Subdifferentiable functions satisfy Lusin properties of class C^{1} or C^{2}

Abstract

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.