A Sard theorem for Tame Set-Valued mappings

Abstract

If $F$ is a set-valued mapping from $\R^n$ into $\R^m$ with closed graph, then $y\in \R^m$ is a critical value of $F$ if for some $x$ with $y\in F(x)$, $F$ 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.