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A bounded function with only finitely many discontinuities is Riemann integrable.
Finite discontinuities imply Riemann integrability: Let and let be bounded. If the set of points where has a discontinuity is finite, then is a Riemann integrable function on .
This gives a practical sufficient condition for Riemann integrability that can be checked by analyzing the set of discontinuities; it is often proved using the oscillation criterion.
A discontinuity point of a function at a point is a point where is not continuous at ; equivalently, is a discontinuity point if there exists such that for every there is with and .
A set of discontinuities of a function , where , is the set
Equivalently, exactly when is a discontinuity point of .
Oscillation criterion: Let and let be bounded. For a partition , let be the oscillation of on the subinterval . Then is a Riemann integrable function on if and only if for every there exists a partition such that
This criterion is equivalent to the usual definition via upper sums and lower sums, and it is especially useful for proving integrability results like finite discontinuities imply integrability.