Scenes from the History of Real Functions

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.

Contenu

1 The place of the theory of functions of a real variable among the mathematical discipline.- 1.1 The subject matter of the theory of functions.- 1.2 Three periods in the development of the theory of functions.- 1.3 The theory of functions and classical analysis.- 1.4 The theory of functions and functional analysis.- 1.5 The theory of functions and other mathematical disciplines.- 2 The history of the concept of a functio.- 2.1 Some textbook definitions of the concept of a function.- 2.2 The concept of a function in ancient times and in the Middle Ages.- 2.3 The seventeenth-century origins of the concept of a function.- 2.4 Some particular approaches to the concept of a function in the seventeenth century.- 2.5 The Eulerian period in the development of the concept of a function.- 2.6 Euler's contemporaries and heirs.- 2.7 The arbitrariness in a functional correspondence.- 2.8 The Lobachevskii-Dirichlet definition.- 2.9 The extension and enrichment of the concept of a function in the nineteenth century.- 2.10 The definition of a function according to Dedekind.- 2.11 Approaches to the concept of a function from mathematical logic.- 2.12 Set functions.- 2.13 Some other functional correspondences.- 3 Sequences of functions. Various kinds of convergenc.- 3.1 The analytic representation of a function.- 3.2 Simple uniform convergence.- 3.3 Generalized uniform convergence.- 3.4 Arzelà quasiuniform convergence.- 3.5 Convergence almost everywhere.- 3.6 Convergence in measure.- 3.7 Convergence in square-mean. Harnack's unsuccessful approach.- 3.8 Square-mean convergence. The work of Fischer and certain related investigations.- 3.9 Strong and weak convergence.- 3.10 The Baire classification.- 4 The derivative and the integral in their historical connection.- 4.1 Some general observations.- 4.2 Integral and differential methods up to the first half of the seventeenth century.- 4.3 The analysis of Newton and Leibniz.- 4.4 The groundwork for separating the concepts of derivative and integral.- 4.5 The separation of differentiation and integration.- 4.6 The Radon-Nikodým theorem.- 4.7 The relation between differentiation and integration in the works of Kolmogorov.- 4.8 The relation between differentiation and integration in the works of Carathéodory.- 4.9 A few more general remarks.- 5 Nondifferentiable continuous functions.- 5.1 Some introductory remarks.- 5.2 Ampère's theorem.- 5.3 Doubts and refutations.- 5.4 Classes of nondifferentiable functions.- 5.5 The relative "smallness" of the set of differentiable functions.- Index of names.