Characterizations of Riemann and Lebesgue Integrability

Prove that the product of Riemann integrable functions on a closed, bounded interval is Riemann integrable, and the product of bounded Lebesgue integrable functions on a set of finite measure is Lebesgue integrable.

Let $E$ be measurable and the set of points at which $f: E \rightarrow \mathbb {R}$ fails to be continuous have measure zero. Show that $f$ is measurable.