Classification of Lipschitz Derivatives in Terms of Semicontinuity and the Baire Limit Functions
Year: 2026
Type: article
Abstract: 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 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 .
Source: Annales Mathematicae Silesianae
Institution: University of Silesia in Katowice
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Funder: Uniwersytet Śląski w Katowicach
OpenAlex ID: W7131127649