$f$ bounded almost everywhere $\Longrightarrow$ $f$ bounded or $f$ discontinuous

Question

Let $f:\mathbb{R}\to\mathbb{R}$ be a function. I am just asking if the following result is true:

$f$ is bounded almost everywhere $\Longrightarrow$ $f$ is bounded or $f$ is discontinuous

Answer 1

If $f$ is continuous and unbounded, then for each $M$ there exists $x_0$ such that $|f(x_0)|>M$ and by continuity, $|f(x)|>M$ in an open neighbourhood $(x_0-\epsilon, x_0+\epsilon)$ of $x_0$. As that interval is not a zero-set, $M$ fals to be an almost-bound of $f$.