Characterization of Riemann integrable functions

I was asked in class today whether there is a characterization of the Riemann integrable functions, and there is a nice one due to Lebesgue: A bounded function on an interval [a,b] is Riemann-integrable if and only if its set of discontinuities has Lebesgue measure zero. For comparison, the set of discontinuities for the Dirichlet function is the entire interval [a,b].