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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.20849 |
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Table of 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 $\LLip f$.