Suppose $f(x):=\left\{\begin{matrix}1/q&\text{ if }x=p/q\in\mathbb Q, p\in\mathbb Z, q\in\mathbb N\text{ and }\gcd(p,q)=1 \\\ 0&\text{ if }x\not\in\mathbb Q.\end{matrix}\right.$

At how many points is $f:\mathbb R\rightarrow\mathbb R$ continuous? Assume the standard topology on $\mathbb R$.