Continuous nowhere differentiable multivariate functions

Let $U$ be an open set in ${\mathbb{R}}^d$. A continuous function $f\text{\hspace{0.17em}⁣}:U\to \mathbb{R}$ is strongly nowhere differentiable if and only if for each $\gamma \in (0,1]$ and for each unit speed $C^{1,\gamma }$ curve $c\text{\hspace{0.17em}⁣}:[a,b]\to U$, the composition $f\circ c\text{\hspace{0.17em}⁣}:[a,b]\to \mathbb{R}$ is nowhere differentiable on $(a,b)$. For bounded $U$, let $\bar{U}$ be the closure of $U$ and $C(\bar{U})$ be the Banach space of continuous real-valued functions on $\bar{U}$ with the sup norm. Theorem. In the sense of the Baire category theorem, almost every $f\in C(\bar{U})$ is strongly nowhere differentiable on $U$.