Classification of Lipschitz derivatives in terms of semicontinuity and the Baire limit functions

Oleksandr V. Maslyuchenko; Ziemowit M. Wójcicki

Published: 2026-02-24

2025

Classification of Lipschitz derivatives in terms of semicontinuity and the Baire limit functions

Oleksandr V. Maslyuchenko

, Ziemowit M. Wójcicki

Abstract

We introduce the generalized notion of semicontinuity of a function defined on a topological space and derive the useful classification of the socalled 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 Lip f respectively) in terms of Baire limit functions. In particular, we prove that lip f is F_σ-lower, Lip f is F_σ-upper, Lip 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 Lip f.

Keywords:

big Lipschitz derivative , little Lipschitz derivative , local Lipschitz derivative , semicontinuous function , Fσ-semicontinuous function

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Maslyuchenko, O. V., & Wójcicki, Z. M. (2026). Classification of Lipschitz derivatives in terms of semicontinuity and the Baire limit functions. Annales Mathematicae Silesianae. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/23715

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2025
Published: 2025-11-02

ISSN: 0860-2107

eISSN: 2391-4238

10.2478/amsil

Publisher

University of Silesia Press

Licence CC

This work is licensed under a Creative Commons Attribution 4.0 International License.

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