Prove or disprove: if the set of continuous points of function $f\colon \mathbb{R}\to\mathbb{R}$ is dense everywhere, then
the set of continuous points of $f$ is uncountable;
the set of discontinuous points of $f$ is countable.
Prove or disprove: if the set of continuous points of function $f\colon \mathbb{R}\to\mathbb{R}$ is dense everywhere, then
the set of continuous points of $f$ is uncountable;
the set of discontinuous points of $f$ is countable.
The following observations give an outline of one way to answer 1.
The following observations give an outline of one way to answer 2.
But you don't need such generalities for 2. You may want to look at your favorite example of a closed uncountable set with dense complement to get a more direct solution. (I'm assuming your favorite is the same as mine.)