Functions continuous on curves in o-minimal structures

TL;DR

This paper characterizes when bounded definable functions in o-minimal structures can be continuously extended along non-oscillatory curves, linking the problem to properties of types and definability.

Contribution

It provides necessary and sufficient conditions for continuous extension of functions along curves in o-minimal structures, translating the problem into type-theoretic conditions.

Findings

All such types are definable.

Conditions for the existence of a definable set with continuous function extension.

Characterization of types allowing continuous extension of bounded functions.

Abstract

We give necessary and sufficient conditions on a non-oscillatory curve in an o-minimal field such that, for any bounded definable function, the germ of the function on an initial segment of the curve can be continuously extended to a closed definable set. This situation is translated into a question about types: What are the conditions on an $n$-type such that, for any bounded definable function, there is a definable set containing the type on which the function is continuous, and can be extended continuously to the set's closure? All such types are definable, and we give the precise conditions that are equivalent to existence of a desired definable set.