Continuous nowhere differentiable multivariate functions

Let $U$ be an open set in $\mathbb{R}^d$. A continuous function $f\colon U \to \mathbb{R}$ is strongly nowhere differentiable if and only if for each $\gamma\in(0,1]$ and for each unit speed $C^{1,\gamma}$ curve $c\colon [a,b] \to U$, the composition $f\circ c \colon [a,b] \to \mathbb{R}$ 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 $U$.