A pasting lemma for Lipschitz functions

TL;DR

This paper establishes a precise condition under which functions that are Lipschitz on two separate compact sets are also Lipschitz on their union, advancing understanding of Lipschitz continuity in piecewise contexts.

Contribution

It provides a necessary and sufficient condition for the Lipschitz property to hold on the union of two compact sets, extending previous partial results.

Findings

Characterization of when separate Lipschitz functions are Lipschitz on unions

Necessary and sufficient condition for Lipschitz extension

Enhanced understanding of Lipschitz continuity in compact sets

Abstract

We give a necessary and sufficient condition ensuring that any function which is separately Lipschitz on two fixed compact sets is Lipschitz on their union.