Differentiable approximation of continuous locally definable maps that preserves the image

Recently, we showed that continuous definable maps defined on compact definable sets can be uniformly approximated by continuous definable maps of class $\mathcal{C}^p$ without changing their image. The aim of this paper is to extend the previous result, this time taking into account the (strong) Whitney topology, to continuous locally definable maps defined on locally compact locally definable sets. The argument is an interplay between o-minimal and PL geometry and makes essential use of Pawłucki's desingularization techniques as well as our aforementioned result for the compact case.