Moderate# Continuity Involving Rationals and Reals

TOPO-3Q3OAG

Suppose $f(x):=\left\{\begin{matrix}x&\text{ if }x\in\mathbb Q \\\ -x&\text{ if }x\not\in\mathbb Q.\end{matrix}\right.$ At how many points is $f:\mathbb R\rightarrow\mathbb R$ continuous? Assume the standard topology on $\mathbb R$.