User:Keith.U/Sandbox/Riemann Integral

Definition
Let $I = \left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $f: \left[{a \,.\,.\, b}\right] \to \R$ be a real function.

Let $S \left({ f; \Delta }\right)$ denote the Riemann sum of $f$ for a  subdivision $\Delta$ of $I$.

Then $f$ is said to be Riemann integrable on $I$ :
 * $\exists L \in \R : \forall \epsilon \in \R_{>0} : \exists \delta \in \R_{>0} : \forall$ subdivisions $\Delta$ of $I : \left\Vert{ \Delta }\right\Vert < \delta \implies \left\vert{ S \left({ f; \Delta }\right) - L }\right\vert < \epsilon$

where $\left\Vert{ \Delta }\right\Vert$ denotes the norm of $\Delta$.

The real number $L$ is called the Riemann integral of $f$ over $I$.