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.