Continuity and differentiability of set-valued maps revisited in the light of tame geometry

Continuity of set-valued maps is hereby revisited: after recalling some basic concepts of variational analysis and a short description of the state-of-the-art, we obtain as by-product two Sard-type results concerning local minima of scalar- and vector-valued functions. Our main result though, is inscribed in the framework of tame geometry, stating that a closed-valued semialgebraic set-valued map that does not necessarily have closed graph is almost everywhere continuous (in both topological and measure-theoretic sense), strictly continuous and strictly differentiable (as set-valued map). The result, depending on stratification techniques, holds true in a more general setting of o-minimal (or tame) set-valued maps. Some applications are briefly discussed at the end. © 2011 London Mathematical Society.