Monotone Real Function is Darboux Integrable

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

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

Then $f$ is Riemann integrable over $\left[{a \,. \, . \, b}\right]$.

Also see

 * Continuous Function is Riemann Integrable