On some Lipschitz-type functions
type: Item
instance of: scholarly article
title: On some Lipschitz-type functions
author: Manisha Aggarwal
author: Stefan Cobzaş
published in: Journal of Mathematical Analysis and Applications
publication date: 2022-09-30T00:00:00Z
review text: This paper deals with a class of locally Lipschitz functions defined between metric spaces; namely, the so-called \textit{locally equi-Lipschitz} functions (also known under the name of \textit{locally \(L\)-Lipschitz} functions). A function \(f: (X,d) \to (Y, d')\) is said to be equi-locally Lipschitz if there exists some \(L>0\) such that for every \(x\in X\) the function \(f\) is \(L\)-Lipschitz on some open ball around \(x\). The main goal of the paper is to see when the class of these functions coincides with the whole family of the locally Lipschitz functions (Theorem 3.1) or contains the class of the uniformly locally Lipschitz functions (Theorem 3.3). It is worth noting that in these theorems there appears, respectively, the condition ``the set \(X'\) of accumulation points of \(X\) is compact'', respectively, ``the set \(X'\) is precompact''. Recall that the condition ``\(X'\) is compact'' is known to be equivalent to saying that \(X\) admits a topologically equivalent metric making it a UC space, that is, a space where every real-valued continuous function is uniformly continuous.
Mathematics Subject Classification ID: 54E40
Mathematics Subject Classification ID: 54C05
Mathematics Subject Classification ID: 54E50
zbMATH DE Number: 7595143
zbMATH Keywords: locally Lipschitz function
zbMATH Keywords: uniformly locally Lipschitz function
zbMATH Keywords: locally equi-Lipschitz function
zbMATH Keywords: UC-spaces
reviewed by: M. I. Garrido
MaRDI profile type: Publication
full work available at URL: j.jmaa.2022.126631
OpenAlex ID: W4293385557
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DOI: 10.1016/J.JMAA.2022.126631
description: scientific article; zbMATH DE number 7595143