx=  s (x) dx, it follows that fs (x) dx = lim  s (x) dx.
J a       J a                 J a       n=ooj a
* Stekloff states in the Comptes Rendus for Nov. 10, 1902, that the theorem was communicated
by Liapounoff to the Kharkow Mathematical Society in 1896. The theorem was also obtained
by de la Vallee-Poussin, Annales de la soc. sci. de Bruxelles, vol. xvii, p. 18.




472]


Properties of the coefficients


717


converges absolutely to the value -  f (x) ( (x)dx; where a,, as, bs are the
Fourier's constants for f(x), and ao', as', bs' those for p (x).
That the theorem holds in the case in which the Fourier's series are
uniformly convergent is shewn by a direct formation of the product, and has
long been known.
To establish the result, we observe that the two series
(ao + a/)2+   {(as + as)2 + (bs + bs )2},
s=1
2 (a - a)2 +   {(a - a,)2 + (b, - b},
8=1
are absolutely convergent, and have for their sums
r     If{/ () + q (X)}2 da,   I () () )}2 d,
7T J -rt               i/ J -- r
respectively. By taking the difference of these two series, the result follows
at once.
This result may be applied to express the Fourier's constants off(x) 1 (x),
the product of two limited integrable functions, in terms of the Fourier's
constants for the two functions f(x), + (x).
00
Thus, if       f(x)   ao + X (a, cos nx + b, sin nx),
(x)    a0 +   (a,' cos nx + b' sin nx),
n=l
00
f (x) (x (x) ~ ao + E (a, cos nx + 3n sin nx),
a=1
we have, from the above theorem,
00
ao= oaoa' + 53 (aa,.' + b,b.').
r=l
Corresponding to the function 4 (x) cos mx, the Fourier's constant
- f    (X) cos mx cos rxdx = I (a'+,r + am,r);
also the Fourier's constant for the function f(x)  (x) cos mx, which corresponds to a, relative to f(x) + (x), is am. We thus obtain an expression for
a,, by employing the two functions f(x), d (x) cos max, and applying the above
theorem. We then find that
a  ao   +         '   + a',,) + b, (bm00+,. b,.)
r=l
In a similar manner, it can be proved that
1m    2ob m' + S { a, (b'mr-bmr)-br(a'm+ra-'mar)}.
r=l




718                  Trigonometrical series                 [CH. VII
THE INTEGRATION OF FOURIER'S SERIES.
473. Iff(x) be any summable function which has a Fourier's series
1 a0 + E (a,. cos rx + b, sin rx),
r=1
the integral  f (x) dx is a continuous function, with limited total fluctuation
-7T
in (- r, 7r). It is therefore representable by a Fourier's series
1 a   ' + E (a,.' cos rx + b,.' sin rx),
r=1
which is uniformly convergent in any interval in the interior of the interval
(- r, 7r). Denoting the function  f (x) dx by b (x), we have
1 [r                1   (  sin rxl    If7,
- |    (x) cos rxdx =    r           -      / () sin r dx,
7r V-J             7rL        r_' rTr
1
or                         ac. = - - b,
r
whenever the function f(x) is such that the formula of integration by parts
is applicable. This has been shewn in ~ 394, to be the case when f(x) is
either a limited summable function, or also when it is integrable, but has
points of infinite discontinuity which belong to a reducible set of points.
In a similar manner, it can be shewn that b' = - ar - - a cos r7r. Therer     r
fore the function b (x) =  f  (x) dx is represented by the Fourier's series
7rT
* 1
ao +    - [- br cos rx + (a,.- co cos r7r) sin rx].
r=l r
To determine ao', we observe that, at the point x  - vr, the sum of the series
must be I f (-T + 0) + 4   (- 0)}, or i 7ra; therefore
oo 
b,
lao'-    - cosr7r -7rao.
r=l r
Also, in the interior of the interval (- r, rr), we have
00 1
-x = E -cos rr sin rx.
r=I r
Therefore 0 (x) is represented, in the interval (- r, 7r), by
a0 (7r + x) + S  {ar sin rx + b, (cos r7r - cos rx)},
which is obtained by integrating the terms of the Fourier's series corresponding to f(x), between the limits -rr, x.




473, 474]          Properties of Poisson's integral                      719
The following theorem has been established:If f(x) be a summable function, which, if unlimited, has points of infinite
discontinuity belonging only to a reducible set of points, then  f (x) dx, where
- r< a < 3     7- r, is represented by the convergent series obtained by integration,
term by term, of the Fourier's series corresponding to f (x).
It will be observed that no assumptions have been made as regards the
convergence of the Fourier's series which represents f(x).
It appears from the theorem of ~ 472, that, when f(x) is a limited
function, integrable in accordance with Riemann's definition, the two series
00     00
aCr2, E br2 are both convergent.   From   this result*, the convergence of
r=l     r=l
1      X 1                          1   2
the two series   X       -larl,  -brl follows.  For ar+ -   - a, hence
r=^    r=lr                          r2  r
- a,. <    a,2 + I -2;
r"' 1 
and thus    -I a,. is, for all values of r, less than some fixed positive number;
whence the convergence of the series E - I a, I follows.
PROPERTIES OF POISSON'S INTEGRAL.
474.   Let the function f(x), defined for the interval (- r, 7r), be either
limited, or unlimited, but such that it possesses a Lebesgue integral in the
interval, which integral is of course absolutely convergent.     It has been
pointed out in ~ 435, that Poisson's integral
_____     ~'bI  -  hY'. 1 - 2h cos (x - x') + h2
where -1 < h < 1, represents the sum of the convergent series
2j     f(x') dx' + E h1 cos nx -      c  n. x  f (x') dx'
-7-             n n=l  I
+        sin nx.  sinx'. f(x') dx.
It will here be shewnt that, as h converges to the limit 1, Poisson's
integral converges to the value
lim  I{f( (x + t)+ f (x- t)},
t=o
* See Bocher's "Theory of Fourier's series," Annals of Math., ser. 2, vol. vnI, p. 108.
t The limit to which Poisson's integral converges has been studied by Schwarz, in two
memoirs; see his Math. Abh., vol. II, pp. 144, 175. Schwarz has considered the case, more
general than that in the text, in which x varies as well as h; but he has confined his attention to
the case in which the function is either continuous, or else has only a finite number of discontinuities. See also Forsyth's Theory of Functions, 2nd ed. p. 450, or Picard's Traite d'Analyse,
vol. I, p. 249. For a more complete treatment of questions connected with Poisson's integral,
see the memoir by Fatou in the Acta Mat., vol. xxx.




720


Trigonometrical series


[CH. VII


where x has any constant value such that this latter limit has a definite
value.
We find, by a direct method, that
r  ^ dO = 2 tan-l_ f tan 1'6
1-2/ cosO + h2 d = 2 tan-1      h a 
and thence that
1 f=       1 - h3
27r    1- 2h-cos 8 +  dO 
Denoting the value of Poisson's integral, for a fixed value of x, by I(x),
we have
I(x) - lim i {f(x + t) +f/( - t)}
t=O
1Cr            i - h2
27r J  1 -2h cos (x- ') + h2() d;
where O(x') denotes
f(x')- lim   {(x + t) + f (x- t)},
t=O
and x has a fixed value such that
lim 2 ft (x + t) +f(x -t)}
t=O
has a definite value.
A positive number 8 can be so chosen that, if 0 <  c 8,
f (x + ) +f (x -  ) - lim {f (x + t) +f (x- )} < e,
t=O
where e is a prescribed positive number. We have then
1 [x~s _____ 1-h2 _7       6 Ef     1-A2 <d4
27r  a- 1   2h cos (x -  ) +h  (  )     <2      1 -  2h cos+ h
and the expression on the right-hand side is less than
6   7r        I - h2
27r J  1 - 2h cos (x - x ) + h2 d', or than e
The remaining part of the integral which represents
I (x) - linm  {f(x + t) -+f(x- t)}
t=O
is                   + L  I                       0 (') dx'1
is  27r [LI+  -  I  - 2h cos (x - x') + 2 t (x') dx';
and this is numerically less than
i1- h2               (') d
1 - 2hcos 83+h2  27rJI




474]


Properties of Poisson's integral


721


or than
I - h3     Flim i f    ~ t)f(x   t)H+ If       f (x') Idx
1- 2h cos 8 + ht) + f(- )                 +   r    I f(*') ' 
-Alh2             1-h            1-h
Also                   <                   <
1 - 2h cos 8 + h < (1 - h) + 4h sin2 S8   4h sin2 8;
if then i be a prescribed positive number, we have
1 - h2             1-h
1-2hcos 8+~h < q, if 4h i      < n V, or if h > (1 + 4 sin2 8)-1,
and this is satisfied if
h > 1 - 4r sin2 18.
The numbers e, q being arbitrarily fixed, 8 can be determined as above;
then, provided 1 - h < 4V sin2 $, we have
(x) - 2 linm {/(x + t) + f(x - t)} I < e + Ar,
t=O
where A is a fixed positive number, for a fixed value of x. Since e, r are
arbitrarily small, we have therefore proved that
lim I (x) = lim ' {f(x + t) + f(x -t)}.
h=l       t=0
For, if ' be arbitrarily fixed, we may first choose e, so that e <  ~, and we
may then choose V < '/2A; hence, if h be sufficiently small, I(x) differs
from the limit by less than n.
The following theorem has now been established:If f(x) be a limited or unlimited function, possessing a Lebesgue integral
in the interval (- r, 7r), then, for any fixed value of x, for which
lim  {f (x + t) + f( - t)}
t=O
has a definite value, Poisson's integral converges to the value of that limit, as
h converges to 1. In particular, at a point of ordinary discontinuity of f(x),
Poisson's integral converges to the value
I2 I (x + O) + f (x - 0)};
and, at a point of continuity of f (x), it converges to the value f (x).
It has been already pointed out in ~ 435, that no conclusion can be drawn
as to the convergence or non-convergence of the Fourier's series at such
a point x. In case however the Fourier's series converges, it follows from
Abel's theorem (~ 356), that it must converge to the same limit as does
Poisson's integral. We therefore obtain the following theorem:If f (x) be any function, for which the Fourier's coefficients exist, as proper
(Lebesgue, or Riemann) integrals, or as absolutely convergent (Lebesgue, or
H.                                                          46




722


Trigonometrical series


[CH. VII


Harnack) improper integrals, then at any point x at which the Fourier's series
is convergent, it must converge to the value
lima { If (x + t) +f (x- t)},
t=O
provided this limit have a definite value.
This theorem has already been established otherwise, in ~ 470.
It may be remarked that, if (a, 13) be any interval in which f(x) is
continuous, the end-points a, /3 being points of continuity, then the number S,
corresponding to a fixed e, may be chosen so as to be independent of x, for
all values of x in (a,,). This follows from the property of uniform continuity of a continuous function (~ 175).
Also A is less than a fixed number, for all values of x in (a, /). It
therefore follows that Poisson's integral converges to the value f(x) uniformly
in the interval (a, 83), in which f(x) is continuozs.
APPROXIMATE REPRESENTATION OF FUNCTIONS BY FINITE TRIGONOMETRICAL
SERIES.
475. If the function f(x), defined for the interval (- r, 7r), be continuous in the interval (a, /3), contained in (- r, 7r), including the end-points
a, /, it has been seen in ~ 474, that Poisson's integral converges to the value
f(x), uniformly in the interval (a, /3), as h converges to the value 1.
Therefore, a value h, of h, may be chosen, corresponding to an arbitrarily
fixed positive number e, so that f(x) differs from the sum of the convergent
series
2r f   (f (x) dxa + ~   h1n cos nx. - If  () cos nx' dx'
-7r J -' \1                  ' _  J -r'
+ sin nx.-   f(x) sin nx' dx
q]J - 7rT
by less than Ie, for all values of x in (a, /). Since the series converges
uniformly for all values of x, an integer m may be so fixed, that the remainder
of the series after the mth term is numerically less than Ie, for all values of x.
In this manner, we obtain* a finite trigonometrical series
A +(A     cos x + Bsin x)...o  + (AA  cos  mx+  B sinnmx)
the sum of which differs from f(x) by less than e, for every value of x in the
interval (a, /3) in which f(x) is continuous.
This mode of approximate representation of f(x), in the interval (a, /), is
clearly not unique, because the values of the function in that part of (- r, 7r)
* See Picard's Traite d'Analyse, 2nd ed., vol. I, p. 275.




474-476]


Finite trigonometrical series


723


which is not in (a, ) may be altered in any manner, subject only to the
integrability of f(x) in (- r, 7r), and the continuity of f(x) at the points
a, /3.
In the above finite series, each of the circular functions can be expanded
in powers of x, and the result rearranged as a power-series, of which the sum
consequently differs from f(x) by less than e, for all values of x in (a, /).
Since the power-series is uniformly convergent, we thus obtain a proof of
Weierstrass' theorem, already established in ~ 373, that a finite polynomial
P (x) can be determined, such that f(x) - P (x) < 2e, for all values of x in
(a, /3); the number e being arbitrarily chosen.
Another method*, not involving the use of Poisson's integral, may be
employed to determine an approximate representation of a function f(x),
continuous in (a, /), by means of finite trigonometrical series. Choose 1, so
that - 1 <a <  < 1. As in ~ 373, a continuous polygonal line can be constructed, such that its ordinate, for each point x in (a, /), differs from f(x)
by less than le. The polygonal line may be extended to the whole interval
(-, 1), so as to be a continuous polygonal line for the whole interval, and to
be such that its ordinates at the points  =l, -I are equal to one another.
In virtue of Dirichlet's theory of Fourier's series, the polygonal line may
be represented, for the whole interval (- 1, I), by a Fourier's series,    t /      n7rx. n7rocx
ao +     acos COS - + bsin-;
n= 1                   I
and, by the theorem of ~ 451, this series converges uniformly in (-, 1), to
the value f(x). The sum of the Fourier's series differs from f(x) by less
than Ie, at every point of (a, 3). The integer m may be so chosen that the
sum of the terms for n > m, is less than  e, for all values of x in (a, /,, on
account of the uniform convergence. Therefore the finite series
a0+    (an cos — +b n sin — l,
n=l \      L          inl
has the required property, that its sum differs from f(x) by less than e, for
all values of x in (a, /). This method may be applied, in the same manner
as in the case of the preceding one, to prove Weierstrass' theorem relating
to the approximate representation of a continuous function by a finite
polynomial.
476. Let f(x) be a function such that both f(x) and { f(x)2 possess
Lebesgue integrals in the interval (- r, 7r); and let snm() denote the sum of
a finite trigonometrical series
n=nm
4Ao + 2 (An cos mx + B sin mx).
n=1
* Volterra, Rendiconti del Circolo mat. di Palermo, vol. xi, 1897, p. 83.
46-2




724


Trigonometrical series


[CH. VII


Let us consider the integral
Im   f   {f () -Sm(X)}2 dx.
We find that
I     - =    I {f(x) dx +  A-7r     f (x) dx
J =1 -IT  Lr J -nn I         2   r J -7T                    2
+     {An --     f (x) cos nxdx  + Y   Bn --     f (x)sin nxdx
n=                              n1 "f  - J =1  7Tr
27r  r7r         '    =1   rr7r
-21 ^ f- f(x) dx} -,r      [   _{f  f(x) cos ndxt
+ {    f (x) sin nxdx}.
If Im be regarded as a quadratic function of
Ao), A   B1... Am, Bm,
it is clear that the value of Im will be least, when
1 C"r                1 [l.~f~,  ~~~ ~~,7r
Ao =   ff(x) dx,     An= -      f() cos nxdx,
7T J -- 7J
Bn =  I f/(x) sin nxdx,
for n= 1, 2, 3,... m; i.e. when A0, An, Bn are the Fourier's coefficients
corresponding to the function f(x). These values of A0, An, Bn are therefore
such that the finite trigonometrical series gives the best approximation to
the value of f(x), in accordance with the standard of the method of least
squares. The following theorem has been now established:If* f(x) be defined for the interval (- r, 7r), and be such that both the
function itself, and its square, possess Lebesgue integrals in the interval, then
the values of the 2m + 1 constants A,, A1, B1... A,, Bm, which are such that
r7r    _         nf    _                     2
I    /L(x)    o- A-   (Am cos mx + B, sin mx   dx,
-_ - Ln=l 
has the smallest value, are the Fourier's coefficients corresponding to the
function f(x).
The minimum value of the integral Im is
-r              r      n=1
{f(X)}2 dx - 7r [\ao +    (an +bn2),
where a0, an, b10 denote the Fourier's constants corresponding to the function
f(x). It follows that this difference is essentially positive, whatever value m
o00
may have, and therefore the series  a02 + Z (a~2+ bn2) is necessarily conn=l
* This theorem was given by Toepler, in a somewhat less general form, see Wiener Anzeigen,
vol. xiII, 1876.




