Removability of exceptional sets for differentiable and Lipschitz functions

TL;DR

This paper investigates the conditions under which certain sets are removable for differentiability and Lipschitz functions, establishing that a set's removability is characterized by the absence of perfect subsets.

Contribution

It provides a complete characterization of removable sets for differentiability and Lipschitz functions in terms of perfect subsets, advancing the understanding of set removability.

Findings

A set is removable if and only if it contains no perfect subsets.

Characterization applies to both differentiability and pointwise Lipschitz conditions.

Sets with no perfect subsets are precisely the removable sets in these contexts.

Abstract

We discuss removability problems concerning differentiability and pointwise Lipschitz conditions for functions of a real variable. We prove that, in each of the settings under consideration, a set is removable if and only if it has no perfect subsets.