Differentiability of continuous functions in terms of Haar-smallness

Abstract

One of the classical results concerning differentiability of continuous functions states that the set $\mathcal{S}\mathcal{D}$ 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 $f:\{0,1{\}}^{\mathbb{N}}\to C[0,1]$ such that $f^{-1}[B+h]$ is Lebesgue's null for all $h\in C[0,1]$. We prove that $\mathcal{S}\mathcal{D}$ 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 $\mathcal{S}\mathcal{D}\cap (C+h)$ 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.