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

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

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

Suppose that:
 * $\displaystyle \underline{\int_a^b} f \left({x}\right) \ \mathrm dx = \overline{\int_a^b} f \left({x}\right) \ \mathrm dx$

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

Then $f$ is said to be (properly) Riemann integrable on $\left[{a \,.\,.\, b}\right]$.