I need to show if the following is true or false.
If the function is continuous in every irrational number x then f is continuous at every number.
Let $\displaystyle x\in \mathbb{Q}$ and write $\displaystyle x = \frac{p}{q}$ where $\displaystyle \gcd(q,|p|)=1$ with $\displaystyle q>0$. Then define $\displaystyle f(x) = \tfrac{1}{q}$ and $\displaystyle f(\alpha) = 0$ if $\displaystyle \alpha \in \mathbb{R} - \mathbb{Q}$. It turns out that $\displaystyle f$ is continous at all the irrational points and discontinous at all the rational points.