\begin{definition}[Definition:Definite Integral/Riemann]
Let $\closedint a b$ be a closed real interval.
Let $f: \closedint a b \to \R$ be a real function.
Let $\Delta$ be a finite subdivision of $\closedint a b$, $\Delta = \set {x_0, \ldots, x_n}$, $x_0 = a$ and $x_n = b$.
Let there for $\Delta$ be a corresponding sequence $C$ of sample points $c_i$, $C = \tuple {c_1, \ldots, c_n}$, where $c_i \in \closedint {x_{i - 1} } {x_i}$ for every $i \in \set {1, \ldots, n}$.
Let $\map S {f; \Delta, C}$ denote the Riemann sum of $f$ for the subdivision $\Delta$ and the sample point sequence $C$.
Then $f$ is said to be '''(properly) Riemann integrable''' on $\closedint a b$ {{iff}}:
:$\exists L \in \R: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall$ finite subdivisions $\Delta$ of $\closedint a b: \forall$ sample point sequences $C$ of $\Delta: \norm \Delta < \delta \implies \size {\map S {f; \Delta, C} - L} < \epsilon$
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:
:$\ds \int_a^b \map f x \rd x$
More usually (and informally), we say:
:'''$f$ is (Riemann) integrable over $\closedint a b$'''.
\end{definition}