Exception sets of intrinsic and piecewise Lipschitz functions

2021/05/25 by Gunther Leobacher, Leobacher, Gunther, Alexander Steinicke +1
Mathematics · #54E40 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Equations Stability Results #General Topology (math.GN)

paper · pdf · doi:10.48550/arxiv.2105.12004

openalex publication_date 2021/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a class of functions defined on metric spaces which generalizes the concept of piecewise Lipschitz continuous functions on an interval or on polyhedral structures. The study of such functions requires the investigation of their exception sets where the Lipschitz property fails. The newly introduced notion of permeability describes sets which are natural exceptions for Lipschitz continuity in a well-defined sense. One of the main results states that continuous functions which are intrinsically Lipschitz continuous outside a permeable set are Lipschitz continuous on the whole domain with respect to the intrinsic metric. We provide examples of permeable sets in ℝ^d, which include Lipschitz submanifolds.

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