This question is out of my curiosity, I have finished my calculus course years ago and unfortunately all the knowledge became rusty, right now I cannot deal even with this simple-looking question.
Let $f : \mathbb{R} \to \mathbb{R}$ be a measurable function such that $F(x) = \int_0^x f(y)\ \mathrm{d}y$ exists and $\|F\|_\infty = M$ is bounded ($\int$ denotes the Lebesgue's integral). Must $F$ be continuous?
If not, could someone sketch the counter-example?