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Under uniform convergence, the limit of Riemann integrals equals the Riemann integral of the limit.
Interchanging limit and integral: Let be Riemann integrable for every , and suppose uniformly on . Then is Riemann integrable and
This is a basic continuity property of the Riemann integral with respect to uniform convergence, and it is the key step behind term-by-term integration of uniformly convergent series.
A sequence of functions from a set into a metric space converges uniformly to a function if for every there exists such that for all and all ,
Equivalently,
A Riemann integral of a Riemann integrable function is the number
defined by
where and are the lower and upper sums over all partitions of .
Uniform convergence and integration: Let be Riemann integrable for all , and let be the partial sums. If uniformly on , then is Riemann integrable and
where the series of real numbers on the right converges.