Differentiable Lipschitz 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.