Riemann integral

Let $I=[a,b]$ be an interval of $\\mathbb{R}$ and let $f\\colon I\\to\\mathbb{R}$ be a bounded function. For any finite set of points $\\{x_{0},x_{1},x_{2},\\ldots,x_{n}\\}$ such that $a=x_{0}<x_{1}<x_{2}\\cdots<x_{n}=b$ , there is a corresponding partition $P=\\{[x_{0},x_{1}),[x_{1},x_{2}),\\ldots,[x_{n-1},x_{n}]\\}$ of $I$ .

Let $C(\\epsilon)$ be the set of all partitions of $I$ with $\\max(x_{i+1}-x_{i})<\\epsilon$ . Then let $S^{*}(\\epsilon)$ be the infimum of the set of upper Riemann sums with each partition in $C(\\epsilon)$ , and let $S_{*}(\\epsilon)$ be the supremum of the set of lower Riemann sums with each partition in $C(\\epsilon)$ . If $\\epsilon_{1}<\\epsilon_{2}$ , then $C(\\epsilon_{1})\\subset C(\\epsilon_{2})$ , so $S^{*}(\\epsilon)$ is decreasing (http://planetmath.org/IncreasingdecreasingmonotoneFunction) and $S_{*}(\\epsilon)$ is increasing (http://planetmath.org/IncreasingdecreasingmonotoneFunction). Moreover, $\\lvert S^{*}(\\epsilon)\\rvert$ and $\\lvert S_{*}(\\epsilon)\\rvert$ are bounded by $(b-a)\\sup_{x}\\lvert f(x)\\rvert$ . Therefore, the limits $S^{*}=\\lim_{\\epsilon\\to 0}S^{*}(\\epsilon)$ and $S_{*}=\\lim_{\\epsilon\\to 0}S_{*}(\\epsilon)$ 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

\\int_{a}^{b}f(x)dx=S^{*}=S_{*}.