On continuous functions definable in expansions of the ordered real additive group

Let $\mathcal R$ be an expansion of the ordered real additive group. Then one of the following holds: either every continuous function $[0,1] \to \mathbb R$ definable in $\mathcal R$ is $C^2$ on an open dense subset of $[0,1]$, or every $C^2$ function $[0,1] \to \mathbb R$ definable in $\mathcal R$ is affine, or every continuous function $[0,1] \to \mathbb R$ is definable in $\mathcal R$. If $\mathcal R$ is NTP$_{2}$ 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 $\mathcal R$ defines a $C^2$ function $[0,1] \to \mathbb R$ that is not affine, it also defines an ordered field on some open interval whose ordering coincides with the usual ordering on $\mathbb R$. Assuming $\mathcal R$ does not interpret second-order arithmetic, we show that the last statement holds when $C^2$ is replaced by $C^1$.