Weakly o-minimal fields have the exchange property but not generic differentiability

Abstract

We answer two open questions about weakly o-minimal fields posed by Macpherson, Marker, and Steinhorn: whether weakly o-minimal fields have the exchange property and whether they have generic differentiability. We construct an ordered field $(K,+,\cdot,\le)$ and a function $f : K \to K$ such that the expansion $(K,+,\cdot,\le,f)$ has a weakly o-minimal complete theory but $f$ is nowhere differentiable. In an appendix, we prove that algebraic closure has the exchange property in any weakly o-minimal theory of ordered fields.