Conditions for integrability

This is commonly called the Riemann-Lebesgue Theorem, or the Lebesgue Criterion for Riemann Integration (the wiki article).

The statement is that a function on $[a,b]$ is Riemann integrable iff

• It is bounded
• It is continuous almost everywhere, or equivalently that the set of discontinuities is of zero lebesgue measure

Just a small clarification (as an answer as I cannot comment):

The way in which mixedmath wrote the answer might lead to confusions, a different way would be:

A bounded function $f$ on $[a,b]$ is Riemann integrable iff it is continuous almost everywhere.

Note that this assumes the function to be bounded and it is not an implication of the theorem (For instance think of $f=\frac{1}{\sqrt{(x)}}$, whose integral in $[0,1]$ is 2 but it is not bounded).