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A friend of mine introduced me to the following question: Does there exist a smooth function $f: \mathbb{R} \to \mathbb{R}$ ($f \in C^{\infty}$), such that $f$ maps rationals to rationals and irrationals to irrationals and is nonlinear?

He has been able to prove that such a polynomial (with degree at least 2) doesn't exist.

The problem has been asked before at least at http://www.artofproblemsolving.com.

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    Did miss that, sorry.2010-12-11

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Sergei Ivanov has given a positive answer for the existence of such functions on MO: https://mathoverflow.net/questions/48910/smooth-functions-for-which-fx-is-rational-if-and-only-if-x-is-rational.