A Sard theorem for tame set-valued mappings

If F is a set-valued mapping from Rn into Rm with closed graph, then y ∈ Rm 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 (respectively a σ-porous set). As a corollary of this result we get that the collection of asymptotically critical values of a set-valued mapping with a semialgebraic graph has dimension not greater than m − 1. 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. © 2007 Elsevier Inc. All rights reserved.