Typical Properties of Lipschitz Functions

Abstract

In terms of Baire category, a typical real-valued Lipschitz function on a finite dimensional space has a local minimum at every point of a dense subset of the domain, and a Dini subdifferential that is either singleton or empty at all points. Moreover, its Dini subdifferential is empty outside a set of first category. Hence a typical Lipschitz function has no points of subdifferential regularity.