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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.

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