Applications of the Painlev\'e-Kuratowski convergence: Lipschitz functions with converging Clarke subdifferentials and convergence of sets defined by converging equations

TL;DR

This paper explores applications of Painlevé-Kuratowski convergence in analysis, focusing on Lipschitz functions, distance functions, and the convergence of zero sets, with implications for singularity and approximation theories.

Contribution

It generalizes classical convergence theorems to Lipschitz functions, proves reverse convergence results for distance functions, and studies convergence of zero sets in real analysis.

Findings

Lipschitz functions with converging derivatives also converge uniformly.

The squared distance function's convergence is characterized in reverse.

Convergence of zero sets is established under local uniform convergence.

Abstract

In this note we investigate three kinds of applications of the Painlev\'e-Kuratowski convergence of closed sets in analysis that are motivated also by questions from singularity theory. Firstly, 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). Secondly, we prove the reverse theorem for the squared distance function. Next, 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 zeroes). 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 and they give some partial results to the problem of when is the limit of a convergent sequence of real algebraic sets algebraic.