Chapter 6 - Lipschitz Functions and the Generalized Gradient

Abstract

The subject of Chapter 6 is the generalized gradient of a Lipschitz function. We study Lipschitz functions, characterizing their regularity, and introducing a new class of Lipschitz functions: the strictly differentiable ones. Among other properties, we characterize the generalized gradient as the convex hull of the set of limits of gradients, assuming without proof Rademacher’s Theorem. This result gives us an idea of how to extend these nonsmooth concepts to vector functions. In this sense we define the Generalized Jacobian, which allows us to set the powerful and general Lipschitz Inverse Function Theorem. We finish the chapter by introducing the graphical derivative and coderivative. These concepts represent another way of defining nonsmooth extensions of the differential in the vectorial case.