Exception sets of intrinsic and piecewise Lipschitz functions

TL;DR

This paper introduces a new class of functions on metric spaces, focusing on their exception sets where Lipschitz continuity fails, and shows that outside permeable exception sets, functions are Lipschitz continuous overall.

Contribution

It defines permeability for exception sets and proves that functions outside permeable sets are globally Lipschitz continuous with respect to the intrinsic metric.

Findings

Permeable sets include Lipschitz submanifolds in .

Functions intrinsically Lipschitz outside permeable sets are globally Lipschitz.

The concept generalizes piecewise Lipschitz functions on polyhedral structures.

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 $\mathbb{R}^d$, which include Lipschitz submanifolds.