Whitney’s extension problem in o-minimal structures

Abstract

In 1934, H. Whitney asked how one can determine whether a real-valued function on a closed subset of $\mathbb{R}^n$ is the restriction of a $C^m$-function on $\mathbb{R}^n$. A complete answer to this question was found much later by C. Fefferman in the early 2000s. Here, we work in an o-minimal expansion of a real closed field and solve the $C^1$-case of Whitney's extension problem in this context. Our main tool is a definable version of Michael's selection theorem, and we include other another application of this theorem, to solving linear equations in the ring of definable continuous functions.