# Analysis problem

Let $x\in \mathbb{Q}$ and write $x = \frac{p}{q}$ where $\gcd(q,|p|)=1$ with $q>0$. Then define $f(x) = \tfrac{1}{q}$ and $f(\alpha) = 0$ if $\alpha \in \mathbb{R} - \mathbb{Q}$. It turns out that $f$ is continous at all the irrational points and discontinous at all the rational points.