Definable smoothing of continuous functions

Abstract

Let $\mathbf{R}$ be an o-minimal expansion of a real closed field. Given definable continuous functions $f:U\rightarrow R$ and $\epsilon:U\rightarrow(0,+\infty)$, where $U$ is an open subset of $R^{n}$, we construct a definable $C^{m}$-function $g:U\to R$ with $\vert g(x)-f(x)\vert <\epsilon(x)$ for all $x\in U$. Moreover, we show that if $f$ is uniformly continuous, then $g$ can also chosen to be uniformly continuous.