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

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

Let $f: \closedint a b \to \R$ be a real function. Let $f$ be bounded on $\closedint a b$.

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

The Darboux integral of $f$ over $\closedint a b$ is denoted:
 * $\ds \int_a^b \map f x \rd x$

and is defined as:
 * $\ds \int_a^b \map f x \rd x = \underline {\int_a^b} \map f x \rd x = \overline {\int_a^b} \map f x \rd x$

where $\ds \underline {\int_a^b}$ and $\ds \overline {\int_a^b}$ denote the lower integral and upper integral, respectively.