Arc-quasianalytic functions

Abstract:

We work with quasianalytic classes of functions. Consider a real-valued function $y = f(x)$ on an open subset $U$ of $\mathbb {R}^n$, which satisfies a quasianalytic equation $G(x,y) = 0$. We prove that $f$ is arc-quasianalytic (i.e., its restriction to every quasianalytic arc is quasianalytic) if and only if $f$ becomes quasianalytic after (a locally finite covering of $U$ by) finite sequences of local blowings-up. This generalizes a theorem of the first two authors on arc-analytic functions.