Nowhere Differentiable Functions

Abstract

Many examples of nowhere differentiable continuous functions are known, the first having been constructed by Weierstrass. One of the simplest existence proofs is due to Banach (1931) [18 p 327]. It is based on the category method. Banach showed that, in the sense of category, almost all continuous functions are nowhere differentiable, in fact it is exceptional for a continuous function to have a finite one-sided derivative, or even to have bounded difference quotients on either side anywhe in an interval.