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No continuous function that switches $\mathbb{Q}$ and the irrationals
Let $f: \mathbb{R} \to \mathbb{R}$ be function satisfying the two conditions: $f(\mathbb{R}\setminus \mathbb{Q}) \subseteq \mathbb{Q}$ and $f(\mathbb{Q}) \subseteq \mathbb{R}\setminus \mathbb{Q}$. Then,
Show that $f$ cannot be continuous.
I'm trying this problem for some time but can't make any useful progress. I will appreciate any help. Even some good hints will do. Regards.