Table Of ContentScience Networks . Historical Studies
Volume 7
Edited by Erwin Hiebert and Hans Wussing
Editorial Board:
S. M. R. Ansari, Aligarh R. Halleux, Liege
D. Barkan, Pasadena S. Hildebrandt, Bonn
H. J. M. Bos, Utrecht E. Knobloch, Berlin
U. Bottazzini, Bologna Ch. Meinel, Regensburg
J. Z. Buchwald, Toronto G. K. Mikhailov, Moskva
S. S. Demidov, Moskva S. Nakayama, Tokyo
J. Dhombres, Nantes L. Novy, Praha
J. Dobrzycki, Warszawa D. Pingree, Providence
Fan Dainian, Beijing W. Purkert, Leipzig
E. A. Fellmann, Basel J. S. Rigden, New York
M. Folkerts, Munchen D. Rowe, Pleasantville
P. Galison, Stanford A. 1. Sabra, Cambridge
1. Grattan-Guinness, Bengeo R. H. Stuewer, Minneapolis
J. Gray, Milton Keynes V. P. Vizgin, Moskva
Springer Basel AG
Fyodori\.~edvedev
Scenes from the
History of Real Functions
Translated from the Russian by Roger Cooke
Springer Basel AG
Author's address
Professor Dr. EA. Medvedev
Institute for History
of Science and Technology
Staropanskii Pereulok, d. 1/5
103012 Moskva, USSR
Originally published as: Ocherki istorii teorii funktsii deistvitel'nogo peremennogo,
Moskva: Nauka 1975
Deutsche Bibliothek Cataloging-in-Publication Data
Medvedev, Fedor A.:
Scenes from the history of real functions I Fyodor A.
Medvedev. Transl. from the Russian by Roger Cooke. - Basel ;
Boston ; Berlin : Birkhăuser, 1991
(Science networks ; VoI. 7)
Einheitssacht.: Ocerki istorii teorii funkcij dejstvitel'nogo
peremennogo <engl.>
ISBN 978-3-0348-9721-1 ISBN 978-3-0348-8660-4 (eBook)
DOI 10.1007/978-3-0348-8660-4
NE:GT
This work is subject to copyright. AU rights are reserved, whether the whole or part
of the material is concerned, specifically those of translation, reprinting, re-use of
illustrations, broadcasting, reproduction by photocopying machine or similar means,
and storage in data banks. Under § 54 of the German Copyright Law where copies are
made for other than private use a fee is payable to »Verwertungsgesellschaft Wort«, Munich.
© 1991 Springer Basel AG
Originally published by Birkhliuser Verlag Basel in 1991
Softcover reprint ofthe hardcover lst edition 1991
ISBN 978-3-0348-9721-1
Translator's Note
The Russian original of the present work, bearing the title O'IepKH JilCTOPHH
TeopHH <l>YHK~H lleHCTBHTeJIhHOrO IIepeMeHHoro (Essays on the History
of the Theory of Functions of a Real Variable), was published in 1975. The
present translation incorporates some references to historico-mathematical
work that has appeared since that time along with corrections of the misprints
that were detected in the original. The new references and the corrections
were both provided to the translator by Prof. Medvedev. These are the only
elements added in the present work, which is otherwise a complete translation
of the original. The following points, however, should be noted:
1. For the sake of euphony and brevity the English title is not a translation of
the Russian, but has been chosen to convey the general contents of the book
as described by Prof. Medvedev in the preface~a selection of topics from the
history of the theory of functions of a real variable.
2. The original work generally referred to Russian translations of mathemat
ical works whenever such an edition existed. These Russian translations have
in every case been replaced by their originals in the bibliography; and when
ever the originals were available to the translator, page citations to them in
the text have been amended accordingly. In the two cases where the originals
were not available-·the references to Cavalieri's Exercitationes in Section 2
of Chapter 4 and the references to Cauchy's Calcul infinitesimale in Section
5 of Chapter 4~the Russian page references were used in the hope that they
may provide at least a clue as to the location referred to.
