Is there a differentiable function $f:\mathbb R \rightarrow \mathbb R$ such that $f(\mathbb Q) \subseteq \mathbb Q$, but $f'(\mathbb Q) \not \subseteq \mathbb Q$? A friend of mine asserted this without giving any examples. I seriously doubt it, but I had hard time trying to disprove it since analysis isn't really my thing. I can't even think of any class of differentiable functions with $f(\mathbb Q) \subseteq \mathbb Q$ other than the rational functions.
Is there a differentiable function such that $f(\mathbb Q) \subseteq \mathbb Q$ but $f'(\mathbb Q) \not \subseteq \mathbb Q$?
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real-analysis