2 7 Ju l 2 00 6 A Sard Theorem for Tame Set - Valued Mappings ∗ †

If F is a set-valued mapping from IR n into IR m with closed graph, then y ∈ IR m is a critical value of F if for some x with y ∈ F (x), F is not metrically regular at (x, y). We prove that the set of critical values of a set-valued mapping whose graph is a definable (tame) set in an o-minimal structure containing additions and multiplications is a set of dimension not greater than m − 1 (resp. a porous set). As a corollary of this result we get that the collection of asymptotically critical values of a semialgebraic set-valued mapping has dimension not greater than m − 1, thus extending to such mappings a corresponding result by Kurdyka-Orro-Simon for C 1 semialgebraic mappings. We also give an independent proof of the fact that a definable continuous real-valued function is constant on components of the set of its subdifferentiably critical points, thus extending to all definable functions a recent result of Bolte-Daniilidis-Lewis for globally subanalytic functions.