Continuous nowhere differentiable multivariate functions

2025/10/15 by Maria Girardi, Ralph Howard, Girardi, Maria +1
Computer Science · #26A16 #26B05 #46E15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Neural Networks and Applications

paper · pdf · doi:10.48550/arxiv.2510.13061

openalex publication_date 2025/10/15 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

Abstract

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

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