476, 477]


Differentiation of Fourier's series


725


vergent. It has been shewn in ~ 472, that, on certain assumptions, the series
converges to the value -   f(x)}2dx. An attempt was made by Harnack*
7r J -r
to establish this fact directly, and to found thereon a theory of the convergence of Fourier's series.
Ow    n
It follows, from the above result, that the series 2 an2, E bn2 are both
1     1
convergent, and therefore that lim an = 0, lim bn = 0, which has already been
n = oo    n = oo
established in ~ 454, independently of the assumption here made, that
If(x)}2 is integrable in (- v, 7r).
THE DIFFERENTIATION OF FOURIER'S SERIES.
477. In general, the series obtained by differentiating a convergent
Fourier's series is not convergent, as may, for example, be seen in the case
00 1
of the series S-sinnx; neither is the series so obtained necessarily the
1 n
Fourier's series corresponding to f' ().
Let f(x) be a limited function, with only a finite number of ordinary
discontinuities; let it also be assumed that f' (x) has a Lebesgue integral in
(- w, 7r), and that, if it have points of infinite discontinuity, such points form
a reducible set. This is consistent with there being a set of points of zero
measure at which f' (x) has no definite value. At the points of discontinuity
off(x), we may regard f' (x) as undefined. We have then
f(x) cos nxdx =   n  f(x) sin nx    -   -     (x) sin nxdx
7-Tr  '           -Tr J-                 T W J-Tr
=     - _1   I {f(a + 0) -f(a - 0)} sin  -   (x) sin nxdx,
the summation Z referring to the finite number of points a of ordinary
discontinuity of (x) in the interior of (-7r, 7r). In a similar manner, we
find that
| f(x) sin nxdx
=   [(- 1) {/f(- 7r + 0) -f(7 - 0)} + I {f(a + 0) -f(a - 0)} cos na]
llrq1
+ -      f (x) cos nxdx.
l7r
Also
f' /()d= l[f(7r-0)-f(- r +)-             {f(a + O)-f(a) - 0)}].
* See two articles in the Math. Annalen, vol. xvIn.




726


Trigonometrical series


[CH. VII


If then, the Fourier's coefficients for the functions f(x), f' (x) be denoted
by a0, a, by, and a,', an', bn' respectively, we have
ao =     [f(r- 0) -f(- 7r + 0)] - -  {f(a +  ) -f(a - 0)
an= nbn - - [(- 1) {f(- r + 0) -f(r - O)} +  {(a +0) -f(a - O)} cos na],
7r
b, = - na,, - I-  f (a + 0) - f(a - 0)} sin na.
7r
In particular, if f(x) be continuous in the interval (- r, 7r), so that the
function obtained by extending f(x) beyond the interval, in accordance with
the rule f(x) =f(x + 27r), is continuous except at the points - r, 7r, we have
ao'=  {f(r) — f(- )}, a/ =nb + (- ) {(w) -f(- )},
bd = - na,,. Unless f(7r)=f(- r), the Fourier's series corresponding tof' (x)
is not obtained by term by term differentiation of the Fourier's series for f(x).
Even when this condition is satisfied, no assertion can in general be made as
to the convergence of the Fourier's series for f' (x). We have thus obtained
the following theorem:If f(x) be continuous in (- 7r, 'r), and if f(- 7r) =f(7r), and f' (x) have a
Lebesgue integral, and have at most a reducible set of points of infinite
discontinuity, the Fourier's series for f'(x), whether it converge or not, is
obtained by the term by term differentition of that corresponding to f(x).
If it be known thatf' (x) has limited derivatives at any point, or if
imf'( + h) -f' (   + 0)  lm f'(x - h) -f'(x- 0)
h=+o         h         ' 7=+0        -h
are definite, or are indeterminate between finite limits of indeterminacy,
then, in accordance with Theorem  III, of ~ 457, the Fourier's series for
f'(x) converges at the point x.
478. In case the function f(x) have derivatives f' (x), f" (x),... of any
number of orders, and f(x), f' (w), f" (x),... are all limited, and continuous
in (- r, 7r), except at a finite number of points at which they have ordinary
discontinuities, the coefficients an, bn may be expressed in a form which
exhibits these discontinuities.
At a point a of discontinuity of f(x), the functionf' (x) may be regarded
as undefined, the values of f' (a + 0), f' (a - 0) being
lim  (a + h)-(   +0)   lim f(a-h)-f(a-O)
h=+o        h        '     =+0     -h
respectively. A similar remark applies to the higher differential coefficients.




477-479]         Differentiation of Fourier's series                727
We find, by integrating twice by parts,
an = -    X {f(c + o) -f ( - 0)} sin n -     I' ( +    -' ( -) cos 1nn7r                             flew ~{f'(,// +O-f'C'3-O)1cosn/3
1  pr
I 1
-2  J   f/I (x) cos nxdx,
b6, =    {f(a + 0) -f(a - 0)} cos na -    I f {f' (/, + 0) -f / (3 - 0)} sin n/3
11   7'
- 27r    f /  (x) sin nxdx,
where - r is now included among the points a of discontinuity of f(x), and
amongst /3, the points of discontinuity of f'(x). The points a in general
occur amongst the points 3.
We may proceed, by further* integration by parts, to express an and bn
in a series proceeding by powers of 1/n, the coefficients of which involve the
measures of discontinuity of the functions at the points ac, 3,....
Conversely, if the Fourier's coefficients forf(x) are given in the forms
a = -   A sin na + -  B cos n3 +...,
n             n2
b   - -  A cos na+ -2 B   sin n/3 -...,
so that the Fourier's series has for its general term
- A sin n (a - x)+ - 2B cos n ( - x)+...,
1                  n1
we have
f( +   )-f(a -   )= -  A, f'(/3 +o) -f'(/3)  - -  )= - B,.
Thus the points of discontinuity, and the measures of discontinuity of
f(x), f' (x),... are determined when an, bn are exhibited as series proceeding according to powers of 1/n.
479. The following further theoremst relating to the differentiation of
trigonometrical series will be stated:If the trigonometrical series
o00
-~ao + E (an, cos nx + bN sin nx)
converge for a particular value c of x, and if the series
00
E (- nan sin nx + nbn cos nx),
n=l
* See Stokes "On the critical values of the sums of periodic series," Math. and Phys. Papers,
vol. I, where this investigation is carried out in detail, and the resulting formulae for the
differentiation of Fourier's series are applied to physical problems.
t See Bocher's "Introduction to the theory of Fourier's series," Annals of Math., vol. vII,
p. 120. The second theorem is substantially due to Lerch, Annales sc. de l'dcole normale, ser. 3,
vol. xi, p. 351.




728


Trigonometrical series


[CH. VII


obtained by term  by term  differentiation, converge uniformly in an interval
(a, /) which contains the point c in its interior, then the original series
converges uniformly in (a, /3), and the function f(x) represented by it has,
throughout the interval, a differential coefficient represented by the derived
series.
oo
If the series           E (a, cos nx + b, sin nx)
n=l
converge for a particular value c of x, which is not zero or a multiple of -r,
and  if lim an = 0, lim b = 0, then throughout an interval (a, /3) which
nn=oo      fl=ci 
contains the point c in its interior, but does not include the point 0, or k7Tr,
where k is any integer, the series converges uniformly in the interval (a, /3),
and the function f(x) represented by it will have a differential coefficient f' (x)
given by
2 sinx. f' () =  E {[(n - 1) an, - (n + 1) a,,n] cos nx
n=+ [(n - 1) b,_, - (n + 1) bn+,] sin nx},
where a_ = b_- = ao = bo = 0, provided this last series converges uniformly in
the interval (a, /3).
It is clear that, for a function f(x) which possesses differential coefficients
of all orders in the interval (- r, 7r), it is not in general possible to obtain
representations of all these differential coefficients by means of successive
term by term differentiation of the Fourier's series which represents f(x).
The following theorem, due to Borel*, gives the means of obtaining the
requisite representation of such functions:Having given a function f(x) which has differential coefficients of all orders
throughout the interval (- r, 7r), the function can be represented by means of a
series of the type
00
I (Anx + an cos nx + bn sin nx);
n=O
and the differential coefficients of f(x), of all orders, are represented by the
series obtained by successive term by term differentiation of this series. All the
series so obtained converge uniformly in the interval (- 7r, 7r).
GENERAL EXAMPLES.
1. The trigonometrical series
b, sin x + bsin 2x +... + b sin nx +...,
is uniformly convergent in any interval not containing the point x =0, or any point
0o
x= +2k7r, (k integral), if lim b, =0, and if also 2 I b, -bn+,  be convergent. For
-n=oo             ~n=1
n-1
2 sinx. sn (x) = b cos  - 2 (b,-b+,+) cos i (2r+1) x-bn b cos ~ (2+ 1) x,
r=l
* See the Lemons sur les foznctions de variables reelles, p. 68, where this theorem is proved.




General examples


729


whence the result follows. The series* converges for all values of x, if lim bn=0, and if
n=-oo
also bn i bn+1, for all values of n greater than some fixed value m; the convergence is
then uniform in any interval which does not contain x=0 or x+= 2kTr, for any integral
value of k.
The series ao+al cosx+a2 cos 2x +..., may similarly be shewn to converge uniformly
in any interval not containing x=0, or any point + 2kwr, if liman=0, and if also
a)00~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~1=0
1a,-an+il be convergent. If liman=0, and an Ran+l, for n>m, the series* conn=1                            n=oo
verges for all values of x, except 0 or + 2k7r.
2. Let f(x)+ be a function, of period 27r, limited and integrable in any interval which
does not contain the point x.=0, or any point x=2k7r; but let f(x) not satisfy these
conditions in the neighbourhood of x = 0. Let it be assumed, (1) that.f(x) +f ( -x) I is
integrable, in (0, 7r), (2) that lim {xf(x)}=-0, and, (3) that xf(x) has its Fourier's series
x=0
convergent at the point x =0. The Fourier's coefficients an', bn, for the function f'(x), then
exist, and lim an=O0. Also it follows from (3), that lim bn=0. For this last condition is
equivalent to
sin A (2n +~ ) xvO
lim   /     (x')tan x' sin 2 (+1) -x
=00   -                sin - x'
which holds if f(x) tan ~x have its Fourier's series convergent at x= 0; and f(x) tan 2x
may clearly be replaced by xf (x).
It can now be seen easily that
Ir sin~r(2n+l)(ix- x')
sinu iv-iv')  f( I) did
_-    sin I (x-)     /
has the limit 0, when n is indefinitely increased, on condition that, in the neighbourhood
of x'=0, those elements which correspond to values of x' of opposite sign, but of equal
values, are made to coalesce. When the conditions (1), (2), (3) are satisfied, the necessary
and sufficient condition that the Fourier's series should converge to f(x) is that that
function which = f (x) in the neighbourhood of the point x, and is elsewhere zero, should
be representable by a Fourier's series.
7r
sin -
X
Let                f(X)==     1 I     1' where 0< x-a<e-e,
x log - log log -
and let f(x) +f(- x) =0. This function satisfies conditions (1), (2), (3), and is representable by a series
7rX. 2ixv
al sin - a2 sin -r+....
a          a
If(x) I is not integrable, although f(x) is so; thus the series is a generalized Fourier's
series.
O sin nxi
3. The convergent series t 2      represents a function which is not integrable, in
n=2 logn
any sense, in an interval containing the point x=0.  The series  2  c     is not
n=2  logn
convergent.


* Schlomilch, Compendium d. hoheren Analysis, vol. i, ~ 40.
+ See Fatou, Comptes rendus, March 26, 1906.




730                       Trigonometrical series                        [CH. VII
00
4. In the series 2 sin (n! rx), the coefficients do not become indefinitely small, and
1
therefore the series is not a Fourier's series. The series converges, however, for all rational
values of x; it also converges for an infinite number of irrational values, for example,
for x=sin 1, cos 1, 2/e, and for multiples of these values; also for odd multiples of e.
This example is due to Riemann, and the series has been considered in detail by
Genocchi *.
00           00
5. Consider the series 2 c cosn2x, 2 c sinn2x, where co, c, c2,... are positive
n=O           n=l
numbers, and such that lim c,=0, but such that 2 cn is divergent. The points of conn=co                  n=O
vergence, and the points of divergence, of these series both form everywhere-dense sets.
These series have been treated in detail by Genocchi.
1
6. The function / (x) =  2 - (nx), where (nx) denotes the excess of nx over the nearest
n=ln
integer, and where (nx)=0 when nx is half an odd integer, is not integrable in accordance
with Riemann's definition. Riemann has however given the series
1,  [-(-     sin) 2n7r,
7r nw=l    n
as representing f (x); where the summation So refers to all the factors 0, of n.
RIEMANN'S THEORY OF TRIGONOMETRICAL SERIES.
480. After the fundamental investigation of Dirichlet, in which sufficient
conditions were obtained for the convergence of the Fourier's series corresponding to a given function, the next great advance in the theory was
made by Riemannt, in his celebrated memoir on the representation of a
function by means of trigonometrical series. This memoir formed the point
of departure, on which much of the subsequent development of the theory
depended.
Denoting such a series by
Ao       +     Ai+A+... + An +...,
where                Ao=   ao,   A= an cos nx + b, sin nx,
it is assumed, for the most. part, that
lim (an cos nx + bn sin nx) = 0,
n = 0o
for each value of x in a given interval. It was proved later by Cantor, that
this assumption necessarily implies that lim an = 0, and lim bn = 0.     In some
n = 00          n = oo
* Atti di Torino, vol. x.
t "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe." This
memoir, originally written as a thesis in 1854, was published in the Abhandlungen d. K. Ges. d.
Wissenschf. zu Gottingen, vol. xIII. See also Riemann's Gesammlelte Werke, 2nd ed., p. 227.




