Applications of the Painlevé-Kuratowski convergence: Lipschitz functions with converging Clarke subdifferentials and convergence of sets defined by converging equations
Daniel Fatuła
TL;DR
The paper investigates Painlevé-Kuratowski convergence (PK) of closed sets and its applications to analysis, notably via Clarke subdifferentials for Lipschitz functions. It generalizes the classical derivative-convergence theorem to Lipschitz functions by employing graphical convergence of Clarke subdifferentials and establishing uniform convergence under suitable conditions. It further shows that PK convergence of closed sets implies convergence of the Clarke subdifferentials of squared distance functions, while highlighting that such convergence may fail for distance functions themselves, and it provides real Hurwitz-type results for sets defined by converging equations alongside relevant counterexamples. The Appendix collects supplementary results, including a simple proof of the Zarankiewicz theorem, a convex-set convergence result from boundaries, and fibre-convergence remarks, underscoring broader implications for approximation theory and singularity analysis.
Abstract
In this note we investigate some applications of the Painlevé-Kuratowski convergence of closed sets in analysis that are motivated also by questions from singularity theory. First, we generalise to Lipschitz functions the classical theorem stating that given a sequence of smooth functions with locally uniformly convergent derivatives, we obtain the local uniform convergence of the functions themselves (provided they were convergent at one point). We use Clarke subdifferentials instead of derivative and Painlevé-Kuratowski convergence of their graphs instead of local uniform convergence. Next we focus on reverse theorem. We show that Painlevé-Kuratowski convergence of closed nonempty sets implies convergence of distance functions and Clark subdifferentials of squared distance functions, but does not imply convergence of Clark subdifferentials of distance functions. Finally we turn to the study of the behaviour of the fibres of a given function. We prove some general real counterparts of the Hurwitz theorem from complex analysis stating that the local uniform convergence of holomorphic functions implies the convergence of their sets of zeros. From the point of view of singularity theory our two theorems concern the convergence of the sets when their descriptions are convergent. They are also of interest in approximation theory.