Takagi-van der Waerden functions in metric spaces and its Lipschitz derivatives

2024/06/09 by O. V. Maslyuchenko, Maslyuchenko, Oleksandr, Ziemowit M. Wójcicki +1
Mathematics · #26A16 #26A21 (Secondary) #46G05(Primary) 46T20 #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2406.05684

openalex publication_date 2024/06/09 · openalex created_date 2024/06/12 · openalex updated_date 2026/07/28

Abstract

We introduce the Takagi--van der Waerden function with parameters a>b>0 by setting f_a,b(x)=∑_n=1^∞ bⁿ d(x,S_n), where S_n is a maximal \frac1aⁿ-separated set in a metric space X. So, if X=\mathbb R and S_n=\frac1aⁿ\mathbb Z then f_2,1 is the Takagi function and f_10,1 is the van der Waerden function which are the famous examples of nowhere differentiable functions. Then we prove that the big Lipschitz derivative Lip f_a,b(x)=+∞ if a>b>2 and x is a non-isolated point of X. Moreover, if the shell porosity p^s(X,x)<λ<1 for some λ and each non-isolated point x∈ X then the little Lipschitz derivative lip f_a,b(x)=+∞ for large enough a>b and any non-isolated point x∈ X. In particular, this is true for any normed space. Finally, we prove that for any open set A in a metric (normed) space X without isolated points there exists a continuous function f such that Lip f(x)=+∞ (and lip f(x)=+∞) exactly on A.

Related

• Centered Takagi--van der Waerden functions and their Lipschitz derivatives
• Infinitesimally Lipschitz functions on metric spaces
• Approximation by Lipschitz functions
• Generalized Lipschitz numbers, fine differentiability, and quasiconformal mappings
• Classification of Lipschitz derivatives in terms of semicontinuity and the Baire limit functions