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

2025/09/25 by O. V. Maslyuchenko, Maslyuchenko, Oleksandr V., Ziemowit M. Wójcicki +1
Economics, Econometrics and Finance · #26A16 #26A21 #46T20 #FOS: Mathematics #Functional Analysis (math.FA) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2509.20849

openalex publication_date 2025/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \LLip 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, \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.

Citations

• On sets where lip(f) is infinite for monotone continuous functions
• Characterization of lip sets
• Big and little Lipschitz one sets
• On sets where lip f is finite
• On Fréchet differentiability of Lipschitz maps between Banach spaces

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