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Zeitschrift für Analysis und ihre Anwendungen


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Volume 17, Issue 4, 1998, pp. 795–803
DOI: 10.4171/ZAA/850

Published online: 1998-12-31

Finite Chainability, Locally Lipschitzian and Uniformly Continuous Functions

G. Marino[1], Grzegorz Lewicki[2] and P. Pietramala[3]

(1) Università della Calabria, Arcavacata Di Rende, Italy
(2) Jagiellonian University, Krakow, Poland
(3) Università della Calabria, Arcavacata di Rende, Italy

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.

Keywords: Metric spaces, finite chainable subsets, uniformly continuous functions, uniformly locally Lipschitzian functions

Marino G., Lewicki Grzegorz, Pietramala P.: Finite Chainability, Locally Lipschitzian and Uniformly Continuous Functions. Z. Anal. Anwend. 17 (1998), 795-803. doi: 10.4171/ZAA/850