TITLE: Proof that if $f$ is integrable then also $f^2$ is integrable
QUESTION [10 upvotes]: Prove this:

Let $f :[a,b] \to \mathbb{R}$ be a bounded and integrable function. Show that $f^2$ is integrable too.

I'm in trouble with this. Can anyone show how to do it?

REPLY [5 votes]: If you are familiar with it, this follows immediately from Lesbegue Integrability Criteria.
As $f$ is bounded, so is $f^2$. Moreover, the set of discontinuities of $f^2$ is a subset of the set of discontinuities of $f$.