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Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a real function, and satisfy that: for all $x\in\mathbb{R}$ $$\lim_{r\to x,r\in\mathbb{Q}}f(r)=f(x).$$ Show that $f$ is continuous on $\mathbb{R}.$

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    you have continuous function $\tilde{f}$ on $\mathbb Q$. Now the question is: what is extension of this functon on $\mathbb R$. Try this http://www.physicsforums.com/showthread.php?t=4300832012-10-09

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