Everywhere differentiable Lipschitz continuous functions can be highly pathological

Abstract

The talk sheds light on the difference between differentiable vs strict differentiable Lipschitz functions from the view point of nonsmooth analysis: while in the latter class, the limiting subdifferential is always reduced to a singleton, the limiting subdifferential of a differentiable Lipschitz function may assume almost every possible value. A concrete example-scheme will be presented revealing that the class of such pathological locally Lipschitz differentiable functions is dense (for the topology of the uniform convergence) and spaceable (for the Lip- norm topology). As a by-product, we obtain the following surprising result: all convex bodies of a finite dimensional space are contained in the range of the subdifferential of some real-valued differentiable locally Lipschitz function.