New constructions for local approximation of Lipschitz functions.II

In this paper, some properties of the set-valued mapping _Dαf(.)$D_{\alpha }f(.)$ connected with the new approximation method of a function f(.)$f(.)$ defined in the first part of the article are given. Continuity and Lipschitz properties of _Dαf(.)$D_{\alpha }f(.)$ are formulated. A continuous extension of the Clarke subdifferential of any function represented as a difference of two convex functions is given. For the convex case, the set-valued mapping _Dαf(.)$D_{\alpha }f(.)$ is similar to the ε$\epsilon$-subdifferential mapping.