Definable functions continuous on curves in o-minimal structures

We give necessary and sufficient conditions on a non-oscillatory curve in an o-minimal structure such that, for any bounded definable function, there exists a definable closed set containing an initial segment of the curve on which the function is continuous. This question is translated into one on types: What are the conditions on an n-type such that, for any bounded definable function, there is a definable closed set containing the type on which the function is continuous. We introduce two concepts related to o-minimal types: that of scale, which measures the “density” of a smaller model inside a larger one at some point, and that of a decreasing type, which allows us to manipulate types more easily than before. We formalize the notion of scale mentioned in [MS94] and refine the characterization there of definable types in o-minimal theories. Then we join it with the notion of decreasing type to achieve our main result. A decreasing sequence has the property that the map taking initial segments of the sequence to the T -convex subrings generated by them preserves inclusion.