Abstract
A set-valued mapping M from a topological vector space E into a normed vector space F is tangentially regular at a point ((x) over bar, (y) over bar) in its graph gph M if the Clarke tangent cone to gph M at ((x) over bar, (y) over bar) is equal to the Bouligand contingent cone to gph M at ((x) over bar,(y) over bar). Recently, after the work of Burke, Ferris and Qian on directional regularity of distance functions associated with subsets, we characterized in several cases the tangential regularity of a set-valued mapping M as the directional regularity of the scalar function Delta (M) defined by Delta (M)(x, y) := d(y, M(x)). In this paper we characterize this tangential regularity, for a new class of set-valued mappings, as the directional regularity of Delta (M) in a weak sense. We also show that the images of a set-valued mapping are tangentially regular whenever its graph is tangentially regular.