3. Prof. Medvedev's Russian translations of quotations from English works
have been replaced by the original English, except for the quotation from
H. G. Wells in the preface, which the translator was unable to locate.
Contents
Preface............................................................. 9
Chapter 1 The place of the theory of functions of a real
variable among the mathematical disciplines
1.1 The subject matter of the theory of functions...... ..... ....... 11
1.2 Three periods in the development of
the theory of functions. . .... ............ ....... ...... . .... ..... 13
1.3 The theory of functions and classical analysis. . . . . . . . . . . . . . . . . . 15
1.4 The theory of functions and functional analysis. . . . . . . . . . . . . . . . 18
1.5 The theory of functions and other
mathematical disciplines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Chapter 2 The history of the concept of a function
2.1 Some textbook definitions of the concept of a function... ...... 25
2.2 The concept of a function in ancient times and in the
Middle Ages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 The seventeenth-century origins of the concept
of a function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Some particular approaches to the concept of a function
in the seventeenth century .................................... . 35
2.5 The Eulerian period in the development of the
concept of a function ......................................... . 40
2.6 Euler's contemporaries and heirs .............................. . 44
2.7 The arbitrariness in a functional correspondence .............. . 50
2.8 The Lobachevskii-Dirichlet definition ......................... . 55
2.9 The extension and enrichment of the concept of a
function in the nineteenth century ............................ . 63
2.10 The definition of a function according to Dedekind ............ . 67
2.11 Approaches to the concept of a function
from mathematical logic ...................................... . 70
2.12 Set functions ................................................. . 73
2.13 Some other functional correspondences ........................ . 77
Chapter 3 Sequences of functions.
Various kinds of convergence
3.1 The analytic representation of a function ...................... . 82
8 Contents
3.2 Simple uniform convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.3 Generalized uniform convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.4 Arzela quasi uniform convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.5 Convergence almost everywhere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.6 Convergence in measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.7 Convergence in square-mean. Harnack's
unsuccessful approach......................................... 128
3.8 Square-mean convergence. The work of Fischer
and certain related investigations. . . .. . . . . . . . . . . . . .. . . .. . . . . . .. 137
3.9 Strong and weak convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
3.10 The Baire classification .............. '. . . . . . . . . . . . . . . . . . . . . . . . . . 157
Chapter 4 The derivative and the integral
in their historical connection
4.1 Some general observations..................................... 168
4.2 Integral and differential methods up to the first half of the
seventeenth century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.3 The analysis of Newton and Leibniz. . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.4 The groundwork for separating the
concepts of derjvative and integral. . . .. . . . .. . . . . . . . . . .. .. . . . .. . 181
4.5 The separation of differentiation and integration. . . . . . . . . . . . . . . 187
4.6 The Radon-Nikodjrn theorem.................................. 198
4.7 The relation between differentiation and integration in the
works of Kolmogorov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
4.8 The relation between differentiation and integration in the
works of Caratbeodory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
4.9 A few more general remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
Chapter 5 Nondifferentiable continuous functions
5.1 Some introductory remarks.................................... 212
5.2 Ampere's theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
5.3 Doubts and refutations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
5.4 Classes of nondifferentiable functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 224
5.5 The relative "smallness" of the set
of differentiable functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Index of names. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
Preface
To attempt to compile a relatively complete bibliography of the theory of
functions of a real variable with the requisite bibliographical data, to enumer
ate the names of the mathematicians who have studied this subject, exhibit
their fundamental results, and also include the most essential biographical
data about them, to conduct an inventory of the concepts and methods that
have been and continue to be applied in the theory of functions of a real
variable ... in short, to carry out anyone of these projects with appropriate
completeness would require a separate book involving a corresponding amount
of work. For that reason the word essays occurs in the title of the present
work, allowing some freedom in the selection of material.
In justification of this selection, it is reasonable to try to characterize
to some degree the subject to whose history these essays are devoted. The
truth of the matter is that this is a hopeless enterprise if one requires such a
characterization to be exhaustively complete and concise. No living subject
can be given a final definition without provoking some objections, usually
serious ones. But if we make no such claims, a characterization is possible;
and if the first essay of the present book appears unconvincing to anyone, the
reason is the personal fault of the author, and not the objective necessity of
the attempt.
It can hardly raise objections in a set of essays on the history of the
theory of functions1 to designate a special section devoted to the history of
the central concept of this subject. One may rather complain that it is too
brief and incomplete, or perhaps unsatisfactory. But here again the reproach
lies against the author rather than the actual state of affairs.
