Classification of Lipschitz derivatives in terms of semicontinuity and the Baire limit functions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915513196609536 |
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| author | Maslyuchenko, Oleksandr V. Wójcicki, Ziemowit M. |
| author_facet | Maslyuchenko, Oleksandr V. Wójcicki, Ziemowit M. |
| contents | We introduce the generalized notion of semicontinuity of
a function defined on a topological space and derive the
useful classification of the so-called Lipschitz derivatives
of functions defined on a metric space.
Secondly, we investigate some connections of the Lipschitz
derivatives defined on normed spaces to the Fréchet
derivative and relations between little, big and local
Lipschitz derivatives (denoted by $\lip f$, $\Lip f$ and $\LLip f$
respectively) in terms of Baire limit functions.
In particular, we prove that $\lip f$ is $\mathcal{F}_σ$-lower,
$\Lip f$ is $\mathcal{F}_σ$-upper, $\LLip f$ is
upper semicontinuous. Moreover, for a function $f$ defined on an open
or convex subset of a normed space, the upper Baire limit function of functions
$\lip f$ and $\Lip f$ are equal to $\LLip f$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20849 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of Lipschitz derivatives in terms of semicontinuity and the Baire limit functions Maslyuchenko, Oleksandr V. Wójcicki, Ziemowit M. Functional Analysis 46T20, 26A16, 26A21 We introduce the generalized notion of semicontinuity of a function defined on a topological space and derive the useful classification of the so-called Lipschitz derivatives of functions defined on a metric space. Secondly, we investigate some connections of the Lipschitz derivatives defined on normed spaces to the Fréchet derivative and relations between little, big and local Lipschitz derivatives (denoted by $\lip f$, $\Lip f$ and $\LLip f$ respectively) in terms of Baire limit functions. In particular, we prove that $\lip f$ is $\mathcal{F}_σ$-lower, $\Lip f$ is $\mathcal{F}_σ$-upper, $\LLip f$ is upper semicontinuous. Moreover, for a function $f$ defined on an open or convex subset of a normed space, the upper Baire limit function of functions $\lip f$ and $\Lip f$ are equal to $\LLip f$. |
| title | Classification of Lipschitz derivatives in terms of semicontinuity and the Baire limit functions |
| topic | Functional Analysis 46T20, 26A16, 26A21 |
| url | https://arxiv.org/abs/2509.20849 |