Centered Takagi--van der Waerden functions and their Lipschitz derivatives

TL;DR

The work addresses the problem of realizing infinite Lipschitz derivatives on a prescribed closed subset $A$ of a metric space by constructing a continuous centered Takagi–van der Waerden function. Using a carefully designed sequence of maximal $1/a^{n}$-separated sets and associated distance sums, the authors prove that $L^{\infty}(f)=\mathbb{L}^{\infty}(f)=A$ for general metric spaces (with $A\subset X^d$ closed and without isolated points), and extend the result to Hermetic spaces to obtain $\ell^{\infty}(f)=L^{\infty}(f)=\mathbb{L}^{\infty}(f)=A$, which also yields $\mathrm{lip} f=\infty$ on $A$. This provides a unified method to encode exact blow-up behavior of big, local, and little Lipschitz derivatives along a chosen set, linking to differentiable structures on metric spaces. The results advance the understanding of how differentiability properties can be prescribed and controlled via explicit constructive functions on general metric spaces.

Abstract

Using a modification of a generalized Takagi-van der Waerden function on a metric space we prove that for any closed subset of a metric space without isolated points there exists a continuous function such that its big and local Lipschitz derivatives are equal to infinity exactly on this set. Moreover, if given space is hermetic (for example, if it is normed) then the little Lipschitz derivative has the same property.

Theorems & Definitions (16)

• Proposition 2.1: MaWo
• Remark 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof