Scenes from the History of Real Functions

Abstract

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 Arzela 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-Nikodym 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 Caratheodory.- 4.9 A few more general remarks.- 5 Nondifferentiable continuous functions.- 5.1 Some introductory remarks.- 5.2 Ampere'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.