  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 is almost everywhere continuous (in both
topological and measure-theoretic sense). 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.
