By an influential theorem of Boman, a function $f$ on an open set $U$ in $\mathbb R^d$ is smooth ($\mathcal C^\infty$) if and only if it is \emph{arc-smooth}, i.e., $f\circ c$ is smooth for every smooth curve $c : \mathbb R \to U$. In this paper we investigate the validity of this result on closed sets. Our main focus is on sets which are the closure of their interior, so-called \emph{fat} sets. We obtain an analogue of Boman's theorem on fat closed sets with H\"older boundary and on fat closed subanalytic sets with the property that every boundary point has a basis of neighborhoods each of which intersects the interior in a connected set. If $X \subseteq \mathbb R^d$ is any such set and $f : X \to \mathbb R$ is arc-smooth, then $f$ extends to a smooth function defined in $\mathbb R^d$. As a consequence we also get a version of the Bochnak-Siciak theorem on all such sets: if $f : X \to \mathbb R$ is arc-smooth and real analytic along all real analytic curves in $X$, then $f$ extends to 查看全文>>