Analytic functions and Nash functions along curves

Abstract

Let X be a real analytic manifold. A function f :X rightarrow mathbb {R} is said to be curve-analytic if it is real analytic when restricted to any locally irreducible real analytic curve in X. We prove that every curve-analytic function with subanalytic graph is actually real analytic. To accomplish this task, we give a criterion for an arc-analytic function to be real analytic. A function is called arc-analytic if it is real analytic along any parametric real analytic arc. We also obtain analogous results for Nash manifolds and Nash functions, in which case the assumption of subanalyticity is superfluous.