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BORNOLOGIES AND LOCALLY LIPSCHITZ FUNCTIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 90 / Issue 2 / October 2014
- Published online by Cambridge University Press:
- 15 May 2014, pp. 257-263
- Print publication:
- October 2014
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- Article
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Let
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\langle X,d \rangle $ be a metric space. We characterise the family of subsets of 
$X$ on which each locally Lipschitz function defined on 
$X$ is bounded, as well as the family of subsets on which each member of two different subfamilies consisting of uniformly locally Lipschitz functions is bounded. It suffices in each case to consider real-valued functions.