479, 480] Riemann's theory of trigonometrical series


731


parts of Riemann's investigations, it is sufficient to make only the wider
assumption that
an, cos nx + b, sin nxl
is less than some fixed positive number K, for all values of n, and for all
values of x in a prescribed interval. It is not assumed that the coefficients
necessarily have the form of the Fourier's coefficients; so that the theory
refers to trigonometrical series in general.
For each value of x in the interval (- r, 7r), for which the series converges,
the limiting sum will be denoted by f(x). This function f(x), defined by
means of the given series, is defined only for those values of x for which the
series converges. In later investigations undertaken by Du Bois Reymond,
it is assumed that the function f(x) exists also at a point x at which the
series oscillates; the function being regarded as multiple-valued at such a
point, with limits of indeterminacy identical with those of the series at
the point.
The question asked and answered by Riemann was as follows:-The
functionf(x) being defined at the points of convergence of the given series,
as the limiting sum of that series, what can be inferred as to the properties
of the function f(x)?
In order to answer this question, Riemann undertook an examination of
the properties of the function F(x), which is represented by the series
1            1          1
C+ C'x+    Aox- - A, - A2       -....
obtained by integrating the terms of the given series twice.
For any value of x for which IA[l is less than some fixed positive number,
for all values of n, the function F(x) exists.
In any interval of x in which I A, j is less than a fixed positive number e,
for all values of x and n, the terms of the series are less, in absolute value,
than those of
jl++2       +...+32+...);
and therefore the series is uniformly convergent. It follows that the function
F(x) is continuous in the interval. This is, in particular, the case if
lim a = 0, lim b = O.
-=co     n = oo
The function F(x) has the properties formulated in the following three
theorems:(1) For any value of x, for which the series Ao + A1 + A +... +An +..
converges, the expression
F (x +a- +  )-  (x + a- ) - F (x - a +3) + F(x - a-  )
4a/3




732                     Trigonometrical series                     [CH. VII
converges to the value f(x), when a, 3 are indefinitely diminished in any manner
such that the ratio of one to the other remains finite.
In particular
F(x + 2)+ F (x - 2a) - 2F(x)
4a2
converges to f(x), as a is indefinitely diminished.  In this theorem, it is
sufficient to assume that, in a fixed neighbourhood of x
an cos nx + bn sin nx I
is less than a fixed positive number e, for all values of n, and for all points
in the neighbourhood.
(2)  For any value of x whatever
F(x + 2a)+F(x - 2a)- 2F(x)
2a
converges to the limit zero, as a converges to zero.
It is unnecessary that the function f(x) should exist for the value of x
concerned, and it is sufficient that, for all values of x,
an cos nx + bn sin nx
should have the limit zero, as n is indefinitely increased.
(3)  If b, c are two arbitrary constants, such that b < c, and if X(x), X'(x)
are functions which are continuous in the interval (b, c), and vanish for
x = b, x= c, and if further X(x), be such that X"(x) is* a limited and integrable
function in the interval (b, c), then the expression
21 F (x) cos   (x - a). \() dx
b
converges to zero, as /u is indefinitely diminished, uniformly for all values of a.
It is necessary that an and bn have the limit zero.
481.  Proceeding to the demonstration of these theorems, it may be
observed that Riemann's theorem (1) can be generalised so as to include the
case in which the series Ao0 + A+ A+ +... does not converge at the point x,
but oscillates, with finite upper and lower limits U, L, of indeterminacy. In
that case it will be shewn that the expression
F(x+ 2a)- 2F(x) + F(x - 2a)
4a2
has a limit, for a = 0, which, whether it be definite or not-, lies between the
numbers
(U+ L)      2(U-L) 1 +       + 
* Riemann restricts X" (x) to have only a finite number of maxima and minima in the interval,
and makes no mention of uniform convergence.
Du Bois Reymond gave the value 2(U+L)  ( +- + )(U-L); see Abh. d. bayerisch.
Akad. vol. xII, p. 136.




480, 481]   Riemann's theory of trigonometrical series             733
In the case U= L, this reduces to Riemann's original theorem. To prove
the generalised theorem, we find that
F(X + 2a)-2F ()+F(x- 2a)      A   A, /sin a + A (sin 2a;2
WA02           =                     /sinlt  a  2a  +
Now A0 + A +-... + An_1 lies between the two numbers
2(U+ L) ~ 2 (U-L) ~ en;
where En is a positive number, such that, if 8 be any arbitrarily small prescribed positive number, a number m can be found, such that for n - m, En < 8.
We may accordingly write
Ao+A1 +... + An_ - (U + L) + I (U- L) f0 +qn
where 1    n >- - 1, and,n is a number such that IJ  < 8, for n  m.,     ( /sin na\2
The series    An    a ) may be written
~   U-L6         ) (sin n - la2  (sinnai\2
+ J =1 (2          {       I  )n-la  na m
and we may divide the series into three parts; first, from n = 1 to n = m,
where I1nl < 8, for n _ m; next, from n = m + 1 to n = s, where s is the integral
part of 7r/a; and lastly from n = s + 1 onwards; we suppose a to be chosen so
small that ma < rr. The first part of the sum consists of a number n, of terms,
where m is independent of a; and this part has the limit zero, when a is
indefinitely diminished. The second part of the sum is such that, in every
term,
(sin n -la)2  (sin na 2
\ n-a          na 
is positive; and therefore this part of the sum lies between the two values
+( U-L +)       sin ma2   (sinsa2'}
- ~ 2,Ir. ma         sa    f
The third part of the sum may be written in the form
281  9     +    1s2  n - la        na    )
n=s+l\  2         / (km-la)            m    If
(U-L     +   ) sin(2-l)asina
m22
n=s+l   2   O +*In
Now                 (sin n- la2   /sin  - la 2
n- la           nu 
is positive, and less than
(rn-1)2 n) a2




734


Trigonometrical series


[CH. VII


sin a sin (2n - I)a.
and   n     2a2       is numerically less than  1; hence the sum of the
n2a2                               n2a'
third part of the series lies between
(  12  +  ) s2a-+(  2 + 8 8
and the same expression with the sign changed.    When a is indefinitely
diminished, sa has the limit Vr; hence the limiting sum of the series
lies between the two numbers
+8          1+- +;:(U+L)+(       2-L +)(1+27r     7r2
or, since 8 can be chosen as small as we please, it lies between
(U+L)       (U-L) (1i +   I + )
In case x is a point of convergence of the series X An, we have U= L =f(x),
and the above limit becomes f(x).
To prove Riemann's original theorem, we have
F(x     + p,) - 2F(x) + F(x - a - 8)  (a + /3 )2 [f() + 81]
F(x + a - /3) - 2F(x) + F(x - a + /3) = (a - 3)2 [f (x) + 82]
where 68, 82 converge to zero, when a +,3, a- /, do so. From these equations
we find
F(x + a + p) - F( + a- p/)- F(x - a + /) + F(x)
4a/3
=f () + (       - ) -   /;
the expression on the right hand side converges to f(x), when a, /3 are
indefinitely diminished so that their ratio remains finite, since o8, 82 converge
to zero; and therefore Riemann's original theorem is established.
482. To prove Riemann's second theorem that, whether,An converges
or not, so long as lim An, for each fixed x, is zero,
F(x + 2a)- 2F(x) + F(x- 2a)
2a
converges to zero, as a does so, we observe that the series E A sin n  can
be separated into three groups of terms. The first group is from n = 0 to
n=m, where m is so chosen that A      I < e, for n> m; the sum  of these
terms remains finite, as a is indefinitely diminished; denote this sum by S.




481, 482]   Riemann's theory of trigonometrical series          735
The second group of terms is from n = m + 1 to n = s, where sa =  a fixed
C
number c<(s +l)a; the sum of these terms is numerically less than e-. The
remainder of the series is numerically
e      1   e
<c2 =+ n2< sa2
and -  differs arbitrarily little from  e, when a is sufficiently small.  It
follows that
F(x + 2a) - 2F(x) + F(x- 20) = 2       si na )
2a                          na
< 2(\Sc a+ec+6)
whence the theorem follows, since e is arbitrarily small.
There now remains Riemann's third theorem, which requires the consideration of the expression
2   F(x) cos, (x - a) X(x) dx,
when b becomes indefinitely great, the function X(x) satisfying the conditions
already stated. Since F(x) is represented by a uniformly convergent series,
we may replace the integral by the sum of the integrals obtained by
substituting, in the integral, the series which represents F(x). We have
then
~2 fF(x) cos ((x-a) X(x) dxa =  2  ( +G '+  Ax 2) +  oos   -  (x) (co  ) ()dx
00  2  f0
--     fC  An cos p (x - a) X(x) dx.
We shall first consider the series
9o,= 2 1
f    | AI f COS (x- a) ) (x) dx.
The general term of this series is
/2    f:
2 an, [cs {(I +  ) - pal + cos {(- n) ) -  }pa] (x) dx
+    b f [sin {(/ + n) x - a} - sin {(/ - n) x - /a}] X(x) dx,
and each of these integrals may be integrated twice by parts, since
' (),   X" (x)




736


Trigonometrical series


[CH. VII


are both limited integrable functions. The general term then takes the
form
/2
22 ( + n)2 an cos 1(u + ~n)x -,Ua} "(c) dx
2        re
- 2q     2 a~n) a  cos (- n) x - a} Xl"(x) dx
-22 (2(+  )2 b  sin {(t + n) - ~a} x"(x) d
+ 22(     )2 b f sin {(u - n) x - a} X" (x) dx.
The series can be divided into four parts, containing respectively the
four sets of terms here expressed; and these four parts will be considered
separately.  Corresponding to an arbitrarily chosen positive number e,
a value pi, of,p can be so chosen that
j cos ( + n) x. X"(x) dx,  sin (a + n) x. X"(x) dx
are both numerically < ~e, for pA H 1u, and for every value of n; this follows
from the theorem in ~ 454. It then follows that
I  cos {(i+n) x - la X"(x) dx < e, when p_- >t.
The terms of the first part of the series are, for,A -,u, numerically less
than the corresponding terms of the series
a2
n=1  n
therefore the sum of the series is numerically less than
n=l n 
which is arbitrarily small. Therefore the sum of the first part of the series
converges to zero, as.L is indefinitely increased. In a similar manner, it can
be shewn that the third part of the series has the same property. Next, let
us consider the second part of the series.
Choose an integer nj, such that
cos (px - pa) '"(x) dx < e, for p  l n,
and also such that an ] < e, for n > n,; it then follows that
00  1                          0  1
X =  1aI an, I< e. We shall denote E - an I by k.
n=n,l+n                          1 n
Let y, be any value of pA greater than 2n,, so that
f cos {(a -n) x -  a} X"(x) dx <, for y AjtL and n _n.




482]       Riemann's theory of trigonometrical series               737
We then divide the series into parts
nl    L-2    {+2   nit+l      oo
Y+         ++I  +             2 +, +
n=1 l      n =l+ l  n= —1  -=L+3  n=n1i+-+l
and consider these five portions separately. We have, at once,
nl I     ~L2    n=1ni1        Ake/l~,2
-   <     -    2 <           2-2- ( a   <<2  2ke; for 1 _ pu.
n=1 i 2(-ni)2     =1 n       2 (l - n
Let the integer t be so chosen, for any particular value of p, that
b< -L- lL+ 1;
we then have
1             l-l
a —2   NKe L-2                  N KE      dx
<                       < 
|n=+l+l  2/A n=n,+l I _     2 n2  2, J      (( -  )2
rb
where K denotes J    X"(x) dx.   For, if the interval of integration be
divided into portions
ni ',     2 '  V 2, 'i n, + I  nj 
/^i  1+I\ k      P       4-2
we obtain the series by giving the integrand its least value, in each portion.
On evaluation of the integral, we have
Y2   <KE(!~         +          _-       +2 log"            }.
m:n14-1 22 Lkn    t       p                    ~- /_- n,  n,  ) -  +I
Am=n,+L  2y (no    -1   /b-b + 1,-ni          ^ l+-b4 1)
1 T /3 3  4              3  2 
<   Ke (2+-log     ) <  e - +, for,_,.
2                        42 1.eI
Next, we have
(&+2   E  f.2   /12..2       2      C
<   {,    + ~  +.   (x) I dx
<4e f   X (x) dx.
rln-  x{ l &.
nl+c+l 1
An upper limit for          may be obtained by a method precisely
n=Lt+3 
t-2
similar to that employed above in the case of  X; it can thus be shewn
n=~n+l
to be less than a certain multiple of e, for all values of >/u-!.
It can also be shewn, as in the case of the first portion of the series, that
2    < 2ke, for /p _-Li.
n = ni+c+l1
The numbers e, ni having been first fixed, /, can then be fixed so that, for
p  p1 > 2n,, the sum
00       U2 
2~n          2 an  COS cos(- n) x -- /a} X"(x) dx
=1    2n2 ( -n)2
H.                                                           47




738


Trigonometrical series


[CiI. VII


is numerically less than a certain multiple of the arbitrarily chosen
number e. Therefore this part of the series converges to zero. The fourth
part of the original series may be considered in the same manner; and thus
the sum of the original series converges to zero, as / is indefinitely increased.
We have now to consider the expression
2    (C+ C'x + Aox2) cos p (x - a) X(x) dx.
J b
It can be verified that
2u" (C + C'X +. 1     d72 r (            Q A             )?({c+c'f+-~AoX2)cosp(x -a) =-         C — -- +C'x + -2AoX2 cos  (x- a)
-2(C'+Ax) sin    (x- a)
hence, on integrating twice by parts, the expression to be considered takes
the form
/b [2 (C0+ Aox) sin  (x- a)"  (x)
-L   - -3A, + c' + 'AoX2 cos i (x - a) X"(x) dx.
By a further application of the theorem of ~ 454, it is seen that p' can
be so determined that this integral is numerically less than e, for, > '.
It has now been shewn that
/2   F (x) cos p (x - a) X(x) dx
converges to zero, as 1/ is indefinitely increased, uniformly for all values of a.
483. Riemann has employed the theorems (1) and (3) of ~ 480, to obtain
the necessary and sufficient conditions that a periodic function may be
representable by means of a trigonometrical series such that the coefficients
tend to the limit zero, as n is indefinitely increased. His theorem may be
stated as follows:
If f(x) be a function, of period 2r, defined for every value of x, the
necessary and sufficient conditions that a trigonometrical series
n
a0O + E (an cos nx + b, sin nx),
such that lim an = 0, lim bn = 0, exists, which, at every point of convergence,
AnW=lo    ln=o00
converges to the value f(x), are the following:(1) That a continuous function F (x) should exist, such that
F(x + a +a/) - F(x + a-  3)-F(x -   + 3) + F(x- a -  )
converges to f(x), as a, / are diminished indefinitely in such a manner that
their ratio has a finite limit.




