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). In this paper we characterize, in several cases, this tangential regularity as the directional regularity of the scalar function Delta(M) defined by Delta(M)(x, y) : = d(y, M(x)). The results allow us to express, in a useful formula, the subdifferential of Delta(M) in terms of the normal cone to the graph of M.