C 1 , ω ( · ) - regularity and Lipschitz - like properties of subdifferential

It is known that the subdifferential of a lower semicontinuous convex function f over a Banach space X determines this function up to an additive constant in the sense that another function of the same type g whose subdifferential coincides with that of f at every point is equal to f plus a constant, i.e., g = f + c for some real constant c. Recently, Thibault and Zagrodny introduced a large class of directionally essentially smooth functions for which the subdifferential determination still holds. More generally, for extended real-valued functions in that class, they provided a detailed analysis of the enlarged inclusion ∂g(x) ⊂ ∂f(x) + γB for all x ∈ X, where γ is a nonnegative real number and B is the closed unit ball of the topological dual space. The aim of the present paper is to show how results concerning such an enlarged inclusion of subdifferentials allow us to establish the C1 or C1,ω(·) property of an essentially directionally smooth function f whose subdifferential set-valued mapping admits a continuous or Hölder continuous selection. The C1,ω(·)-property is also obtained under a natural Hölder-like behaviour of the set-valued mapping ∂f . Similar results are also proved for another class of functions that we call ∂1,φ(·)-subregular functions. When X is a Hilbert space, the latter class contains prox-regular functions and hence our results extend old and recent results in the literature.