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arxiv: 2405.06314 · v4 · pith:XZFME4RYnew · submitted 2024-05-10 · 🧮 math.GT · math.CA· math.MG

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

Daniel Fatu{\l}a This is my paper

Pith reviewed 2026-05-24 00:44 UTC · model grok-4.3

classification 🧮 math.GT math.CAmath.MG
keywords Painlevé-Kuratowski convergenceClarke subdifferentialLipschitz functionsHurwitz theoremzero setssingularity theoryapproximation theory
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The pith

Lipschitz functions converge locally uniformly when their Clarke subdifferentials converge in the Painlevé-Kuratowski sense

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends a classical theorem on sequences of smooth functions to Lipschitz functions. If the Clarke subdifferentials converge locally uniformly in the Painlevé-Kuratowski sense, then the functions converge locally uniformly provided they converge at one point. It establishes the converse for the squared distance function. It also proves real versions of the Hurwitz theorem in which local uniform convergence of functions implies convergence of their zero sets. These address questions from singularity theory on when convergent descriptions produce convergent sets, along with partial results on limits of real algebraic sets.

Core claim

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 prove some general real counterparts of the Hurwitz theorem from complex analysis.

What carries the argument

Painlevé-Kuratowski convergence of closed sets, applied to graphs of Clarke subdifferentials of Lipschitz functions and to zero sets of functions

Load-bearing premise

The Clarke subdifferential is well-defined for the Lipschitz functions and the Painlevé-Kuratowski convergence applies directly to the graphs or epigraphs of the subdifferentials.

What would settle it

A sequence of Lipschitz functions whose Clarke subdifferentials converge in the Painlevé-Kuratowski sense but the functions themselves fail to converge locally uniformly even though they agree at one point.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript applies Painlevé-Kuratowski convergence of closed sets to three problems. It generalizes the classical local-uniform convergence result for C^1 functions with locally uniformly convergent derivatives to the Lipschitz case by replacing gradients with Clarke subdifferentials. It proves a converse statement for the squared-distance function. It also establishes real-analytic counterparts of Hurwitz's theorem on convergence of zero sets when the defining equations converge, with remarks on implications for singularity theory and limits of real algebraic sets.

Significance. If the stated theorems hold, the results supply standard, usable extensions of classical convergence theorems into the nonsmooth setting. They directly address questions about convergence of sets defined by converging equations and give partial answers to when limits of algebraic sets remain algebraic. The approach relies on well-established notions (Clarke subdifferential, PK convergence) without introducing new ad-hoc objects.

minor comments (3)
  1. [Abstract] The abstract states the three applications clearly but does not name the precise hypotheses or conclusions of each theorem; adding one sentence per result would improve immediate readability.
  2. [§2] Notation for the Painlevé-Kuratowski limit (e.g., PK-lim, limsup, liminf) should be fixed once at the beginning of §2 and used uniformly thereafter.
  3. [Theorem on zero-set convergence] In the statement of the real Hurwitz-type result, the precise relation between the subdifferential condition and the non-vanishing of the limit function on the boundary of the domain should be restated explicitly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, accurate summary of the manuscript's contributions, and recommendation to accept. No major comments require a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper extends classical uniform convergence results for C^1 functions to the Lipschitz case by replacing gradients with Clarke subdifferentials and applying Painlevé-Kuratowski set convergence to their graphs. It also gives real-analytic counterparts of Hurwitz's theorem on zero-set convergence under the same hypotheses. These steps rely on externally defined, standard notions (Clarke subdifferential, PK-convergence) whose properties are invoked directly rather than derived from the paper's own fitted quantities or prior self-citations. No equations reduce a claimed prediction to a fitted input by construction, no uniqueness theorem is imported from the author's own work, and no ansatz is smuggled via citation. The central claims therefore remain independent of the inputs they are applied to.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background from nonsmooth analysis and set convergence without introducing fitted parameters or new entities.

axioms (2)
  • domain assumption Clarke subdifferential is well-defined and closed for locally Lipschitz functions
    Invoked for the first generalization; standard in the field but required for the set-convergence statements to apply.
  • domain assumption Painlevé-Kuratowski convergence preserves relevant properties of epigraphs or graphs under local uniform limits
    Central to both the function convergence and zero-set results.

pith-pipeline@v0.9.0 · 5717 in / 1286 out tokens · 30719 ms · 2026-05-24T00:44:59.255716+00:00 · methodology

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Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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