Efroymson's approximation theorem for globally subanalytic functions

Abstract

Efroymson's approximation theorem asserts that if $f$ is a ${\mathcal{C}}^0$ semialgebraic mapping on a ${\mathcal{C}}^{\infty }$ semialgebraic submanifold $M$ of ${\mathbb{R}}^n$ and if $\epsilon :M\to \mathbb{R}$ is a positive continuous semialgebraic function then there is a ${\mathcal{C}}^{\infty }$ semialgebraic function $g:M\to \mathbb{R}$ such that $|f-g|<\epsilon$. We prove a generalization of this result to the globally subanalytic category. Our theorem actually holds in a larger framework since it applies to every function which is definable in a polynomially bounded o-minimal structure (expanding the real field) that admits ${\mathcal{C}}^{\infty }$ cell decomposition. We also establish approximation theorems for Lipschitz and ${\mathcal{C}}^1$ definable functions.