  Let $Y\subset{\mathbb R}^n$ be a triangulable set and let $r$ be either a
positive integer or $r=\infty$. We say that $Y$ is a
$\mathscr{C}^r$-approximation target space, or a
$\mathscr{C}^r\text{-}\mathtt{ats}$ for short, if it has the following
universal approximation property: For each $m\in{\mathbb N}$ and each locally
compact subset $X$ of~${\mathbb R}^m$, any continuous map $f:X\to Y$ can be
approximated by $\mathscr{C}^r$ maps $g:X\to Y$ with respect to the strong
$\mathscr{C}^0$ Whitney topology. Taking advantage of new approximation
techniques we prove: if $Y$ is weakly $\mathscr{C}^r$ triangulable, then $Y$ is
a $\mathscr{C}^r\text{-}\mathtt{ats}$. This result applies to relevant classes
of triangulable sets, namely: (1) every locally compact polyhedron is a
$\mathscr{C}^\infty\text{-}\mathtt{ats}$, (2) every set that is locally
$\mathscr{C}^r$ equivalent to a polyhedron is a
$\mathscr{C}^r\text{-}\mathtt{ats}$, and (3) every locally compact locally
definable set of an arbitrary o-minimal structure is a
$\mathscr{C}^1\text{-}\mathtt{ats}$ (this includes locally compact locally
semialgebraic sets and locally compact subanalytic sets). In addition, we
prove: if $Y$ is a global analytic set, then each proper continuous map $f:X\to
Y$ can be approximated by proper $\mathscr{C}^\infty$ maps $g:X\to Y$. Explicit
examples show the sharpness of our results.
