Continuous nowhere differentiable multivariate functions
Maria Girardi, Ralph Howard
TL;DR
The paper extends the classical one-dimensional phenomenon of nowhere differentiable functions to higher dimensions by introducing strong nowhere differentiability defined via all unit-speed $C^{1,\gamma}$ test curves. It constructs univariate building blocks that are precisely $C^{0,\alpha}$ and assembles them into multivariate functions so that their restrictions along every admissible curve are nowhere differentiable; the main result shows that, in the Baire category sense, almost every $f\in C(\overline U)$ is strongly nowhere differentiable on $U$. The proofs combine a concrete auxiliary-function construction, a curve-compactness/Arzelà–Ascoli framework, and a Baire-category argument to yield a dense $G_\delta$ set of typical functions with the desired property; the work also discusses open questions and provides Katzourakis' concrete example of precisely $C^{0,\alpha}$ functions in the appendix.
Abstract
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 $γ\in(0,1]$ and for each unit speed $C^{1,γ}$ 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 $U$.