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?
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2You wrote that the domain of $f$ is an open subset. If the domain is not $\Bbb R^n$, then the expression $f(\Bbb Q^n)$ does not make sense. I have edited the question to avoid this problem. – ajotatxe Dec 28 '17 at 16:17
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1I'd say at first glance that the answer is "no", but finding a counterexample seems hard... – ajotatxe Dec 28 '17 at 16:19