# Efroymson's approximation theorem for globally subanalytic functions

## TL;DR

This paper generalizes Efroymson's approximation theorem from semialgebraic to globally subanalytic functions, extending to broader o-minimal structures and including Lipschitz and -definable functions.

## Contribution

It extends Efroymson's approximation theorem to globally subanalytic and o-minimal structures, broadening the class of functions that can be approximated by smooth definable functions.

## Key findings

- Approximation theorem holds for globally subanalytic functions.
- Results apply to functions definable in polynomially bounded o-minimal structures.
- Includes approximation results for Lipschitz and -definable 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.