Approximations Involving Lipschitz Functions

Abstract

In this chapter we study the problem of the uniform approximation of some classes of functions (e.g. uniformly continuous) by Lipschitz functions, based on the existence of Lipschitz partitions of unity or on some extension results for Lipschitz functions. A result due to Baire on the approximation of semi-continuous functions by continuous ones, based on McShane’s extension method, is also included. The chapter ends with a study of homotopy of Lipschitz functions and a brief presentation of Lipschitz manifolds.