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Every monotone function on a closed interval is Riemann integrable.
Monotone functions are Riemann integrable: Let . If is monotone, then is a Riemann integrable function on .
This theorem is a cornerstone of the Riemann theory, providing integrability for many non-continuous examples; it complements criteria based on discontinuities and the oscillation criterion.
A monotone function on an interval is a function that is either nondecreasing (if implies ) or nonincreasing (if implies ); it is strictly increasing or strictly decreasing if the inequalities are strict whenever .
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 .
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.