On some Lipschitz-type functions
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.
scientific article; zbMATH DE number 7595143
scientific article