A Sard Theorem for Set-Valued Mappings∗†

If F is a set-valued mapping from IRn into IRm with closed graph, then y ∈ IRm 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 setvalued mapping has dimension not greater than m−1, thus extending to such mappings a corresponding result by Kurdyka-Orro-Simon for C1 semialgebraic mappings. Finally, as a by-product of the proof of the theorem, we get 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.