Continuous nowhere differentiable multivariate functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918160867786752 |
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| author | Girardi, Maria Howard, Ralph |
| author_facet | Girardi, Maria Howard, Ralph |
| contents | 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 $U$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_13061 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Continuous nowhere differentiable multivariate functions Girardi, Maria Howard, Ralph Classical Analysis and ODEs Functional Analysis 26B05, 46E15, 26A16 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 $U$. |
| title | Continuous nowhere differentiable multivariate functions |
| topic | Classical Analysis and ODEs Functional Analysis 26B05, 46E15, 26A16 |
| url | https://arxiv.org/abs/2510.13061 |