McShane's extension theorem revisited (Q2189627)

On a metric space \((X, d)\) where \(X\) has at least two points, a function \(f : (X, d) \to \mathbb R\) is called \(\lambda\)-Lipschitz if for some \(\lambda > 0\), and all \(x_1, x_2 \in X\), we have \(|f(x_1) - f(x_2)| \leq \lambda d(x_1, x_2)\). The author shows that a \(\lambda\)-Lipschitz function on a non-empty subset of \(X\) has a \(\lambda\)-Lipschitz preserving extension to \(X\), thus arriving at McShane's extension theorem from 1934 [\textit{E. J. McShane}, Bull. Am. Math. Soc. 40, 390 (1934; JFM 60.0224.09)] for a bounded real-valued uniformly continuous function defined on \(A \ne\varnothing\) in \(X\) since a bounded uniformly continuous function on \(A\) is shown to be Lipschitz with respect to a metric uniformly equivalent to \(d\). The author then uses the results to give an elementary proof within the metric space framework to characterize metric spaces in which the real-valued uniformly continuous functions form a ring, independent of the tools used in such characterizations by \textit{J. Cabello-Sánchez} [``\(U(X)\) as a ring for metric spaces \(X\)'', Filomat 31, No. 7, 1981--1984 (2017)] and by \textit{A. Bouziad} and \textit{E. Sukhacheva} [Topology Appl. 254, 132--144 (2019; Zbl 1410.54020)]. Analysts are encouraged to watch out for Lipschitzian behavior underlying important results in analysis.