Separable reduction of local regularity

Marian Fabian

We prove several regularity results for set-valued mappings. In some cases, we improve also the statements known already for single-valued mappings. Linear openness of the set-valued mapping in question is deduced from the properties of its suitable approximation. This approach goes back to the classical Lyusternik-Graves theorem saying that a continuously differentiable single-valued mapping between Banach spaces is linearly open around an interior point of its domain provided that its derivative at this point is surjective. In this paper, we consider approximations given by a graphical derivative, a contingent variation, a strict pseudo $H$-derivative, and a bunch of linear mappings.