A bounded function is Riemann integrable exactly when its discontinuities form a null set.Suggest a slogan
Let \(a<b\) and let \(f\colon[a,b]\to\RR\) be bounded, with set of discontinuities \(D_f\subseteq[a,b]\). Then \(f\) is Riemann integrable on \([a,b]\) if and only if \(D_f\) has Lebesgue measure \(0\).