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Find a function $f(x)$ which is defined at every real number but is continuous at $0$ and is not continuous at every other number.

This was a bonus question for our Calculus 1 homework on limits a few months ago and no one could figure it out.

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The canonical example is $$ f (x)=\begin{cases}x,&\ x\in\mathbb Q\\0,&\ x\in\mathbb R\setminus\mathbb Q\end{cases}$$

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