Real analysis of calculus

Question

Give an example of a function $f : R \rightarrow R$ which is continuous at all irrationals points and discontinuous at all rational points of $R$.

Answer 1

Define $f(x) = \begin{cases} 0 & x \in \mathbb{R}\setminus \mathbb{Q} \\ \frac{1}{q} & x = \frac{p}{q} \end{cases}$

at each irrational $f(x) = \lim\limits_{y \to x} = 0$, while at rationals any sequence of irrationals which converge to $x$ produce $0$.