  Let $W$ be a subset of the set of real points of a real algebraic variety
$X$. We investigate which functions $f: W \to \mathbb R$ are the restrictions
of rational functions on $X$. We introduce two new notions: ${\it
curve-rational \, functions}$ (i.e., continuous rational on algebraic curves)
and ${\it arc-rational\, functions}$ (i.e., continuous rational on arcs of
algebraic curves). We prove that under mild assumptions the following classes
of functions coincide: continuous hereditarily rational (introduced recently by
the first named author), curve-rational and arc-rational. In particular, if $W$
is semialgebraic and $f$ is arc-rational, then $f$ is continuous and
semialgebraic. We also show that an arc-rational function defined on an open
set is arc-analytic (i.e., analytic on analytic arcs). Furthermore, we study
rational functions on products of varieties. As an application we obtain a
characterization of regular functions. Finally, we get analogous results in the
framework of complex algebraic varieties.
