Note that for all statements of "integrability" below I am referring to Reimann integrability.
Please read: I have already seen the proof where you first prove that $f^2$ is integrable. This I understand. However, I have proven a theorem that says that if $f$ is integrable and $g$ is continuous, $g \circ f$ is integrable. I was told that this could be used to prove that if $f$ and $g$ are integrable, $fg$ is integrable.
However, I'm stuck on how to apply this fact. The fact that neither $f$ nor $g$ is necessarily continuous tells me I will need to create some special function that is continuous in order to utilize the theorem, but I'm out of ideas. Any advice or hints would be appreciated.