Functions that are Lipschitz in the small
Let \(X\) and \(Y\) be metric spaces. A function \(f:X\rightarrow Y\) is said to be Lipschitz in the small (\(f \in \mathrm{LS}(X,Y)\) for short) if there are \(r> 0\) and \(K<\infty \) so that \(d(f(u),f(v)) \leq Kd(u,v)\) for any \(u,v \in X\) with \(d(u,v) \leq r\). A subset \(A \subset X\) is called small determined if \(f|_A\) is Lipschitz for every function \(f \in \mathrm{LS}(X, \mathbb R)\). Answering a question by \textit{G. Beer} and \textit{M. I. Garrido} [J. Math. Anal. Appl. 428, No. 2, 804--816 (2015; Zbl 1341.46018)], the authors give an intrinsic description of small determined sets in terms of \(\varepsilon\)-step territories. Apart from this, the authors present characterizations of those \(X\) for which \(\mathrm{LS}(X, \mathbb R) \) is linearly order isomorphic to \(\mathrm{Lip}(Y)\) for some metric space \(Y\).
scientific article; zbMATH DE number 6679335