On real-valued functions of Lipschitz type
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908480529498112 |
|---|---|
| author | Gutev, Valentin |
| author_facet | Gutev, Valentin |
| contents | The classical McShane-Whitney extension theorem for Lipschitz functions is refined by showing that for a closed subset of the domain, it remains valid for any interval of the real line. This result is also extended to the setting of locally (pointwise) Lipschitz functions. In contrast to Lipschitz and pointwise Lipschitz extensions, the construction of locally Lipschitz extensions is based on Lipschitz partitions of unity of countable open covers of the domain. Such partitions of unity are a special case of a more general result obtained by Zdeněk Frol\'ık. To avoid the use of Stone's theorem (paracompactness of metrizable spaces), it is given a simple direct proof of this special case of Frol\'ık's result. As an application, it is shown that the locally Lipschitz functions are precisely the locally finite sums of sequences of Lipschitz functions. Also, it is obtained a natural locally Lipschitz version of one of Michael's selection theorems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17825 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On real-valued functions of Lipschitz type Gutev, Valentin General Topology 26A16, 54C20, 54C60, 54C65, 54E35, 54E40 The classical McShane-Whitney extension theorem for Lipschitz functions is refined by showing that for a closed subset of the domain, it remains valid for any interval of the real line. This result is also extended to the setting of locally (pointwise) Lipschitz functions. In contrast to Lipschitz and pointwise Lipschitz extensions, the construction of locally Lipschitz extensions is based on Lipschitz partitions of unity of countable open covers of the domain. Such partitions of unity are a special case of a more general result obtained by Zdeněk Frol\'ık. To avoid the use of Stone's theorem (paracompactness of metrizable spaces), it is given a simple direct proof of this special case of Frol\'ık's result. As an application, it is shown that the locally Lipschitz functions are precisely the locally finite sums of sequences of Lipschitz functions. Also, it is obtained a natural locally Lipschitz version of one of Michael's selection theorems. |
| title | On real-valued functions of Lipschitz type |
| topic | General Topology 26A16, 54C20, 54C60, 54C65, 54E35, 54E40 |
| url | https://arxiv.org/abs/2411.17825 |