Does an analytic $f$ need be polynomial to close $\mathbb{Q}$

Question

If an analytic function $f : \mathbb{R}\to\mathbb{R}$ satisfies $f(\mathbb{Q}) \subseteq \mathbb{Q}$, can we conclude that $f$ is a polynomial?

Answer 1

$f(x)=\frac{1}{x^2+1}$ is analytic but not a polynomial.