482, 483] Riemann's theory of trigonometrical series


739


(2)  That, if b, c be any two numbers,
JL2  F(t) cos L (t - X) (t) dt
should converqe to the limit zero, as,Jv is indefinitely increased; where X(t)
is a continuous function such that X'(t) is continuous, and that X"(t) is a limited
function, and such that X(t), X'(t) vanish at b and c.
That the conditions are necessary has been already established in ~~ 481,482.
To prove that the conditions are sufficient, let F(t + 27r)-F(t) be denoted
by   ( (t), then, from the condition (1), we deduce that, since f(t + 27r) =f(t),
lim  b (t+ h) +  )(t- h)- 2(t) = 
7h=0          h
Applying Schwarz's theorem, established in ~ 211, it follows that b (t)
must be a linear function of t. It is now seen that Ao and C' can be so
determined that F(t)- C-'t- ~Aot2 is periodic, and of period 27r.
The condition (2) holds, not only for F (t), by hypothesis, but also if F (t)
be replaced by C't +  Aot2, as has been proved in ~ 482. Denoting by
(t) the periodic function F (t) - C't -  At2, it follows that
lim  J f  (t) cos  (t - x)  (t) dt ( = 0.
Let* b < -7r, c > 7r; and let X (t) = 1, in the interval (-7r, 7r); then
lim /z2      (t) cos (t) t - x) dt + L2  k (t) cos I (t - x) X (t) dt
+ 2      (t) cos  (t - ) X (t) d  = 0.
Now let I/ be an integer n; we have then
lim   2 |  ~ (t) cos n (t - x) dt + n2  (t)cos nr (t- x) \ (t) dt = 0,
=oo      -7r                       +27r
where X, (t) = X (t - 2r), in the interval (b + 27r, 7r) of t; and where X, (t) = X (t)
in the interval (7r, c). The function X, (t) satisfies the conditions in (2), for
the interval (b + 27r, e); hence we have
lim n2 J    (t) cos n (t - x) X (t) dt  0,
n=o    b +27r
and therefore also
7r
lim nf2    (t) cos  (t - x) dt = 0.
Now let
2     (t)dt =0, 1   '  (t) cos n (t - x)dt
* Weber's notes on Riemann's memoir have here been utilized.
47-2








740                    Trigonometrical series                [CH. VII
where A, is of the form an cos nx + bn sin nx; then, in virtue of the result
just established, lim An, = 0. It follows that the Fourier's series
AlA          A,
12   22  *"   n2
is convergent for every value of x, and therefore converges to the value  x (x),
this function being continuous and periodic.
The function F(x) defined by the series
A,       A,,
C + C'x + 2 AoX2 -Al A 2 
2 O' OW    -22     "   2...
satisfies, with respect to the function f(x), the conditions of theorems (1) and
(3) of ~ 480, and lim An = 0, for every value of x. It therefore follows, from
theorem (1), that the series Ao + Al + A2 +... + An +..., for any value of x
for which it is convergent, converges to the value f(x).
It will be observed that the theorem gives no information as to whether
the series A, + A, + A2 +. + A+ +... is a Fourier's series or not; neither
does it make any assertion as to the values of x for which the series is
convergent.
When f(x) is a given periodic function, defined for all values of x, and
which satisfies the conditions of the theorem, the process of forming the
trigonometrical series which represents f(x) at each point at which that series
converges is as follows:-The function F (x) is first determined, so as to
satisfy the condition (1), in relation to the given function f(x), and then the
periodic function + (x) can be determined. This latter function is then
replaced by the Fourier's series which represents it; and thus a series which
everywhere represents F(x) is obtained. The required series is then obtained
by differentiating twice the terms of the series which represents F(x).
Conversely, Riemann's method gives a process of summation of a given
trigonometrical series Ao + Al +... + A, +..., such that lim An = 0, for every
n= oo
value of x. The series being given, the function F(x) is defined by the
convergent series
~  A,  A,        A,
C+ C'x + ~AoX2.
+G 2    12   22,...n..
The convergent series
A,+    An   - (sin inh,
~=l   \ nh /
of which the sum is
F( + 2h) + F(x - 2h) - 2F(x)
4h2




