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Does there exist a continuous function $f : \Bbb R \to \Bbb R$ which takes irrational values at rational points and rational values at irrational points?

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    $\begingroup$ No. See this. $\endgroup$ – David Mitra Dec 29 '13 at 23:10
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By Baire category theorem, we can assume that such a function takes a constant rational value on a dense (comeager) subset of some interval. But then it is constant there.

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