An Almost Everywhere Extension Theorem for Continuous Definable Functions in an O-minimal Structure

Abstract

Let $\mathcal{R} = (R, <, \mathcal{S})$ be an o-minimal expansion of an ordered group. In this thesis, we define the class $\mathcal{C}$ of asymptotically monotone cells and we show they have the property that, for any cell $C \in \mathcal{C}$ and for any definable, continuous, bounded function $f : C \rightarrow R$, it is always possible to continuously extend $f$ "almost everywhere" to the frontier of $C$. We make this notion precise using a theory of dimension for sets definable in an o-minimal structure. This result is a generalization of a known fact about continuous extensions of definable, continuous, bounded functions on open cells; we show by way of counterexample that the original result does not generalize to the class of all cells and hence that the assumption that our cells are asymptotically monotone is required. Background on o-minimality and the theory of dimension for definable sets is provided.