Riemann integral

Let I=[a,b] be an interval of and let f:I be a bounded function. For any finite setMathworldPlanetmath of points {x0,x1,x2,,xn} such that a=x0<x1<x2<xn=b, there is a corresponding partitionPlanetmathPlanetmath P={[x0,x1),[x1,x2),,[xn-1,xn]} of I.

Let C(ϵ) be the set of all partitions of I with max(xi+1-xi)<ϵ. Then let S*(ϵ) be the infimumMathworldPlanetmath of the set of upper Riemann sums with each partition in C(ϵ), and let S*(ϵ) be the supremum of the set of lower Riemann sums with each partition in C(ϵ). If ϵ1<ϵ2, then C(ϵ1)C(ϵ2), so S*(ϵ) is decreasing (http://planetmath.org/IncreasingdecreasingmonotoneFunction) and S*(ϵ) is increasing (http://planetmath.org/IncreasingdecreasingmonotoneFunction). Moreover, |S*(ϵ)| and |S*(ϵ)| are bounded by (b-a)supx|f(x)|. Therefore, the limits S*=limϵ0S*(ϵ) and S*=limϵ0S*(ϵ) exist and are finite. If S*=S*, then f is Riemann-integrable over I, and the Riemann integral of f over I is defined by

Title Riemann integral
Canonical name RiemannIntegral
Date of creation 2013-03-22 11:49:24
Last modified on 2013-03-22 11:49:24
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 14
Author bbukh (348)
Entry type Definition
Classification msc 28-00
Classification msc 26A42
Related topic RiemannSum
Related topic Integral2
Defines Riemann integrable