Does $f(\mathbb{Q}^n) \subseteq \mathbb{Q}$ hold only for rational functions?

Question

Let $f$ be a $C^{\infty}$-function from an open subset $U$ of $\mathbb{R}^n$ to $\mathbb{R}$.
If $f(\mathbb{Q}^n\cap U) \subseteq \mathbb{Q}$ can we conclude that $f$ is a rational function?