Bornologies and locally Lipschitz functions. (English) Zbl 1318.26004
The authors offer a characterization of the family of subsets of a metric space \(X\) on which every locally Lipschitz function \(f:X\to \mathbb{R}\) is bounded. Also, they consider the family of subsets on which each member of two distinct sub-families consisting of uniformly locally Lipschitz functions is bounded.
The authors note also, that there are other approaches also to prove some of their results, but they have offered here simple direct constructions. Results of the type proved in this paper can be used to show that an unbounded function with a particular continuity property defined on some subsets of \(X\), cannot be extended to a globally defined function with the same continuity property.
The authors note also, that there are other approaches also to prove some of their results, but they have offered here simple direct constructions. Results of the type proved in this paper can be used to show that an unbounded function with a particular continuity property defined on some subsets of \(X\), cannot be extended to a globally defined function with the same continuity property.
Reviewer: József Sándor (Cluj-Napoca)
MSC:
| 26A16 | Lipschitz (Hölder) classes |
| 46A17 | Bornologies and related structures; Mackey convergence, etc. |
| 54E35 | Metric spaces, metrizability |
Keywords:
Lipschitz functions; function preserving Cauchy sequences; bornology; totally bounded set; Bourbaki bounded setReferences:
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