The article Level sets of nonsmooth functions, Part 1: Lipschitz and piecewise-differentiable Rank Theorems (click here for free access until May 1, 2024) presents a piecewise-differentiable (PC)^r Rank Theorem and extends a previously stated Lipschitz Rank Theorem, with the goal of characterizing the level sets of nonsmooth functions f.

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) locally as the graph of a nonsmooth function, within a homeomorphic transformation of the same class as f. In this work, the authors define (PC)^r 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, sufficient conditions that are more easily verifiable for practical applications are developed and related to existing full-rank conditions from the literature.