Relations with Other Classes of Functions

Abstract

We have seen (see Sect. 1.3.2) that any Lipschitz function is uniformly continuous. In this chapter we shall present Lipschitz properties of convex functions and convex operators, and equi-Lipschitz properties for families of convex vector-functions. In the vector case, meaning convex functions defined on a locally convex space with values in a locally convex space ordered by a cone, one emphasizes the key role played by the normality of the cone. Other considered topics involve the existence of an equivalent metric making a given continuous function Lipschitz, and metric spaces where every continuous function is Lipschitz. An old result of G. Fichtenholz (from 1922) on the relation between absolutely continuous and Lipschitz functions is included. The chapter ends with a discussion on the differentiability properties of Lipschitz functions—Rademacher-type theorems—in finite and in infinite dimension.