Nondifferentiable continuous functions

Abstract

The problem of nondifferentiable continuous functions has been studied by a very large number of mathematicians and historians of mathematics. Of the papers in which the history of this problem has been traced in more or less detail one can, for example, mention the following: Pascal [1, pp. 91–128], Brunschvicg [1, pp. 337–340], Pasch [3, pp. 122–129], and Hawkins [1, pp. 42–54]. These historical excursions are far from complete, understandably so due to the difficulty of the problem. We do not propose to give anything like a complete description of its history. Our goal will be to trace the general features of a single idea: the change in the view of mathematicians on the relationship between a function and its derivative from certainty as to the existence of a derivative for every function to the establishment of the fact that the set of functions having an ordinary derivative at even one point is in a sense negligibly small in comparison to the whole set of continuous functions, that the main bulk of continuous functions, so to speak, consists of functions not having a derivative at any point. A few more detailed remarks will be made, but only in passing.