On Continuous Functions Definable in Expansions of the Ordered Real Additive Group

Let R be an expansion of the ordered real additive group. Then one of the following holds: either every continuous function [0, 1] → R definable in R is C2 on an open dense subset of [0, 1], or every C2 function [0, 1] → R definable in R is affine, or every continuous function [0, 1] → R is definable in R. If R is NTP2 or more generally does not interpret a model of the monadic second order theory of one successor, the first case holds. It is due to Marker, Peterzil, and Pillay that whenever R defines a C2 function [0, 1] → R that is not affine, it also defines an ordered field on some open interval whose ordering coincides with the usual ordering on R. Assuming R does not interpret secondorder arithmetic, we show that the last statement holds when C2 is replaced by C1.