  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.
