Structural Analysis of Nonsmooth Mappings, Inverse Functions, and Metric Projections

It is known that a locally Lipschitz continuous function ƒ: Rn → Rn which is strongly B-differentiable at a point x0 ∈ Rn is a Lipschitz homeomorphism in a neighborhood of x0 if and only if its B-derivative at x0 is a Lipschitz homeomorphism. We show that strongly B-differentiability is a rather restrictive requirement for piecewise differentiable functions. However, it turns out that generically a piecewise differentiable function can be locally transformed into a strongly B-differentiable function by means of a piecewise differentiable homeomorphism. The corresponding criteria yield structural inverse function theorems for piecewise differentiable functions. The results are applied to the metric projection onto a convex set.