483, 484] Riemann's theory of trigonometrical series


741


is then formed. Whenever the given series is convergent, its sum is then
given as the limit, when h = 0, of the sum of the series
A + A1 si     h  A2 (sin 2h 2
AO4A   h)     A    2h ( /-+....
484. The following theorem gives the necessary and sufficient condition
that the trigonometrical series obtained in accordance with the method of
~ 483, actually converges to f(x), for a particular value of x.
Let b < x < c _ b + 2rr, and let p (t) be a function such that p (t), p' (t), are
continuous in the interval (b, c), and both vanish at b and at c; further let
p" (t), p"' (t), pIV (t) be continuous in the interval (b, c). For t = x, let p (t) = 1,
p' (t) = 0, p" (t) = 0. The necessary and sufficient condition that the series
Ao + A1 + A2 +..., may be convergent, in which case it necessarily converges
to the value f(x), is that
1   f     t  ds2sin(2n +l)(    t 1 ) ( 
2 | F(t) p(t) d     sin  l(x    t) dt
27Jb    P     dt2   sin-I(x - t)
should converge to a definite limit, as n is indefinitely increased.
We have
1 *a -2ir               r=n
A, + A2 +... +- An= -  f    (t) t t C't-  Aot2}  - r2 cos r (x- t) dt
Tr J a                  r=l
a + 947a d2 sin I ( d           2n + l)(-t)
-=  fJ,    {F(t)- C't-Aot} dt2 si(2 +n1) (x - t) dt.
Let a be so chosen that the interval (a, a+ 27r) encloses the interval
(b, c), and consider the integral
+ ' =        d2 sin  (2n +i 1) (x-  t  t
I   ~ (t) X (t) adt2  sin I (x - t) d;
where X(t) = 1 - p(t), when t is in (b, c), and X(t) = 1 in the remainder
of (a, a + 27r), and thus X(x) = 0; also let X(t) be periodic, and of period 27r.
The integral may be expressed in the form
2n   1\f2 a+27
-  n2  )  i   frJ(t) xl(t) sin I (2n + 1) (x - t) dt
- (2n + 1)   fr(t) X(t) cos, (2n + 1) (x - t) dt
f a+
+    ir   t     sin z(2n + 1) (x - t)
+      +(t) X3(t)                 a 2_.dAt,
~~~where          X2(t) = X(t)  cosec (x  - t),
X2(t) = X(t) / cosec  (x -    )
d2
Xs(t) = X(t) sin 1- (x - t), cosec (x - t).




742


Trigonometrical series


[CH. VII


Since X (x)=0, X'(x)= 0, X" ()= 0, and since  "' (t), XIV(t) are continuous
functions in the interval of t, it follows that X, (t), X2 (t) satisfy the condition
of the theorem (3) of ~ 480. Therefore the first two terms of the above
expression converge to zero, when n is indefinitely increased.
The function f (t) X3 (t) has a limited differential coefficient in the interval
(a, a + 2r), and therefore it is expressible by a Fourier's series which everywhere
converges to the value of the function. The sum of the first n + 1 terms is
expressed, at the point x, by the third integral in the above expression,
multiplied by 1/27r; and this converges, as n is indefinitely increased, to the
value of the function at the point x, that is to zero.
It has now been shewn that
i      fa 2      d2 sin  (2n + 1) (x - t) 
2n- oo  a        dt2    sin  (x-t)
r  13=  rr Z/ d2 sin -(2n+l)(t
= lim 2 |      (t) p (t) dt2-sin( -x7- -( t) d
provided the limit on the right-hand side exists.
By partial integration, we have
1f0(c't + -Aot2) p (t) d2sin -2n ~l)(v-t)dt
I j6                d2 sin-}(2n+ 1)(x - t)
2= t      [(C't +  Aot2) p (t) -2       t) 
o1 f    (d2       +  l') 1( t)].n (n1)"t     - "
2-2       t ( +t)] sin  ( +A  1           dt.
2w] b rdiL^12~    si0sin I(          - t) - d
converges func to the value Ao(t), as n is indefinitely inc(t), p(t) respectively, in the 1,
interval (b, c), and each equal to zero in the remainder of the interval
p' (~)= o, p"(x) o.
It has the+ 2refore been shewn that A + A. convergent Fourier's series, since to a
definite limit, provided derivatives in the whole interval (,     r). 
follows that
1        d2i (2t      +      1   )    (x t) )
2r     F(t)     dt2        sin (x - t) d
converges to the value A, as is indefinitorem has accordincreased; for pestablished.
Since the interval,                     is arbitrarily small, this theorem
put has thevidenfore been hefact that A + the+ A + convergence, at the point, of the tonodefinite limit, provided
1 f^WnW dts^2~)(d2 sin  
y-  F ^ (t) p (t)  2T,   ---:-Y ) —x-_  dt
27r; &    *   dt2   sin, (x - t)
converges to a definite limit. The theorem has accordingly been established.
Since the interval (b, c) which contains x, is arbitrarily small, this theorem
puts in evidence the fact that the convergence, at the point x, of the trigonometrical series obtained by Riemann's process, corresponding to a given
function f(x), depends only upon the nature of f (x) in an arbitrarily small
neighbourhood of the point x.




484, 485]  Investigations subsequent to Riemanrn's


743


INVESTIGATIONS SUBSEQUENT TO THOSE OF RIEMANN.
485. The important discovery, made by Seidel and by Stokes, of the
fundamental distinction between series which converge uniformly, and those
which converge non-uniformly in a prescribed interval, remained for a long
time without influence upon the development of the theory of series in
general, and in particular of trigonometrical series.  It was shewn by
Weierstrass that the legitimacy of term by term integration of a convergent
series is dependent upon the uniform convergence of the series; by previous
writers no such restriction upon the universal validity of the process had
been recognized. It was first pointed out by Heine* that a full recognition
of the consequences of the theory of uniformity of convergence made it
necessary to undertake a re-examination of the foundations of the theory
of trigonometrical series. The investigations of Dirichlet and others had
established that a function which satisfies certain conditions can be represented by means of a trigonometrical series in which the coefficients have
the form given by Fourier; unless however it be assumed that a series so
obtained converges uniformly, it cannot be immediately proved that it is
the only trigonometrical series by which the function can be represented.
The customary proof that a function is capable only of a single representation
by means of a trigonometrical series was based upon the assumption that, if
a convergent series  ao +  (an cos nx + b, sin nx) converge to zero for all
1
values of x in the interval (- r, vr), it is legitimate to multiply the series
by cos nx or sin nx, and then to integrate term by term, between the limits
- 7r, r; thus shewing that an =0, bn = 0, for every value of n. If however
it is not known that the series converges uniformly, the process of term by
term integration is not necessarily legitimate, and thus the proof is invalid.
In fact it is conceivable that a non-uniformly convergent series might exist
whose sum is zero for every value of the variable. It thus appeared that,
when a Fourier's series exists which represents a function f(x), it cannot be
immediately inferred that no other trigonometrical series exists which
represents the same function.
A Fourier's series that represents a function f(x) which has discontinuities, is certainly non-uniformly convergent in the neighbourhood of such
continuities, and in default of proof to the contrary, it may also be nonuniformly convergent in the neighbourhood of points at which f(x) is
continuous. Thus, for example, if f(x) is continuous in its whole domain,
and is representable by a Fourier's series, it cannot be assumed that the
series is uniformly convergent.  The value of the representation of a
function f(x) by a series la0o +  (an cos nx + bn sin nx) would be seriously
impaired, if it were not known that the series was, at all events in general,
* Crelle's Journal, vol. LXXI, 1870; see also Kugelfunctionen, vol. i, p. 55.




744


Trigonometrical series


[CH. VII


uniformly convergent. For it could not be assumed that, if r (x) denotes a
continuous function, the integral  f (x)  (x) dx would be represented by
the series
rb            rb
2 a.    (x) dx + E j  (.an cos nx + bn sin nx) r (x) dx;
the employment of Fourier's series in physical and other investigations
would consequently be much restricted.
These considerations gave rise to a series of investigations with the view
of establishing the uniqueness of the representation of a function by means
of a trigonometrical series, and of investigating whether the coefficients in
the series are necessarily expressible in the Fourier form.  Heine* proved
that the Fourier's series which represents a limited function that satisfies
the conditions known as Dirichlet's, viz. that it has only a finite number of
discontinuities and is in general monotone, is uniformly convergent in the
portions of the interval (- r, vr) which remain when arbitrarily small neighbourhoods of the points of discontinuity are removed from   the interval.
This property of the series, of being in general uniformly convergent,
suffices to remove, in the case of a most important class of functions, the
restriction which has been above mentioned relating to those applications
of Fourier's series which involve a term by term integration. It having
thus been shewn that a function satisfying Dirichlet's conditions is representable by a series which converges in general uniformly, Heine proved
that, if a function is representable at all by a series which converges in
general uniformly, there can exist only one such series. This is equivalent
to the theorem that, if a series converges in general uniformly in the interval
(-?r, 7r), and represents zero, then all the coefficients vanish, and the sum
of the series is therefore zero for all values of the variable. Heine proved
further that this theorem holds even when, for a finite number of values of
the variable, the series is not known to converge, or when it is at least not
assumed that its sum is zero for such values of the variable. The possibility
remained however, that when a function is thus uniquely represented by
means of a series which is in general uniformly convergent, other series
not possessing this property of uniform convergence may exist, which also
represent the same function.
It was next proved by G. Cantort that if the expression an cos nx + bn sin nx
be such that, for every value of x in a given interval (a, p3), the limit
lim (acos x + bn sin nx) is zero, then an, bn converge to zero, as n is
t~oo
indefinitely increased, and hence that the series
-2-a0 +  C (an cos nx + bn sin nx)
* Crelle's Journal, vol. LXXI.
f Crelle's Journal, vol. LXXII, also in a simplified form in Math. Annalen, vol. iv (1871).




485]


Investigations subsequent to Riemann's


745


can only converge for all values of x in (a, /) if an, bn have the limit zero, as
n is increased indefinitely. This theorem is independent of any assumption
that the convergence is uniform.  Cantor* then deduced that, if a trigonometrical series -a +  (an cos nx + bn sin nx) converges to zero, for every
value of x with the exception of a finite number of values, for which it is
unknown whether the series converges, all the coefficients an, bn must vanish.
Kronecker* shewed that this theorem can be proved without assuming the
previous one. These proofs depend upon the use of Schwarz's theorem that
if F(x) denotes a function which is such that
limF( x+)- 2F(x)+F(x - e)
e=0            -2
then F (x) must be a linear function of x.
The next stept- was made by G. Cantor in extending the proof of the
uniqueness of the representation of a function by means of a trigonometrical
series to the case in which the function may have an indefinitely great
number of points of discontinuity, these points forming a set of the first
species. Starting with Weierstrass' theorem, that an infinite set of points
possesses at least one limiting point, Cantor developed the theory of the
successive derivatives of a set of points, and proved that if a limited function
has discontinuities which form a set, one of whose derivatives contains only
a finite number of points, then if the function is representable by a
trigonometric series at all, there can be only one such series. In this
connection the theory of sets of points was first considered, and thus the
whole development of this subject, and of the more abstract theory of
transfinite numbers, arose historically from the requirements of the theory
of trigonometrical series. Proofs were given by Dinij and Ascoli~ that, for
restricted classes of functions, a series which represents such functions must
be Fourier's series.
An important advance in the theory was made by Du Bois Reymondll,
who proved that, if a function f(x) can be represented by a series
-a0 + S (an cos nx + bn sin nx),
which is such that a,, bn have the limit zero, as n is indefinitely increased,
the coefficients must always have the form
IP 1r            I                          l
a = -    f (x) dx, an = -   f(x) cos nxdx, bn = -   f(x) sin nxdx,
ql _        -          r7r' -                Tr
whenever these expressions have a meaning. This theorem   includes the
* Crelle's Journal, vol. LXXIII (1871).
t Math. Annalen, vol. v (1872).
+ Sopra la serie di Fourier. Pisa, 1872, p. 247.
~ Annali di Matematica, ser. 2, vol. vi, p. 252, also Math. Annalen, vol. vi, 1873.
I1 Abhandlungen der bayerischen Akademie, vol. XII, 1875.




746


Trigonometrical series


[CH. VII


theorem as to the uniqueness of the representation of such integrable functions
as are representable by series for which lim an = 0, lim bn = 0.
n=-oo     n=oo
The most general formulations of the theorems as to the uniqueness of
the representation of a function by a trigonometrical series are due to
Harnack and Holder; an account of their results will be here given.
THE LIMITS OF THE COEFFICIENTS OF A TRIGONOMETRIC SERIES.
486. The most general form of the theorem of Cantor, that a series
00
ao + Z (an cos nx + bn sin nx)
which converges for every value of x in an interval, with the exception of a
certain set of points, must be such that lim an = 0, lim bn = 0, is due to
~n=as      r~n=oo
Harnack*, who proved that the theorem holds, provided that, if 8 be any
arbitrarily chosen positive number, a sub-interval exists in the given interval,
such that, at every interior point of that sub-interval, the measure of divergence of the series is less than 8. The term " measure of divergence of a
series at a point" is used to denote the excess of the upper, over the lower,
limit of indeterminacy at the point. For each point x, at which the measure
of divergence of the series is less than 8, there is a value m, of n, such that, if
n   m, then an cos nx + bnsin nx < 8:
we suppose that a sub-interval exists at every point of which this condition
is satisfied. If x be any fixed point within this sub-interval, a neighbourhood
(x - r, x + r), of x, can be found, such that
I ancosn(x + )     + bnsin n (x+e)l< 8, ancos n (x -e) + bn sin n(x- e) 1< 8,
for n > m, provided e <: the value of m will depend in general upon the
value of e. We deduce at once
I (a cos nx + bn sin nx) cos ne I < 8, (an sin nx - bn cos nx) sin ne < 8;
and thence, on multiplying by cos nx sin ne, sin nx cos ne, and adding the two
expressions in the inequalities, we have I an sin 2ne I< 48, and similarly
I bnsin 2ne < 48, where n - m. These inequalities hold for every 8, the
values of V, m depending on the value of 8.
Let 2e= a, 48 = 8'; then for each value of a in a certain interval (a, b),
n can be found, such that
Ian sin nat, I a,+ sin n + la I,... I a+ssin (n + s) a...
are all < '. Suppose, if possible, that a sequence an,, an2, an... can be
found, all of whose members are numerically A 8", where 8" > 8'; it will then
be proved that a certain value of a in (a, b) can be found, such that the
sequence an, sin n,a, an2 sin n2a, a3 sin n3a... is such that it contains an
indefinitely great number of members, each of which is numerically greater
* Bulletin des sciences mathematiques, series 2, vol. vi, 1882, also Math. Annalen, vol. xIx.




485, 486]          The limits of the coefficients                 747
than 8'. This being contrary to the hypothesis that, for each value of a,
a sin na < ', for all values of n which are sufficiently great, leads to a
contradiction; and thus it is impossible that such a sequence as an,, a,2
a,,... can exist.
To establish this, we shew that, out of the sequence an,, an2, an..., a
sequence an,,, an,... can be chosen, which is such that, for a certain value
a, of a, in (a, b), a' a, n2, na na... all differ from an odd multiple of 27r by less
than an arbitrarily chosen small positive number f.
If              na > y  -,   and   a < y,  +,
7T   o       7r (q
then                            < a<    f;
9n            n
now suppose n and y, such that
2             2 
a<,   b >    --—,
n?         1?
which is equivalent to
2                      2
- (na + ) < y, < (nb - ) 2
a'7
There exists a possible value of yi which is an odd integer, provided
n (b-a)-2} 2 > 2, or if n     7   2>;
77'              b-a;
take the least of the numbers n1,, n,... which is > r + 2-, and denote it
b-a
by n,'. We can then find a corresponding odd integer y~, and we take a
to lie in the interval (a', b'), where
7r           7r
this interval lies within (a, b), and is of length. Next, an odd integer y2
can be determined such that
(n2al' +  ) - < Y2 < (in2' b -) -,
provided                n2,r>2' +n /
2  b' a'     2      ';
and n,' can be chosen from the sequence in, n,... so as to satisfy this condition, if a lies in the interval (a", b"), where
Y2 -           y2 +g
a 2d2      b     2/




748


Trigonometrical series


[CH. VII


thus (a", b") lies within (a', b'), and is of length 2. If we proceed in this
manner, we obtain a sequence n,', n2',... of numbers all belonging to the
sequence n,, n2, n3,..., and such that if a is the point which lies within all
the intervals (a, b), (a', b'), (a", b")..., then na,', n2'a,... all differ by less
than ~ from odd multiples of 2. Since r can be chosen arbitrarily, we can
find n/', n2'... such that | an, sin n,'a, a an2 sin n,'a... are all _ 8', which is
contrary to the hypothesis in accordance with which I an sin noa is, for every
sufficiently large value of n, < '. It has thus been shewn that no sequence
an,, an2, an... exists, all of whose terms are numerically >_ "; and, if 8" is
first chosen, we may afterwards choose 8'. Therefore, from and after some
value of n, an must be numerically less than 8'; and since this holds for
every 8', we must have lim a =0. In a similar manner it is seen that
= 00
lim b,, = 0.
n=00
The theorem has thus been established that if the series
2 a, + S (an cos nx + bn sin nx)
be such that, for each number 8 (> 0), there exists an interval in (- 7r, wr) at every
point of which the measure of divergence of the series is < 8, then a,, bn
diminish indefinitely, as n is indefinitely increased. In particular, if the points
at which the measure of divergence is  8 form a non-dense set, then lim an = 0,
Xn= o
lim bn = 0.
n=0co
It follows from this theorem that, if a trigonometrical series converges for
an everywhere-dense set of points, and is such that an, bn do not converge to
zero, as n is indefinitely increased, then, for some positive value of 8, the set of
points at which the measure of divergence is _ 8, must be everywhere-dense.
No assumption has been made as to the form of the coefficients an, bn in the
trigonometric series. That lim an = 0, lim bn = 0, in the particular case of a
n==00      n= o
Fourier's series, has been already established in ~ 454.
PROOF OF THE UNIQUENESS OF THE TRIGONOMETRICAL SERIES
REPRESENTING A FUNCTION.
487. Let us assume that the series   ao +  (a, cos nx + bn sin n) conn=l
verges to zero at every point of the interval (- r, 7r), with the exception of a
reducible, and therefore enumerable, set of points, at which it is not assumed
that the series converges.




486, 487]   The uniqueness of a trigonometrical series           749
In accordance with the theorem of ~ 486, the coefficients an, bn converge to
zero, as n is indefinitely increased. Accordingly the condition is satisfied
that Riemann's function
F (cx)a = ~a^ox-t a cos nx + b sinnx
n=           n      n 
is a continuous function. In accordance with Riemann's theorems,
lim  ( + h)+ F(x -  - -2F(x)
h=O             h2
is zero, for every value of x for which the given series converges; and further,
at every point, without exception,
li F (x + h) + F(x - h) -2F (x) 
7=0o            h
In accordance with the extension of Schwarz's theorem, given in ~~ 211 -213, since
lim F (x + h) + F (x - h) - 2 (x)
h=0             h2
vanishes at every point in the interior of an interval of the everywhere-dense
set of those intervals which are complementary to the given reducible set of
points, and since
lim F (x + h)+ F( - h)- 2F(x)   0
h=0             h
at all points, the function F(x) is a linear function ax +b, in the whole
interval (- r, 7r), and consequently in any interval whatever.
Since the function ax + b- {aox2 is represented everywhere by the series
ancos nx + b, sin nx
=n=1       n2
it follows that ax + b -  ax2 must be a periodic function, which can only be
the case if a = 0, and a0 = 0.
Since the series is uniformly convergent, we may multiply by cos nx, or
by sin nx, and integrate term by term; we thus see that a,,= 0, b = 0.
The coefficients of the series therefore vanish identically.
Since the series may be taken to be the difference of two given trigonometric series, we obtain the following theorem:No two distinct trigonometric series ean exist, which converge to the same
value for all points of (- r, 7-), with the exception of a reducible set of points
at which the series are not known to converge to the same value, or to converge
at all.
This extension of Cantor's theorem, relating to the uniqueness of the
representation of a function by a trigonometric series, is obtained by considering the difference of two trigonometric series which might be assumed to
represent, in the sense defined, the same function. It has not been assumed
that the series is necessarily a Fourier's series.




750


Trigonometrical series


[CH. VII


THE REPRESENTATION OF INTEGRABLE FUNCTIONS.
488. With a view to proving that, in a wide class of cases, a trigonometric series, which represents a given function, is necessarily a Fourier's
series, we proceed to the consideration of an extension of Schwarz's theorem,
given in ~ 211-213, due to Du Bois Reymond, and which has been otherwise
proved and extended by Holder and Lebesgue.
Let F(x) be a function which is continuous in the interval (x - a, xi + a),
and let F(x,+a) +F(x, -a)-2F (x) be denoted by Aa2F(x,).     Let us
A,2 F (x)
further suppose that, for each value of x in the given interval,  2
either converges to a fixed value f(x), as e is indefinitely diminished, or else
has two finite limits of indeterminacy f(x), f(x).  If, for any value of x,
U and L are the upper and the lower limits of   2(x) for all values of
e, such that 0 < e - e, then f(x), f(x) are the limits to which U and L converge respectively, when el is indefinitely diminished. We consequently
assume that, in the whole interval (x1 - a, xl + a),
lim   I F (x) 
lim    2   =/ (W),
e=0 
where f(x) is considered to be determinate at points at which the limit is
definite, and to be indeterminate between the limits f(x), f (x), at points in
which the limit is indeterminate.
It will be further assumed that the upper limit of f(x) in the whole
interval is finite, and equal to U; it will also be assumed that the lower
limit of f(x) is finite and equal to L. The function f (x) is therefore limited
in the interval (x1 - a, x4 + a), with U, L for its upper and lower limits.
Let
(x) F (x) - F(x, - a) - x    + a [F(x1 + a) - F(x1 - a)]
2a
+  C (x - xi + a) (Xi + a- x)
where C is a constant. We see that
+(X)   a2 L A 2F (xi)j
and thus  (xi) 0, according as C     a     Let C be so chosen as to
exceed A2F(x)  Since b (x) is continuous in the interval (x1-a, x + a),
exceed     2     Since ( (x) is continuous in the interval (xi - a, xi+ a),
a2
and vanishes at the points x - a, x, + a, there must be at least one point z, in
the interval, at which 4 (x) has a maximum, and is positive.




488, 489] The representation of integrable functions


751


A2() (z) =A2 (__
We find that            e            (z   C
~2        62
and, since +(z) is a maximum    of ( (x), A,20 (z) is never positive, for all
sufficiently small values of e. Therefore the limits of indeterminacy of
AE 2T (z)
the limit of      (z - )C, for =0, are neither of them    positive; hence
62
f(z) f(z) < c.
Now L    f(z), and U   f(z); hence L < C, and this holds for any value
of C that may be chosen, subject to the condition C >  a.
a~2
It follows that L    (       In a similar manner, by choosing
Aa2 F (x1)
<   a2 F2(x) and considering the minimum of    (x), it can be shewn that
U_   2 F (x )  The following theorem* has accordingly been established:If F(x) be contintuous in the interval (x1 - a, x, + a), and' at every point
of the interval,
im F(X + e)+   F(x - e) - 2F(x) 
lim              2               f (X),
e=O             C
where f (x) is either determinate, or indeterminate between definite limits of
indeterminacy, at each point x of the interval, and is a limited function in
the interval, then
F(x, + a) + F(x1 - a) - 2F (x1)
a2
lies between the upper and lower limits of f(x), in the interval (cx - a, xc + a).
489. Let us assume that the series
00
2 ao + S (an cos nx + bn, sin nx)
has, throughout the interval (a, b), the sum f(x), where f(x) is limited in the
interval (a, b), and has a determinate value at every point of the interval,
except at the points of a set E of zero measure, where the values off(x)
may be indeterminate, between finite limits of indeterminacy.  Also let it
be assumed that the conditions of the theorem of ~ 486, are satisfied, so
that lim an and lim bn are zero, or more generally, that I an cos nx + bn sin nx
n=00        = 00
is limited for all the values of n and x.
* See Holder, Math. Annalen, vol. xxiv, p. 183. The theorem has also been established otherwise by Lebesgue, for the case in which f (x) has a definite value at each point; see the Annales
sc. de e'cole normale superieure, ser. 3, vol. xx.




752                   Trigonometrical series                [OH. VII
In accordance with the theorem proved in ~ 481, if F(x) denote the
continuous function defined, for the interval (a, b), by
F (x) = iax2 -  an cos nx + bn sin nx
n~z     FF=(x)
A "'2 F (x'
lim -a2 (   is equal to f(x), at all points of (a, b) which do not belong to E;
x=0O
and, at a point of E, the limit, whether it be definite or not, lies between the
two numbers
f( +f()+ ~      {f() -f(x)} ( 1 +  +  2)
where f (x), f (x) denote the upper, and lower, limits of indeterminacy off(x)
at the point.
Since A  F(x) is, in accordance with the theorem of ~ 488, limited in the
interval, for all values of x and a, we have*
f (x) dx = lim f: F(x + a) + F( - a)- 2F(x) dx,
a= c             O[2
where c is a fixed point in (a, b), and x, is any point in (a, b). The integral
of f(x) exists, in any case, as a Lebesgue integral.
Denoting by F, (x), the indefinite integral of F(x), we have
f          (x) dx =  lim A  [F ( -F (c)]
C ~     a-=O       a
Next, let F2 (x) denote the indefinite integral of F, (x); we then have,
dx     f (x))dx = lima  F2 (x)    F2(c)  ( - c)   F (c)
J c  J<7   a=2 =0  a2   aJ
where x is any point in (a, b). Since
li    aF2(x)  F(x), lim    F2(c))
a=O a0               =O    a=
we have            F(x) =   dxl    /(x) dx + Ax + B,
where A and B are constant for the whole interval (a, b).
Next, let the series
a, + E (a, cos nx + b, sin nx)
1
be such that its sum-function f(x) has indefinitely great values at, or in the
neighbourhood of, points belonging to an enumerable closed set, and therefore
a reducible set G, of points in (- r,?r). In any interval (a, b), contained in
* The extension of the theorem of ~ 384, given in ~ 472, footnote, is here employed.




489]      The representation of integrable functions


753


the interior of an interval complementary to G, the function f(x) is limited,
although it may be indeterminate at points of a set E of zero measure. It
will be further assumed that, either f(x) has a Lebesgue integral in (a, b), or
else that it has a non-uniformly convergent improper integral in accordance
with Harnack's definition, or with the extension of that definition given in
~ 392. Also, in the latter case, it will be assumed that f(x) cos nx,f(x) sin nx
are also integrable in accordance with Harnack's definition, or its extension.
It will now be assumed that lim an = 0, and lim b, = 0.
n=oo          n=00
Since  (x)-    dxf   f (x) dx is linear in any interval contained in one
of the intervals complementary to G, it is linear in that complementary
interval, since it is continuous at the ends of that interval. Moreover c need
not be interior to the interval, but may be any fixed point, say the point
x = 0, in the interval (- r, 7r). Therefore the continuous function
J (x) = F(x) -  dxi ff(x) dx
is linear in each interval complementary to the reducible closed set G.
We have now
A2     ) _aF (x)  1 P+      xl         1 [X XI  XI
)_ e) F()_     dx    f (x)- dx f() -  d  f+  ( d  x;
A 2 F (xand, by Riemann's theorem (2), of ~ 480, we have lim  e F()= 0, at every
e=0    E
point of (- r, 7r) without exception. The function  ff(x) dx being a
continuous function of xi, we see that
-I {fxed di  f (x) dx -   dxfo f(x) dx}
cannot exceed, in absolute value, the fluctuation of ' f(x)dx in the interval
(x - e, x + e) of x1; and therefore the limit of the expression, for e = 0, is zero.
It has now been proved that lim  e  (' ) = 0, at every point of (- 7r, 7r),
e=O0 
without exception.
As in ~ 213, it now follows that the function + (x) is linear in the whole
interval (- 7r, 7r). For, if A1x + B1, A2x + B2 be the values of ' (x) in the
intervals which abut on one another at a point ~ belonging to G, we have
A1i + B = A2 + B2; and since lim A   () = 0, we find that A, = A2, and
e=0 
therefore Aix + B, and A2x + B2 are one and the same linear function.


H.


48




754


Trigonometrical series


[CH. VII


The following theorem has now been established:
If the sum-function f(x) of the series
00
~ ao + ~   (an cos nx + bn sin nx),
1
for which lim an O0, lim bn = O, be limited, or else have indefinitely great
values at, or in the neighbourhood of, points in (-7r, 7r), belonging to a reducible
closed set G, the function being limited in any interval contained in the interior
of an interval complementary to G, and being determinate at each point of such
interval, with the possible exception of the points of a set of zero measure; and
if f(x) either (1) have a Lebesgue integral in (- r, w), or (2) be such that
f(x), f(x) cos nx, f(x) sin nx have non-absolutely convergent integrals in
(-7r, Wr), in accordance with Harnacc's definition, or its extension, then
F(x)- -  dx1   f (x) dx
is a linear function in (- r, 7r); where F(x) denotes Riemann's function
4a0x2 -  2 (a, cos nx + bn sin nx).
n 2
rXI
490. Let x (xi) denote the function  f (x) dx, where x, is in the interval
(-7r, 7r). The functions F(x), f(x) are defined for all values of x, as periodic
functions, of period 27r; if then we defne X(x) for values of x, not in the
interval (-7r, 7r), by  f (x) dx, we have
f-T
X(x, + 2r7r) = % (x,) + r  f(x) dx = % (x,) + r {X (7r) - X (- 7r)},
where r is some integer.
fx
The function F(x) - J   (x))dx is continuous for all values of x; and
changing x into 2rr + x only adds a constant to the function. Therefore
F(x) -     (x)dx is equal to the same linear function, for all values of x;
J o
let this linear function be Xx + fL. The function
f x (x) dx -  aox2 + > +,
is periodic, and of period 27r. The differential coefficient X (x)-  aox + X, of
this function, is also periodic, and therefore its values for x = 7r, x =- 7r are
identical, and thus
ao   { (7r) - x(- 7r)} =-   f(x) d.
7r                 77 J -7_
Since F(x) - 4ax2 is represented by the uniformly convergent series
a,O cos nx + bn sin nx
1         n2,,




489, 490]  The representation of integrable functions


755


a.
we have              {F(x) - aOX2} cos nxdx = -   -i r,
-7r 
{F(x) -    aoX2} sin nxdx =- _  vT.
In these expressions, we substitute X (x) + X for F' (), after integrating
by parts; we thus find that
f {x (x) + X_ -  a} sin nsdx = an,
I   {x (x) + x -  ao} cos nxdx =- b.
Let us denote the integrals
f /(x) cos nxdx, f  () sin nxdx
by Cn(x), Sn(x); in accordance with our assumption, these integrals exist,
either as Lebesgue integrals, in which case they are absolutely convergent,
and their existence follows as a necessary consequence of the existence of
Co(x), or else as non-absolutely convergent improper integrals.
In an interval (a, b) which contains no points of infinite discontinuity of
f(x), we have, by integration by parts, in accordance with ~ 394,
fib
J {X (x)- gaoxI sin rndx
=  -{ (x) -I aox}     l] + -  f {f(x) - aO0} cos nxdx.
Let now the function b (x) be defined by means of the equation
{x(x) - aoX} sin nxdx= [-{(x) - ia0} c sxj
- 2                             2?~ nj 7r
+       ( {Cn (X) (-) - (  (-  )} -  cos axdx +  ();
the function 0 (x) is continuous in the interval (- r, 7r), and it is constant in
the interval (a, b). The interval (a, b) being any interval contained in a
complementary interval of a reducible closed set, it follows (see ~ 206) that
((x) is constant throughout (a, b); and since 0(-7r)= 0, the constant is
zero. Let x= =r, we then have
1                    If7f
I|   (*) - ~a0x} sin nxdx = - {an(T)-a) (- a)};
therefore   an =      (7) - { Cn () -  )} = -   (x) cos nxtdx.
7r                   - J -_7
In a similar manner, it can be shewn that
bn = -     (x) sin nxdx.
7rJ - 7
It has therefore been shewn that, on the assumptions made, the coefficients
of the trigonometrical series necessarily have the form of Fourier's coefficients.
48-2




756


Trigonometrical series


[CH. VII


The series is therefore either a Fourier's series, or a generalized Fourier's
series, in accordance with the terminology introduced in ~ 439.
The following theorem has now been established:Let the function f(x) be defined by the series
00
4ao +, (a, cos nx + bn sin nx),
in the sense that, at every point of convergence of the series, the sum is the value
of f(x), and at every point at which the sum of the series oscillates between
limits, f(x) is multiple-valued between those limits, and at every point of
divergence of the series f(x) is indefinitely great.  Then, in the following cases,
the coefficients of the series necessarily have the forms
1  r"              1  ('                    1  [rT
ao = -    f(x) dx, an= - I   f(x) cos nxdx, bn=       f (x) sin nxdx,
ql' -_r         -- V  -_r                   7rT -,r
so that the series is either a Fourier's series, or else a generalized Fourier's
series:(1)  If f(x) be everywhere definite and limited.  The function is then
necessarily summable, and the series is a Fourier's series.
(2)  Iff((x) be limited, but not everywhere definite, and satisfy the condition
of the theorem of ~ 486, so that lin an = 0, limn bn = 0; and further if f(x)
'n= noo=0
have a Lebesgue integral in (- r, 7r), then the series is a Fourier's series.
(3) If f (x) have infinite discontinuities at a reducible set of points, and
possess a Lebesgue integral in (- r, 7r), and the series be such that lim an = 0,
n=oo
lim bn = 0, then the series is a Fourier's series.
(4) If f(x) have infinite discontinuities at a reducible set of points,
and f(x), f (x) cos nx, f (x) sin nx possess non-absolutely convergent improper
integrals in (a, b), the series being such that lim an = O, lim bn = 0; then the
n=oo       n= oo
series is d generalized Fourier's series.
THE CONVERGENCE OF A TRIGONOMETRICAL SERIES AT A POINT.
491.  If the series  a0 + E (an cos nx + bn sin nx) converge at a point x,
and if the sum-function of the series have definite limits f(x + 0), f(x - 0), on
the right, and on the left, at the point x, it does not necessarily follow that
the series converges at points in a neighbourhood of the point x, at which
the series converges. From   the existence of f(x + 0), f(x - O), it follows
however that, corresponding to an arbitrarily small positive number 8, a
neighbourhood of the point x can be determined, such that the measure of
divergence of the series is, at every point in that neighbourhood, less than 8.
This has been shewn in ~ 486, to be a sufficient condition to ensure that the




490, 491] Convergence of trigonometrical series at a point        757
limits of an and bn are zero, when n is indefinitely increased. Hence we see
from Riemann's theorem, that if
100
F(x) = aoC2 - 2 -2 (an cos nx + b, sin nx),
in
then             li  F(x + e)- 2F(x) + F(x-) ()
then            lhm              2             =/(X),
e=0             c
at the given point x, of convergence of the series.
We now have
2f(x)= lim {4 F(x + 2e)-2F ()+ F (- 2e)   2 F(x +e)-2F() + F(x-e)}
e=0               462                            62
im {F(x + 2) - 2F(x + e) + F(x)  F(x) - 2F(x - e) + F(x - 2e)
"'t~!           e2                          62 '
In accordance with the theorem of ~ 488,
F ( + 2e) - 2F (x + ) + F (x)
2
lies between the extreme values of
liF( + a) - 2F(z)+ F(z- a)
a=O            a
for x < z < x + 2e. It has been shewn in ~ 481 that, for each value of z, this
limit lies between values which depend on the limits of indeterminacy of
f(z) at z. It follows that, for a given positive number 8, the positive number
e can be so determined, that
f(x+ 0)- a< lim F(z+ a)-2F(z) +F(z -a)<            +
a=O            a2
for every value of z, such that x < z < x + 2e.
We thus see that
lim  (x + 2) - 2F(x + e) +F) =f( + 0)
2
Similarly, it can be shewn that
lim F (x - 2) - 2F (x - e) + F (x));
e=O             c2
and therefore we have
f(x) =- {f/(  O + 0) + f(- O)}.
The following theorem has now been established:If a trigonometrical function converge at a point, then the value to which it
converges is the mean of the limits of the sum-function, on the right, and on the
left, of the point, provided those limits exist.
This theorem holds for every trigonometrical series, whether it be a
Fourier's series or not.




758


Trigonometrical series


[CH. VII


FOURIER'S INTEGRAL REPRESENTATION OF A FUNCTION.
492. It has been shewn in the course of the investigation of conditions for the convergence of Fourier's series at a point x, that
1 - sinmz              I f' sin mz
1   si  z f(x + 2z) dz + l -  z(x - 2z) dz
- - ( 2^+- d  — f(x -z)dz
converges to the value
{ f ( + o)+ f (x-o)},
when the positive number m, which is not necessarily integral, is indefinitely
increased; provided f(x) has an integral in the interval (- r, 7r), and satisfies
some one of a number of sufficient conditions in the neighbourhood of the
point x, at which it is assumed that f(x + 0),f(x- 0) exist. The numbers e,
e' are such that
0<6e<2T,     0<dE<Tr.
The above result is represented by the equality
lim1 /f (x') sin (x - -) dx'=  ( f (x + O) +  (x- 0)},
n = o 7rJ a     X    -
where x- r_ a< x< B3-x+7r, and where x is in the interval (-'r, 7r). If
x, x be changed into hrx/l, rx'/l, andnged into ged into ul/r, the function
f(7rx/l) being replaced by f(x), we see that the equality holds for points x
within the interval (- I, 1), where a, 3 now satisfy the conditions
x- lca <x <,83x+l.
When x= a, or x =i, the value of the limit is
f((a + 0), or -f ( - ),
provided the function f(x) is such that the limit exists, and also satisfies
one of the sufficient conditions already referred to. For a given point x, and
for given values of a, /, the number I can always be so chosen that the
conditions x- < a < x < 8 3x + 1 are satisfied.
Now let f (x) be defined for the unlimited interval (- o, oo ), and be not
necessarily periodic, but be such that in any finite interval whatever, f(x)
has a Lebesgue integral.  It will further be assumed that
f If(x)I dx
has a definite double limit, for b= + oo, a= —oo.
When f(x) has a Riemann integral in (a, b), this double limit is, in
accordance with the definition in ~ 292, denoted, whenever it exists, by
fJ    f(x) dx.
rb
We shall use this notation, even when  |f (x) dx exists only as a
Lebesgue integral.  Also, we shall denote by
f(x) dx,
00 C




492]     Fourier's integral representation of a function


759


the double limit, when it exists, of
rb
f f(x) dx, for b = oo, a  - oo,
when the integral through the finite interval exists as a Lebesgue integral.
This amounts to an extension of the definition given in ~ 292, of an improper
integral through an infinite interval, to the case of Lebesgue integrals.
If /3'> >  x, we have
'f (x') sin u ('-x) dx' <    f    () 00x;
J S     x'  - x  -x                  '
therefore, for a fixed point x, /3 may be chosen so great that the integral on
the left-hand side is arbitrarily small. A similar remark applies to the lower
limit a.  We see therefore, that when f(x') is such that
I'
I \f(')Idx'
J -00
exists, in accordance with the above definition,
fQ0 O   s U, ()' - x 
f f(x ) si —x x)    dx
exists, and differs from  f (x sin u(x'-x)dx,
by less than an arbitrarily chosen positive number -e, provided a and 8 are
numerically sufficiently great, for every value of u.
It follows that, when the assumed conditions are satisfied,
1    [(x y/ f, smu(x-a)d
qrj,)/ sin u (x' - x)
f f(x)  xi     dx'
does not differ from I {/(x + 0) + f (- 0)} by more than e, provided u is
equal to, or greater than, some positive number. Since e is arbitrarily small,
we have
lim      f /(x') sinu(' -) dx'   f f(x + O)+ f(x -)}.
qt=o - r -oo      Xt — X
The following theorems have now been established:If f(x) be a function which has a Lebesgue integral in the interval (a, A),
then, at a point x, in the interior of (a, /3), at which f (x + 0), f(x - 0) exist,
and such that one of the known conditions for the convergence of Fourier's
series is satisfied by f (x) in the neighbourhood of that point x,
li1n  f              -       =   f( +0)+    (-0)}
n= p c,   s     h s at ey pit    i    r to te ie         /, 
In particular, this holds at every point x interior to the interval (a, 3), if
f (x) be a function with limited total fluctuation in (a, /3). At the points x = a,
x =3, the values of the limit are i-f(a+ 0), If(/3- 0), provided these
functional limits exist, and f (x) satisfy one of the known sufficient conditions
in a neighbourhood of a or /, on the side towards the interior of the interval.




760


Trigonometrical series


[CH. VII


If x be exterior to the interval (a, 13), the limit is zero.
If f (x) have a Lebesgue integral in every finite interval, and
If(x) )  dx
_00
have a definite value, then
lim       f (X  in(-x) dx'=      If(x + 0) +f(x - 0)},
U=  r 00  00      X   X
for any point x, at which f (x + 0), f(x - 0) exist, and in the neighbourhood of
which f (x) satisfies one of the sufficient conditions for the convergence of
Fourier's series. In particular, this result holds for all values of x, iff(x)
have limited total fluctuation in every finite interval, and also satisfies the
condition that
If(x) I dx exists.
J -30
These theorems contain Fourier's* representation of a function by means
of a single integral.
493. Since       sin -      = x  cos v (x  - x) fv
x -x      J
the single integral  f(xsin u (x'-)dx     may be replaced by
-oo  X) -x    X
dx'     f (x') cos v (' - x) dv'
Therefore the theorem in ~ 492 may be taken to refer to
lim -    dx' f(x') cos v (x- x) dv'.
B==oo T  -)  Jo
It will now be shewn that the repeated integral may be replaced by the
one in which the integrations are taken in the reverse order.
Let ~r (a, 13, v) denote
f f (ax) cos v (x' - x) dx',
and let 4 (v) denote
_00
f   f (x') cos v (x' - x) dx'.
On the hypothesis that lf f(x')ldx' exists, as the limit of a Lebesgue
integral, the integral r (v) also exists. Since I * (a, 3, v) is less than some
fixed positive number, for all values of a, 3, and for all values of v in the
interval (0, u), it follows, by using the theorem of ~ 384, that  r (v) dv is
See the Torie de la Chaleu, Chap. x, ~ 41.
* See the Theorie de la Chaleur, Chap. ix, ~ 416.




492, 493] Fourier's integral representation of a function         761
the limit of the integrals  f(a, /, v) dv, when a and / have the values
in sequences which diverge to +, - o respectively. Also   r (v) dv has
a value which is independent of the particular sequences of a and /. For
dvJ f(x') cos v (x'- x) d',
or                        f(x')sin u (x' - x)dx'
~~~or             | f (c)     (_a)x'  dx
is numerically arbitrarily small, for a sufficiently great value of /3, independently of the value of  '(>,/); and a similar remark applies to
r/0  ra
I dv f (x') cos v (x' - x) dx'.
JO   Ja
We have, therefore,
dv      f (x') cos v (' - x) dx' =  lim  dv f (') cos v (x'- x) dx'
J~O  J~-aQOv~=w, =a= -oo             a
=    lim      dx'  /(') cos v (x' - x)dv
-3= o, a=-oc  a 
-=    dx'   f (x') cos v (x' - x) dv.
-00    0
It follows that
lim     A dvf  /f(') cos v (x'- x) dx'
= lim - f   dx'  f(x') cos v (x' -  ) dx',
ql=: o 7"   - 00  0
whenever the limit on the right-hand side exists.
Therefore       lim - f dv     f (x') cos v (x' - x) dx'
t=a x 7r Jo  -
has the value  {f (x  + ) +f(x - 0)}, provided the function/ (x) satisfies one
of the sufficient conditions already referred to.
The following theorem has now been established:If f (x) have a Lebesgue integral in every finite interval, and be such that
I f (x) l dx has a definite value, as  lim  I f (x) l dx, then
J  oo,3=, a=-o  a
A f dvf   f(x') cosv(x'-x) dx' =     f(x 0)+  f) (x - O)}
for any point x, at which f(x + 0), f(x - 0) exist, and in the neighbourhood of
which f (x) satisfies one of the sufficient conditions for the convergence of
Fourier's series. In particular, this result holds for every value of x, if f (x)
have limited total fluctuation in every finite interval, and also satisfies the
condition that f  f (x) I dx exists.
48-5




