On real-valued functions of Lipschitz type

TL;DR

This paper refines the McShane-Whitney extension theorem for Lipschitz functions, extends it to locally Lipschitz functions, and provides new constructions and applications, including a version of Michael's selection theorem.

Contribution

It introduces a new approach to locally Lipschitz extensions using Lipschitz partitions of unity and simplifies existing proofs, expanding the theoretical framework.

Findings

Refined extension theorem valid for any interval of the real line.

Locally Lipschitz functions characterized as sums of Lipschitz functions.

A locally Lipschitz version of Michael's selection theorem obtained.

Abstract

The classical McShane-Whitney extension theorem for Lipschitz functions is refined by showing that for a closed subset of the domain, it remains valid for any interval of the real line. This result is also extended to the setting of locally (pointwise) Lipschitz functions. In contrast to Lipschitz and pointwise Lipschitz extensions, the construction of locally Lipschitz extensions is based on Lipschitz partitions of unity of countable open covers of the domain. Such partitions of unity are a special case of a more general result obtained by Zden\v{e}k Frol\'{\i}k. To avoid the use of Stone's theorem (paracompactness of metrizable spaces), it is given a simple direct proof of this special case of Frol\'{\i}k's result. As an application, it is shown that the locally Lipschitz functions are precisely the locally finite sums of sequences of Lipschitz functions. Also, it is obtained a natural locally Lipschitz version of one of Michael's selection theorems.