Continuous real function/Strictly increasing/Q to Q/Zero/Exercise

Show that for every real number ${}x\in \mathbb {R}$, there exists a strictly increasing continuous function

$f\colon \mathbb {R} \longrightarrow \mathbb {R}$

such that ${}x$ is the only zero of ${}f$ and such that for every rational number ${}q$, also ${}f(q)$ is rational.

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