Differentiable approximation of continuous definable maps that preserves the image

Recently Pawłucki showed that compact sets that are definable in some o-minimal structure admit triangulations of class ${\mathcal{C}}^p$ for each integer $p\ge 1$. In this work, we make use of these new techniques of triangulation to show that all continuous definable maps between compact definable sets can be approximated by differentiable maps without changing their image after the approximation. The argument is an interplay between o-minimal geometry and PL geometry and makes use of a `surjective definable version’ of the finite simplicial approximation theorem that we prove here.