Clearly a section on sequences of functions is equally legitimate. The
various kinds of convergence of functional sequences play too prominent a
role in the theory of functions to be denied their share of attention, even in
the most modest essays.
The connection between the operations of differentiation and integration
led to the rise of classical analysis. Classical analysis in turn is responsible in
large degree for the development of the theory of functions. And if a breach
between the concepts of derivative and integral is characteristic of the latter,
nevertheless the attempt to establish a connection of some kind between them
1 Here and below the words theory of functions are used for brevity to denote theory of
functions of a real variable.
10 Preface
was one of the most important stimuli for the advancement of mathematical
thought in this area. To trace the vicissitudes of the relationship between the
two fundamental infinitesimal operations is the task of the fourth essay.
The study of nondifferentiable functions, begun in the 1870's and con
tinuing down to the present, led to an essential re-examination of certain en
trenched conceptions. In the concluding part of the present work an attempt
is made to describe particular aspects of the history of this problem.
The essay/chapters just named by no means cover the subject of the
history of the theory of functions, even if one imagines them written with
exhaustive completeness: Enormous gulfs remain almost entirely beyond the
scope of this book, such as the constructive theory of functions in the sense of
Chebyshev-Bernshtein, the theory of quasianalytic and almost-periodic func
tions, and much more. The theories of differentiation and integration could
be the subject of huge separate books, and in the fourth essay only particular
moments of the history of these operations are presented. The same should be
said about the relationship between the history of the theory of series and the
third essay. In short, there are more than enough gaps to make the present
book on the history of the theory of functions a good exercise book for the
critically inclined reader. Perhaps what is written in this book will provoke
even more critical comments. To this one could reply jokingly with the fol
lowing words of H. G. Wells: "Anyone who does not like this book is free to
choose another one or to write a better one according to his own taste and
measure."
Now a few words about the bibliography at the end of the book. First
of all, each article or book mentioned there was studied by the author to
some extent. Furthermore, we could easily have made the bibliography several
times larger than it is while preserving the quality mentioned in the preceding
statement, and only the desire to reduce the volume of the book forced us to
make a selection, whose correctness it is left to the reader to decide. The year
of first publication of a work is noted at the end of each entry. In citations
only the most fundamental work is cited: preliminary communications on the
same theme are not mentioned when it is not necessary to do so.
Chapter 1
The place of the theory of functions of a real
variable among the mathematical disciplines
1.1 The subject matter of the theory of functions
Examining the monographs and textbooks on the theory of functions, one may
notice that, as a rule, they do not give any comparatively brief definition of
the subject matter of this mathematical discipline, as is done, for example, for
topology-the study of the properties of topological spaces that are invariant
under homeomorphisms-or algebra-the study of operations on elements of
sets of arbitrary nature. The definition of the theory of functions occasionally
encountered, as the study of the general properties of functions, is essentially
a tautology.
The subject matter of the theory of functions is most often characterized
by enumerating the problems that are studied in it. Such an enumeration
is either carried out explicitly (usually in the introductions to books) or is
implicit in the table of contents. We now give several examples of what we
are talking about.
In the first treatise on the theory of functions, Dini's book Fondamenti
per la teorica delle funzioni di variabili reali [4], the list of chapters is as
follows: "Irrational numbers" 1; "Groups of numbers and points, their upper
and lower limits" 2; "The concept of a limit. Infinitely small and infinitely
large quantities"; "The concept of a function. Continuity and discontinuity";
"Functions continuous on a given interval"; "Functions having an infinite num
ber of discontinuities"; "Derivatives"; "Theorems on series"; "The principle of
condensation of singularities"; "Functions having a definite finite derivative
nowhere"; "Other considerations relating especially to the existence of the
derivative of finite and continuous functions"; "Definite integrals."
In the introduction to the book of Hobson The Theory of Functions of
a Real Variable and the Theory of Fourier Series (1907), in particular it is
stated that, "The theory of functions of a real variable, as developed during
1 Dedekind's theory of real numbers [2J is meant.
2 This chapter contrullS an exposition of the earliest results on the theory of point sets ob
truned by mathematicians up to 1872. Dini calls the least upper bound and greatest lower bound
the upper limit and lower limit respectively.
