Pathological differentiable Lipschitz functions

Abstract

We focus on the difference between differentiable versus strict differentiable locally Lipschitz functions from the view point of nonsmooth analysis: while in the latter class, all limiting Jacobians are singletons, we show that there exists a differentiable locally Lipschitz function for which the image of the limiting Jacobian map contains all nonempty compact connected subsets of matrices. In the particular case of real valued functions, we obtain differentiable functions with surjective limiting and Clarke subdifferentias. In this case, our concrete example-scheme will also reveal 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).