Annals of Pure and Applied Logic 165 (7-8):1339-1351 (2014)

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 has a definable extension to a closed 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, the germ of the function on the type has a definable continuous global extension? Certain categories of definable types have this property, and we give the precise conditions that are equivalent to existence of the global extension
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DOI 10.1016/j.apal.2014.04.005
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References found in this work BETA

T-Convexity and Tame Extensions.Dries Lou Van Den & H. Lewenberg Adam - 1995 - Journal of Symbolic Logic 60 (1):74 - 102.
Expansions of the Real Field with Power Functions.Chris Miller - 1994 - Annals of Pure and Applied Logic 68 (1):79-94.
Definable Types in o-Minimal Theories.David Marker & Charles I. Steinhorn - 1994 - Journal of Symbolic Logic 59 (1):185-198.
Definable Types in $Mathscr{O}$-Minimal Theories.David Marker & Charles I. Steinhorn - 1994 - Journal of Symbolic Logic 59 (1):185-198.

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