Continuous nowhere differentiable multivariate functions

Abstract

Let $U$ be an open set in ${\mathbb{R}}^d$. A continuous function $f:U\to \mathbb{R}$ is strongly nowhere differentiable if and only if for each ${\hat{I}}^3\in (0,1]$ and for each unit speed $C^{1,{\hat{I}}^3}$ curve $c:[a,b]\to U$, the composition $f\circ c:[a,b]\to \mathbb{R}$ is nowhere differentiable on $(a,b)$. For bounded $U$, let $\overline{U}$ be the closure of $U$ and $C(\overline{U})$ be the Banach space of continuous real-valued functions on $\overline{U}$ with the sup norm. Theorem. In the sense of the Baire category theorem, almost every $f\in C(\overline{U})$ is strongly nowhere differentiable on $U$.