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oai:docta.ucm.es:20.500.14352/1179232025-03-18T14:40:21Zcom_20.500.14352_14col_20.500.14352_15

Subdifferentiable functions satisfy Lusin properties of class C^{1} or C^{2} Azagra Rueda, Daniel Ferrera Cuesta, Juan García Bravo, Miguel Gómez Gil, Javier Lusin property of order 2 Proximal subdifferential Fréchet subdifferential Funciones (Matemáticas) Análisis matemático 12 Matemáticas Let f : Rn →Rbeafunction.Assumethat for a measurable set Ω and almost every x ∈ Ω there exists a vector ξx ∈ Rn such that Lim inf h→0 f (x +h)− f(x)−⟨ξx,h⟩ / |h|2 >−∞. Then we show that f satisfies a Lusin-type property of order 2 in Ω, that is to say, for every ε > 0 there exists a function g ∈ C2(Rn) such that Ln({x ∈ Ω : f(x) ̸= g(x)}) ≤ ε. In particular every function which has a nonempty proximal subdifferential almost everywhere also has the Lusin property of class C2. We also obtain a similar result (replacing C2 with C1) for the Fréchet subdifferential. Finally we provide some examples showing that these kinds of results are no longer true for Taylor subexpansions of higher order. Ministerio de Economía, Industria y Competitividad Depto. de Análisis Matemático y Matemática Aplicada Fac. de Ciencias Matemáticas Instituto de Ciencias Matemáticas (ICMAT) Instituto de Matemática Interdisciplinar (IMI) TRUE pub 2025-02-07T15:40:10Z 2025-02-07T15:40:10Z 2018 journal article https://hdl.handle.net/20.500.14352/117923 XXXX-XXXX 10.1016/j.jat.2018.03.001 eng info:eu-repo/grantAgreement/MINECO//MTM2015-65825-P/ES/ANALISIS FUNCIONAL NO LINEAL Y GEOMETRICO/ restricted access application/pdf Elsevier