762


Trigonometrical series


[OH. VII


It should be observed that, although
r00  rx
dvj      f (W) cos v (- x) dx'
is here employed to denote
lim   d    f(x') cos v (x' - x) dx',
in accordance with the definition of  X (v)dv as the limit of the Lebesgue
rh
integral  X (v) d, when h is indefinitely increased, provided the limit exists,
it is not necessarily the case that
dv      f (x') cos v (x' - x) dx'
~0    I J  -00
exists. For the existence of    X %(v)dv does not necessarily imply that
of   }% (v) dv.
The above theorem contains Fourier's* representation of a function by
means of a repeated integral, frequently spoken of as Fourier's double
integral, although it is not a double integral in accordance with the
terminology employed in this work.
Let f(x) be zero for all values of x which do not lie in the finite interval
(a, /), we then obtain the following result:
If f (x) have a Lebesgue integral in (a, /), then
dv f(       ') cos v (x'  ) dx'
has the value            {f (x + O) + f(x - )}
at an interior point of the interval (a, 13), and has the values
( a+(o),     f(/~0- o),
at the points a, /3; provided, in each case, the limit exists, and f(x) satisfies
one of the requisite conditions in the neighbourhood of the point. If x be
exterior to (a, 8), the repeated integral is zero.
It should be observed that the repeated integral
J dx' f     (x') cos v (x' - x) dv
does not exist, because     fcos v (x' - x) dv
fo
has no definite meaning.
There is no difficulty in obtaining sufficient conditions for the uniform
convergence of Fourier's repeated integral to the value of the function, in an
interval in which the function is continuous, as in the case of Fourier's series.
* See the Theorie de la Chaleur, Chap. ix, ~ 1.




