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

Question

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.

Answer 1

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