Let W be a subset of the set of real points of a real algebraic variety X. We investigate which functions f: W→ R are the restrictions of rational functions on X. We introduce two new notions: curve-rational functions (i.e., continuous rational on algebraic curves) and 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.
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