  In this work we approach the problem of approximating uniformly continuous
semialgebraic maps $f:S\to T$ from a compact semialgebraic set $S$ to an
arbitrary semialgebraic set $T$ by semialgebraic maps $g:S\to T$ that are
differentiable of class~${\mathcal C}^\nu$ for a fixed integer $\nu\geq1$. As
the reader can expect, the difficulty arises mainly when one tries to keep the
same target space after approximation. For $\nu=1$ we give a complete
affirmative solution to the problem: such a uniform approximation is always
possible. For $\nu \geq 2$ we obtain density results in the two following
relevant situations: either $T$ is compact and locally ${\mathcal C}^\nu$
semialgebraically equivalent to a polyhedron, for instance when $T$ is a
compact polyhedron; or $T$ is an open semialgebraic subset of a Nash set, for
instance when $T$ is a Nash set. Our density results are based on a recent
${\mathcal C}^1$-triangulation theorem for semialgebraic sets due to Ohmoto and
Shiota, and on new approximation techniques we develop in the present paper.
Our results are sharp in a sense we specify by explicit examples.
