Level sets of nonsmooth functions, Part 1: Lipschitz and piecewise-differentiable Rank Theorems

Abstract

We present a piecewise-differentiable (PCr) Rank Theorem and extend a previously stated Lipschitz Rank Theorem, with the goal of characterizing the level sets of nonsmooth functions f:Rn→Rm. When the appropriate conditions are satisfied by the generalized derivatives of f, the Rank Theorems allow us to express a given level set f−1(c)⊂Rn locally as the graph of a nonsmooth function, within a homeomorphic transformation of the same class as f. We define PCr and Lipschitz submersions, immersions and maps of constant rank in terms of the most general conditions under which the corresponding Rank Theorems are applicable. Moreover, we develop sufficient conditions that are more easily verifiable for practical applications and relate them to existing full-rank conditions from the literature.