APPENDIX.
ON TRANSFINITE NUMBERS AND ORDER-TYPES.
A brief reference will be made to some criticisms and remarks* which have
been made relating to the views expressed in the general discussion of the theory
of transfinite numbers and order-types, contained in ~~ 152-163. Some other
writings on the subject, which have recently appeared, and were not mentioned in
Chapter III, will also be noticed.
The term "norm" as used in the definition of an aggregate (~ 153) has been
treated by Russell as synonymous with the term "propositional function" which
he himself employs. This proceeding has led to some misconception as to the
scope of the definition in ~ 153. It was intended that a "norm" should always be
of such a character as to leave no doubt as to the existence of elements in the
aggregate which it defines, whereas the corresponding implication is apparently
not made when the term "propositional function" is employed. Thus it is incorrect to assert, as does Russell (loc. cit. p. 40), that the existence of certain
classes is denied, although such classes possess unimpeachable norms. On the
contrary, the existence of the classes in question was denied, on account of the
absence of the necessary norms, in the sense in which the term "norm" is employed.
It may be the case that an unimpeachable propositional function is present, and
yet that there may be no corresponding class. No opinion is here expressed as to
the most suitable terminology in the general logic of classes.
That aggregates exist which have no cardinal number has been maintained in
~ 157, and is also in accordance with the view of Jourdain (loc. cit. p. 266). The
aggregate TV, which contains all the ordinal numbers, is the first example of such
an aggregate.   To this aggregate the term   "inconsistent" has been applied
by Cantor and Jourdain, but it has been proposed later to abandon this
adjective as descriptive of the aggregate W, on the cogent ground that there is
no inconsistency in the recognition of the existence of the aggregate, but only in
the attribution to it of a cardinal number, and of an order-type. The difference
of view as regards the aggregate W, held by Jourdain from that here maintained,
relates not to the existence, but to the constitution of WV. In accordance with
* These criticisms and remarks refer to the article published in the Proc. Lond. Math. Soc.,
ser. 2, vol. IIn, p. 170, which is substantially identical with ~~ 152-163. They are due to
Russell, Proc. Lond. Math. Soc., ser. 2, vol. Iv, p. 29; to Hardy, in the same volume, p. 10; and
to Jourdain, in the same volume, p. 266.




764                              Appendix
the ideas expressed in ~ 157, so far as our present knowledge reaches, every
segment of W is enumerable; whereas, in Jourdain's view, there exist segments of
which the cardinal numbers are Aleph-numbers of unending variety of order.
As soon as general agreement is attained as to the fundamentals of this subject,
it will be possible to introduce a more precise and suitable terminology than is
at present possible. It will probably be desirable to restrict the term "aggregate"
to such collections of objects as possess a cardinal number, and if they are
ordered, also an ordbr-type. Some other term would then be applied to such
cases as the class TV, when no cardinal number, or order-type exists. A complete
theory must contain, in the first place, precise definitions of the terms "cardinal
number" and "order-type," and would then set forth the necessary and sufficient
conditions to be satisfied by a collection, that it might possess a cardinal number, and
if ordered, then also an order-type, in accordance with the definitions of these terms.
A definition of cardinal number has been given in ~ 155, and it has there been
pointed out, that an aggregate possesses a cardinal number, only when it is one of
a plurality of equivalent aggregates essentially distinct from one another. It has
been suggested by Russell (loc. cit. p. 40), and also by Jourdain (loc. cit. p. 273),
that if we have one series A, we can always obtain another series similar to A, by
interchanging two terms of A, or by replacing a term of A by something else, or,
in the case of a normally ordered aggregate, by removing some of the terms at the
beginning; and thus that the criterion, given in ~ 155, is in all cases satisfied.
The elements of the new aggregates so obtained would however not be essentially
distinct from the original one, and the existence of such new aggregates is stated
in ~ 155 to be insufficient as a reason for attributing the existence of an ordertype, or of a cardinal number, to the original aggregate. In order to leave no
doubt upon this point, it is desirable to render the statement of the conditions for
the existence of a cardinal number, or of an order-type, more precise, by postulating that the given aggregate M must be one of a plurality of equivalent, or of
similar, aggregates, such that the elements m' of one of these M', are not only
essentially distinct and different from the elements n, of M, but are also such
that the existence of these elements m' cannot be deduced as a direct consequence
of the existence of the elements m of lM. The finite numbers perform the function
of counting objects of all sorts and descriptions; the analogous function of transfinite ordinals or cardinals, would be that of counting sets of objects which are at
least not confined to belong to a very special class of objects, viz. transfinite numbers
and order-types.
The criticisms contained in ~~ 159-162, of those proofs of theorems which
depend upon the assumption that an infinite number of arbitrary acts of choice is
a valid process, have been discussed by Russell, Hardy, and Jourdain, in relation to
principles known as the "axiom of Zermelo," and the "multiplicative axiom."
Russell has signified (loc. cit. p. 47) his complete agreement with the views
expressed in ~~ 159-162; but Hardy and Jourdain maintain the validity of the
"multiplicative axiom," in accordance with which, when an infinite number of
aggregates have been defined, there exists a new aggregate, each element of




Appendix


765


which consists of an aggregate of elements one of which belongs to each of the
given aggregates. It is held that this principle is valid, although it may be
impossible to assign rules by which one element is to be chosen out of each of the
given aggregates, and when it is consequently impossible to define any single
element of the so-called multiplicative class. This principle is maintained by
Hardy (loc. cit. pp. 14-17), and by Jourdain (pp. 281-282), as a valid postulate, partly by an appeal to authority, and on account of its utility in much
interesting mathematics, and partly because it is assumed to be so self-evident
that a denial of it is paradoxical. It seems however difficult to assign a precise
meaning to the existence of an aggregate, when not even a single element of it is
capable of real definition. Whenever the multiplicative class is, in any particular
case, capable of definition, the axiom is unnecessary. The existence of an abstract
mathematical object would appear to be entirely dependent on a precise definition
of such object; in default of such definition, such hypothetical object is not
identifiable. Thus the hypothetical elements of a multiplicative class, whenever
the use of the axiom is necessary, are not identifiable as individuals, and are
indistinguishable from one another.
The axiom employed by Zermelo in his attempt to prove that every aggregate
is capable of being normally ordered, and which consists of the assumption that
each and every part of a given aggregate can be correlated with a single element
contained in that part, has been discussed* by Hadamard, Lebesgue, Baire, and
Borel, in connection with Zermelo's proof (~ 161). It is there pointed out by
Hadamard, that the real question at issue is whether it is possible to demonstrate
the existence of mathematical entities which cannot be precisely defined; a
question which Hadamard himself answers in the affirmative, but with which
answer the other writers do not appear to be in agreement. The tentative
character of some investigations in which use of an infinite number of acts of
choice has been made is pointed out by Borel.
A propos of a criticism t by Levi, of Bernstein's proof (see ~ 150), that the
aggregate of closed sets of points in the n-dimensional continuum has the power c
of the continuum, Bernstein t has endeavoured to avoid the difficulty involved in
the use of a correspondence which cannot be defined, by introducing the conception
of "multiple equivalence."  Thus, if there are two aggregates M and N, for which
an aggregate F ={4+} of reversible (1, 1) correspondences ~b exists, in which no
element is special (ausgezeichnet), then the two aggregates are said to be multiply
equivalent. The cardinal number ' =f is then termed the multiplicity of the
correspondence b. In case the multiplicity is unity, the aggregates are said to
have a one-valued equivalence. The difficulty of this conception is the same as
the one referred to above, in the case of the multiplicative axiom, viz., that the
aggregate q is such that no single element of it is capable of definition, and that
the elements are consequently indistinguishable from one another.
* "Cinq lettres sur la thoorie des ensembles," Bulletin de la Soc. Math. de France, 1905.
+ "Intorno alla teoria degli aggregati," Lomb. Ist. Rend. (2), 1902, pp. 863-869.
+ "Bemerkung zur Mengenlehre," Nachrichten der Ges. d. Wissensch. zu Gottingen, 1904,
p. 557.




766


Appendix


The difficulties which have arisen in connection with the aggregate W, of all
ordinal numbers, on the assumption that it possesses segments with transfinite
cardinal numbers of infinite variety, have been dealt with* by Bernstein.       He
postulates that the ordinal numbers of the series W have the following properties:(1) they are order-types of normally ordered aggregates; (2) if a be one of them,
there always exists a next greater one, a + 1; and thus a is always the order-type
of segments of normally ordered aggregates. He then defines W to be the aggregate
of all order-types of the segments of normally ordered aggregates, and deduces that
W is itself normally ordered.   He next assumes that, although W      satisfies the
condition (1), it does not satisfy the condition (2); and thus that W is itself not
a segment of a normally ordered aggregate.    This assumption as to the nature of
W, which appears as a postulate, amounts to the assertion that no aggregate (W, e)
can exist, in which e has higher rank than all the elements of IF. That the
validity of such postulation is at least doubtful, appears clearly, if we consider
that the hypothetical object W is incapable of being substituted for the element
A in an aggregate (A, e) in which e is regarded as of ordinally higher rank than A.
It would appear that, if e is of higher rank than A, it might also be of higher
rank than every element contained in A, when A is composite, or in any aggregate
which may be substituted for A.
The failure of attempts to define a set of points of the continuum which shall
have a (1, 1) correspondence with all the ordinal numbers of the first and second
classes is connected with the fact that there exists no systematic representation of
all the numbers of the second class. The usual symbolism o, e, &c. breaks down at
certain points, where new symbols have to be employed for the representation of
higher numbers of the class. It is however easily seen that no enumerable set of
new symbols will suffice for a mode of representation of all the numbers of the class.
For only an enumerable set of numbers could be represented by means of an
enumerable set of symbols w, E,..., all possible forms obtained by use of these
symbols being employed.
An attempt has been made by Kdnigt to distinguish between those elements
of the arithmetic continuum which are "finitely defined," and those which are not
capable of finite definition. It can however be shewn + that the distinction introduced by Konig is an invalid one. A full discussion of this matter would require
a larger amount of space than can here be given to it; reference must therefore be
made to the literature in which the point is discussed.
* " Ueber die Reihe der transfiniten Ordnungszahlen," Math. Annalen, vol. LX, p. 187. An
article by Schonflies " Ueber die logischen Paradoxien der Mengenlehre," Jahresber. d. Deutsch.
Math. Ver., vol. xv, 1906, may be referred to; also an article by Harward, Phil. Mag. Oct. 1905,
p. 457.
t "Ueber die Grundlagen der Mengenlehre und das Kontinuumproblem," Math. Annalen,
vol. LXI, 1905; also-Acta Math., vol. xxx, p. 329. See also an article by Dixon " On a question
in the theory of aggregates," Proc. Lond. Math. Soc., ser. 2, vol. iv. The same idea has been
discussed in the shape of a paradox, by Richard, Acta Math., vol. xxx, p. 295. For a further
development of Konig's ideas, see Math. Annalen, vol. LXIII, p. 217.
+ See Hobson, "On the Arithmetic continuum," Proc. Lond. Math. Soc., ser. 2, vol. iv,
where the matter is fully discussed. A reply by Dixon, "On well ordered aggregates," is given in
the same volume.




LIST OF AUTHORS QUOTED.
[The numbers refer to pages.]


