User:Keith.U/Sandbox/SubSandbox 2/SubSubSandbox 1

Definition
Let $\closedint a b$ be a closed real interval.

Let $f: \closedint a b \to \R$ be a real function. Let $\map S {f; \Delta}$ denote the Riemann sum of $f$ for a subdivision $\Delta$ of $\closedint a b$.

Let $f$ be Riemann Integrable over $\closedint a b$.

That is, suppose that there exists some $L \in \R$ such that:
 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall$ subdivisions $\Delta$ of $\closedint a b: \norm \Delta < \delta \implies \size {\map S {f; \Delta} - L} < \epsilon$

where $\norm \Delta$ denotes the norm of $\Delta$.

The real number $L$ is called the Riemann integral of $f$ over $\closedint a b$ and is denoted:
 * $\ds \int_a^b \map f x \rd x$