Finite Chainability, Locally Lipschitzian and Uniformly Continuous Functions

Abstract

We present a notion of a finitely chainable subset of a metric space $X$. We show that $Y$ is a finitely chainable subset of $X$ if and only if $f(Y)$ is a bounded subset of $\mathbb{R}$ for any uniformly locally Lipschitzian or uniformly continuous real-valued function $f$ on $X$. As a corollary we reprove the Atsuji theorem in a slightly stronger form.