I have sort of a silly question, since I feel confused.
Let $f(x) = \begin{cases} 1, & x\in \mathbb{Q}\\ 0, & x\in \mathbb{R} \setminus \mathbb{Q} \end{cases}$
Intuitively, this function has a discontinuity at every rational number, and this set is countable. On the other hand, I could also say that it has a discontinuity at every irrational number, and this set is uncountable.
Which one of those statements is true?