Abel, 496, 497, 501
Ampere, 265
Archimedes, 40
Arzela, 122, 426, 474, 481, 494, 537, 540, 543
Ascoli, 52, 63, 339, 745
Baire, 65, 93, 114, 146, 237, 239, 327, 525,
530, 532, 533, 765
Baker, 653
Beman, 9
Bendixson, 57, 90, 141, 541, 562
Bernoulli (D.), 637, 639
Bernstein, 156, 159, 185, 189, 765, 766
Bessel, 641
Bettazzi, 51, 246
Birkhoff, 478
Bocher, 482, 649, 719, 727
Bolzano, 58
Bonnet, 359, 666
Borel, 65, 86, 90, 102, 106, 122, 128, 159, 210,
228, 522, 524, 533, 707, 728, 765
Broden, 97, 116, 117, 217, 219, 249, 285, 296,
298, 620, 695
Bromwich, 466, 589, 592
Burali-Forti, 195, 201
Burkhardt, 635, 637
Burmann, 21
Cantor (G.), 9, 21, 30, 47 et seq., 62 et seq.,.71, 74, 76, 79, 82, 94 et seq., 123, 148,
158, 163, 170, 181 et seq., 194, 198 et seq.,
206, 211, 275, 330, 480, 618, 744, 745
Cantor (M.), 19, 35
Cathcart, 316, 606
Cauchy, 37, 222, 339, 365, 463, 500, 504,
510, 641
Cesaro, 707
Couturat, 9


Dini, 235, 243, 266, 274, 285, 294, 298, 343,
349, 366, 378, 461, 471, 478, 564, 607,
620, 626, 666, 694, 701, 745
Dircksen, 641
Dirichlet, 216, 231, 247, 337, 497, 641, 664
Dixon, 766
Du Bois Reymond, 9, 36, 52, 57, 62, 64, 90,
207, 265, 290, 345, 359, 360, 366, 378,
408, 422, 430, 455, 477, 510, 585, 592,
626, 652, 701, 732, 745
Euclid, 19
Euler, 47, 636, 638
Fatou, 701, 719, 729
Fejer, 708
Forsyth, 719
Fourier, 57, 215, 639, 649, 760, 761
Frege, 1
Genocchi, 302, 321, 730
Gibbs, 653
Gibson, 637
Goursat, 87
Grave, 275
Hadamard, 765
Hahn, 251, 351
Hamilton, 10
Hankel, 20, 98, 241, 243, 247, 361, 607, 626
Hardy, 187, 191, 207, 210, 444, 445, 462, 479,
497, 590, 600, 607, 763
Harnack, 57, 90, 98, 101, 109, 244, 248, 316,
367, 426, 725, 746
Harward, 766
Heine, 9, 21, 29, 222, 229, 672, 743, 744
Helmholtz, 2, 9, 10
Hermite, 68
Hilbert, 38, 195, 333, 335
Hobson, 131, 194, 342, 396, 442, 456, 474,
483, 486, 490, 494, 537, 540, 566, 586,
683, 702, 713, 763, 766
Holder, 51, 53, 360, 388, 695, 751


D'Alembert, 636
Dantscher, 323, 515
Darboux, 339, 477, 481, 620
Dedekind, 2, 20, 21, 25, 151




768


List of authors quoted


Huntington, 169
Hurwitz, 715
Husserl, 10
Jacob, 20
Jordan, 65, 107, 136 et seq., 256, 367, 417,
426, 432, 442, 468, 583, 666, 701
Jourdain, 171, 183, 195, 209, 763
Kneser, 649
Konig, 284, 766
Kopcke, 296, 298, 299, 626
Kowalewski, 360
Kronecker, 21, 360, 695, 698


Lagrange, 504, 637
Lambert, 47
Lebesgue, 102, 226, 259, 295,
522, 525, 551, 554, 556,
674, 708, 715, 751, 765
Legendre, 47
Leibnitz, 1
Lerch, 522, 620, 727
Levi, 765
Liapounoff, 716
Lindemann, 68
Liouville, 68, 97
Lipschitz, 694, 701
Locke, 1
London, 464
Loria, 329
Love, 653


332, 342, 390,
576, 580, 646,


Pringsheim, 9, 19, 47, 145, 266, 269, 305, 360,
405, 426, 429, 464, 501, 505, 512
Reiff, 477
Richard, 766
Riemann, 247, 338, 341, 405, 637, 679, 730
Roche, 504
Rudio, 47
Russell, 9, 23, 30, 169, 195, 204, 763
Sachs, 637, 676, 701
Scheoffer, 265, 273, 275, 290, 323, 513, 515,
521
Schepp, 55
Schlflii, 652
Schlomilch, 504, 729
Schonflies, 90, 116, 135, 139, 205, 248, 251,
284, 299, 335, 351, 370, 379, 417, 423,
428, 484, 586, 766
Schroder, 156
Schubert, 9
Schwarz, 277, 319, 676, 701, 719
Seidel, 477
Smith (H. J. S.), 64, 97, 416
Stackel, 672
Steinitz, 299
Stekloff, 716
Stephanos, 48
Stokes, 458, 477, 482, 727
Stolz, 52, 54, 98, 269, 316, 319, 321, 323,
372, 381, 417, 430, 432, 442, 515, 520,
585, 592
Study, 252, 256
Tannery, 31, 480
Thomae, 302, 314, 339, 416, 428
Toepler, 724
Vallee-Poussin (de la), 368, 379, 440, 566,
571, 576, 590, 603, 715, 716
Veblen, 169
Veltmann, 145
Veronese, 51, 55
Volterra, 247, 357, 481, 522, 723


Meray, 21
Mertens, 500
Meyer, 360
Michelson, 653
Mill, 1
Mittag-Leffier, 64, 81, 522
Moore, 335, 381, 384
Netto, 329, 360
Neumann (C.), 222, 263, 360, 666
Newton, 19


Osgood, 113, 480, 482, 485, 486, 540, 543
Pasch, 98, 235
Peacock, 20
Peano, 9, 23, 107, 248, 319, 323, 330
Pereno, 299, 626
Picard, 522, 719, 722
Pierpont, 426, 448
Pincherle, 21
Poincare, 653
Poisson, 640
Pompeiu, 620


Wallis (John), 35
Weber, 739
Weierstrass, 21, 58,
522, 625
Whitehead, 153
Wiener, 625


224, 264, 359, 360, 477,


Young (W. H.), 84, 90, 102, 111, 113, 118
et seq., 128, 185, 186, 342, 345, 395,
487, 540, 541
Zermelo, 156, 209




GENERAL INDEX.
[The numbers refer to pages.]
Abel's theorem on power-series, 497
Adherence, 125
Aggregate; General notion of, 1; Elements of, 1; Finite, 3; Similar, 4, 162, 200; Equivalent,
6, 150; Rank of elements in, 6; Ordinal number of, 7; Definition of finite, 8; Perfect,
49; Straight line as, 56; Power of, 70; Definition of, 149, 197; Comparable and Incomparable, 152; Closed, 166; Dense in itself, 166; Everywhere-dense, 166; General theory
of, 194; Normally ordered, 170; Parts of, 151; Limiting elements of, 165; of ordertypes V, 0, 7r, 167; Segments of normally ordered, 172; Simply ordered, 163, 198;
Structure of simply ordered, 165
Aleph-numbers, 154; General theory of, 181, 209
Archimedean system, 40
Archimedes, Axiom of, 54
Arithmetization, 13; Kronecker's scheme of, 21
Baire's, theorem on representation of functions by series of continuous functions, 525; classification of functions, 532
Cardinal numbers, Definition of, 8, 150, 199, 204; Addition and Multiplication of, 152; as
exponents, 153; Division of, 159; exceeded by other cardinal numbers, 158; of continuum,
183, 187; of second class of ordinals, 180; Relative order of, 151; Smallest transfinite,
154; of aggregate of continuous functions, 227
Categories, First and Second, 114
Coherence, 125
Condensation of singularities, 607; Cantor's method of, 618
Congruency, Axiom of, 53
Connexity of aggregate of real numbers, 49
Content, of sets of points, 77; of closed sets, 108
Continuity, of functions, 221; Uniform, 229; of sum-function, 475
Continuous functions, 225; two definitions not equivalent, 226; defined at points of a set,
226; Oscillating, 281, 294; Construction of, 296; with respect to each of two variables,
324; Non-differentiable, 620
Continuum, of real numbers, 49; given by intuition, 52; Straight line as, 53; Arithmetic, 72;
Cardinal number of, 183, 187; of n-dimensions, 190; Order-type of, 169
Convergence, General principle of, 36; of series, defined, 453; Absolute and conditional, 457;
Absolute, of double series, 468; Uniform, 470; Simply-uniform, 471; Non-uniform, 474;
Tests of uniform, 477; Measure of non-uniform, 484; Distribution of points of nonuniform, 485; of power-series, 495; of product of two series, 500
Correspondence, General notion of, 2
Curves filling space, 330




770


General Index


Derivatives, of sets of points, 61, 98; of functions, 263; Upper and lower, 265; Progressive
and regressive, 265; Limited, 283; General properties of, 285
Differential coefficients of functions, 263; Successive, 275; Partial, 311; Higher partial, 316
Differentiation, of series, 561; of integrals with respect to parameters, 599; of Fourier's
series, 725
Diriclilet's, integral, 656, 695; investigation of Fourier's series, 658
Discontinuity, Infinite, 220; of functions, 233; of functions, classification, 234; of first and
second kinds, 235
Discontinuous functions, Classification of, 240; Point-wise, 243; Construction of point-wise, 246
Divergence, of series, 455; Measure of, 746
Double integrals, 416, 440; Properties of, 430; Improper, 432, 438; over infinite domains,
442; Transformation of, 445
Enumerable aggregate, definition, 66; rational numbers, 68; algebraical numbers, 68; isolated
aggregates, 83
Equivalence theorem, 119
Extent, Exterior and Interior, 108
Fluctuation, of function, 220; Inner, 220
Fourier's series; definition, 640, 646; Formal expression of, 641; cosine and sine series, 643;
Generalized, 646; Dirichlet's investigation of, 658; Dirichlet's conditions for convergence
of, 664; Uniform convergence of, 669, 683; Limiting values of coefficients in, 671;
Convergence of coefficients in, 675; Sufficient conditions of convergence of, 680; Nonconvergence of, 701; Series of arithmetic means related to, 707; Properties of coefficients
of, 715; Integration of, 718; Differentiation of, 725; Limits of coefficients in, 746
Fractional numbers, 12
Frontier of set of points, 108, 136
Functions, Dirichlet's definition, 216; Homonomically and heteronomically defined, 217;
Upper and lower limits of, 219, 224; Limited, definition, 219; Unlimited, 219; Upper
limit of, in an interval, 220; Continuity of, 221; continuous in an interval, 222; Limits
of, at a point, 230; Maxima and minima of, 234, 259; Semi-continuous, 237; Ordinary,
299; Classification of discontinuous, 240; Point-wise discontinuous, 243; Monotone, 245,
263; defined by extension, 249; Most nearly continuous, 251; with limited total fluctuation,
252; with limited total variation, 257; with limited derivatives, 283; with one derivative
assigned, 294; of two or more variables, 299; of two variables, 301; Double and Repeated
limits of, of two variables, 303, 304; Existence and equality of repeated limits of, 306;
Uniform convergence of, 309; Maxima and minima of, 309; linear in each interval of
a set, 357; Summable, 391; represented by series, 469; Maxima and minima of, of one
variable, 512, of two variables, 515; Representation of, by series of continuous functions,
522; Baire's classification of, 532; Construction of continuous non-differentiable, 620;
Differentiable, everywhere-oscillating, 626
Functional image, 223
Functional limits; Aggregate of, 233
Functional relation, 214
Generation, Cantor's principle of, 76
Graphs, Functions defined by, 215
Heine-Borel theorem, 86
Incrementary ratios, 286; Limit of, 290
Indeterminate forms, Evaluation of, 268
Infinitesimals, Non-existence of, 40; Veronese's theory of, 55
Integers, Sets of sequences of, 146
Integrable functions, Particular cases of, 343; null-functions, 347
Integral numbers, Operations on, 10




General Index


771


Integral Calculus, Fundamental Theorem of, 349, 355, 387, 550
Integral, Definite, of limited functions, 338; Riemann's definition of, 338; Upper and lower,
339; Conditions of existence of, 341; Properties of definite, 344; Indefinite, 350; Improper,
364; Harnack's definition of improper, 367; de la Vallee-Poussin's definition of improper,
368; Absolutely and conditionally convergent, 369, 371, 377, 395; Lebesgue's definition of,
390; Equivalence of definitions of, 396; with infinite limits, 398; Change of variable in
an, 410; Double, definition of, 416; Conditions for existence of double, 418; Repeated,
421; Improper double, 432, 433, 440; Non-absolutely convergent improper Lebesgue, 537;
Repeated improper, 566; Repeated Lebesgue, 576; Repeated, of unlimited function, 582;
Restricted Jordan double, 583; Repeated, over infinite domain, 586; Limit of, containing
a parameter, 594; Poisson's, 640, 719; Dirichlet's, 656, 695; Fourier's, 758; Fourier's
single, 760; Fourier's repeated, 762
Integration, Geometrical interpretation of, 388; Double, 415; Lebesgue's theory of, 390; by
parts, 407, 559; of series, 534; of limit of a sequence, 538; with respect to a parameter,
582; of Fourier's series, 718
Intervals, Open and closed, 50; Unenumerable sets of, 89
Irrational numbers, 19; Development of theory of, 21; Dedekind's theory of, 18; Cantor's
theory of, 26
Limits, Arithmetical theory of, 35; Method of, 36
Lines of invariability, 260
Magnitude, Theory of, 19, 52
Maxima and Minima, 234, 259; Proper and improper, 266; of functions of one variable, 512;
of functions of two variables, 515
Mean value theorem, of Differential Calculus, 267; of Integral Calculus, 358; Application of,
to Fourier's series, 666
Measure, of sets of points, 81, 107, 143; Exterior and interior, 103, 142; of limits of sequences
of closed sets, 113; of discontinuity defined, 233
Measurement, 51
Negative numbers, 51
Null-function, Point-wise discontinuous, 350
Order, Notion of, 2
Order-functions, 185
Order-types, 163; Addition and multiplication of, 163; of continuum, 169, 206;?,, 0, r, 167
Ordinal-numbers, Definition of, 4, 175; Transfinite, 74; of second class, 76, 177; Arithmetic
of, 183; Limiting, 177; Sum and product of, 176
Oscillation, of function at a point, 235; of function in an interval, 236, 261
Power-series, 495; Interval of convergence of, 496; Abel's theorem on, 497
Rational numbers, Aggregate of, 18; Section of aggregate of, 23
Real numbers, Dedekind's definition, 23; Cantor's definition, 29; Operations on, 31; Convergent sequences of, 33; Equivalence of two definitions of, 38; Representation of, 45;
everywhere-dense, 49
Remainder-function, Transformed, 483
Repeated, limits of functions, 304; Existence and equality of, limits of functions, 306;
integrals, 421; improper integrals, 566; Lebesgue integrals, 576; integrals of unlimited
functions, 582; integrals over infinite domains, 586; Fourier's, integral, 762
Riemann's theory of trigonometrical series, 730
Saltus, 233, 300; -function, 238
Schwarz's theorem, 277; on partial differential coefficients, 319
Semi-continuous functions, 237




772                             General Index
Sequences, Simple, 6; Convergent, 26; Ascending and descending, 164; Functions defined
by, 453; Double, 464; Repeated limits of double, 465; Monotone, 465; Conditions of
equality of repeated limits of, 466; Integration of limits of, 538
Series, Limiting sum of, 453; Remainder in a, 453; Convergent, 453; Non-convergent
arithmetic, 455; Divergent, 455; Oscillating, 455; Limits of indeterminacy of, 455;
Absolutely and conditionally convergent, 457; of transfinite type, 461; Double, 463, 464,
467; Absolutely convergent double, 468; Uniform convergence of, 470; Simply uniform
convergence of, 471; Taylor's, 501, 514; of continuous functions, 522; Integration of,
534; Differentiation of, 561; Special cases of trigonometrical, 638, 648; Fourier's series,
640, 646; Generalized Fourier's, 646; Formal expression of Fourier's, 646; of arithmetic
means related to Fourier's, 715; Properties of coefficients of Fourier's, 715; Special cases
of trigonometrical, 638; Finite trigonometrical, 722; Riemann's theory of trigonometrical,
730; Uniqueness of Fourier's, 748; Convergent trigonometrical, 756
Sets of intervals; Limiting point of, 60; Properties of, 81; External, internal, and semiexternal points of, 84; Unenumerable sets of, 89
Sets of points, 57; Analysis of, 123; Closed, 64, 84; Common points of system of, 92;
Connex, 138; Degrees of points in, 123; dense-in-themselves, 64; Derivatives of, 61, 98;
Detached, 137; Everywhere-dense, 65; Exterior points of, 107; Frontier of, 108, 136;
Improper limiting points of, 61; Inner limiting, 127; Interior points of, 107; Irreducible,
98; Isolated, 64; Limiting points of, 60; Measurable, 104; Non-dense, 65; Non-dense
closed, 90; Non-dense perfect, 90, 94; of first species, 62, 83; of second species, 63;
of first and second categories, 144; Perfect, 65; Reducible, 98; Separated, 65; Sequences
of closed, 110, 117; Sets of closed, 93; Transfinite derivatives of, 79; Unbounded, 59;
Upper and lower boundaries of, 58; of cardinal number of continuum, 191, 210
Square, Representation of, in linear interval, 329
Sum-function, Continuity of, 475, 489; Transformed, 483; Limits of a, 487
Taylor's series, 501; for functions of two variables, 514
Total fluctuation, 252
Total variation, 257
Unenumerable; Proof that continuum is, 69
Unity, Notion of, 1
Variable, Real, 213
Vibrating strings, 636


CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS.