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 $\varepsilon:M\to \mathbb{R}$ is a positive continuous semialgebraic function then there is a $\mathcal{C}^\infty$ semialgebraic function $g:M\to \mathbb{R}$ such that $|f-g|<\varepsilon$. 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.