Let be an interval of and let be a bounded function. For any finite set of points such that , there is a corresponding partition of .
Let be the set of all partitions of with . Then let be the infimum of the set of upper Riemann sums with each partition in , and let be the supremum of the set of lower Riemann sums with each partition in . If , then , so is decreasing (http://planetmath.org/IncreasingdecreasingmonotoneFunction) and is increasing (http://planetmath.org/IncreasingdecreasingmonotoneFunction). Moreover, and are bounded by . Therefore, the limits and exist and are finite. If , then is Riemann-integrable over , and the Riemann integral of over is defined by
|Date of creation||2013-03-22 11:49:24|
|Last modified on||2013-03-22 11:49:24|
|Last modified by||bbukh (348)|