On real analytic functions on closed subanalytic domains

Abstract

We show that a function \(f: X \rightarrow {\mathbb {R}}\) defined on a closed uniformly polynomially cuspidal set X in \({\mathbb {R}}^n\) is real analytic if and only if f is smooth and all its composites with germs of polynomial curves in X are real analytic. The degree of the polynomial curves needed for this is effectively related to the regularity of the boundary of X. For instance, if the boundary of X is locally Lipschitz, then polynomial curves of degree 2 suffice. In this Lipschitz case, we also prove that a function \(f: X \rightarrow {\mathbb {R}}\) is real analytic if and only if all its composites with germs of quadratic polynomial maps in two variables with images in X are real analytic; here it is not necessary to assume that f is smooth.