The Concept of Function at the Beginning of the 20th Century

A Historiographical Approach

Abstract

The evolution of the concept of function at the beginning of the 20^th century in France after the definitions by Dirichlet and Riemann and the introduction of several pathological functions is studied. Some young mathematicians of those years (Baire, known for his classification of discontinuous functions, Borel and Lebesgue famous for their new theories on measure and integration) made several attempts to propose a large class of functions as “accessible” objects. Their discussions, their purposes and polemics are reported often by their own words supported by a large bibliography. The contribution of some Italian mathematicians, as Vitali, is also underlined.  Some of such discussions are linked to the growth of measure and function theories, others will find mathematical answers in the modern theory of computability for real functions.