Differentiability of continuous functions in terms of Haar-smallness

TL;DR

This paper investigates the Haar-smallness properties of the set of somewhere differentiable functions in the space of continuous functions, extending the analysis to measure-theoretic and multidimensional contexts, and poses open questions about Takagi's function.

Contribution

It proves that the set of somewhere differentiable functions is not Haar-countable and explores differentiability on measure-positive sets, almost everywhere, and in higher dimensions.

Findings

The set of somewhere differentiable functions is not Haar-countable.

Differentiability on sets of positive measure and almost everywhere is studied.

Multidimensional differentiability and open questions on Takagi's function are addressed.

Abstract

One of the classical results concerning differentiability of continuous functions states that the set $\mathcal{SD}$ of somewhere differentiable functions (i.e., functions which are differentiable at some point) is Haar-null in the space $C[0,1]$. By a recent result of Banakh et al., a set is Haar-null provided that there is a Borel hull $B\supseteq A$ and a continuous map $f\colon \{0,1\}^\mathbb{N}\to C[0,1]$ such that $f^{-1}[B+h]$ is Lebesgue's null for all $h\in C[0,1]$. We prove that $\mathcal{SD}$ is not Haar-countable (i.e., does not satisfy the above property with "Lebesgue's null" replaced by "countable", or, equivalently, for each copy $C$ of $\{0,1\}^\mathbb{N}$ there is an $h\in C[0,1]$ such that $\mathcal{SD}\cap (C+h)$ is uncountable. Moreover, we use the above notions in further studies of differentiability of continuous functions. Namely, we consider functions differentiable on a set of positive Lebesgue's measure and functions differentiable almost everywhere with respect to Lebesgue's measure. Furthermore, we study multidimensional case, i.e., differentiability of continuous functions defined on $[0,1]^k$. Finally, we pose an open question concerning Takagi's function.