Monotone functions are Riemann integrable

Monotone functions are Riemann integrable: If $f:[a,b]\to\mathbb{R}$ is bounded and monotone (nondecreasing or nonincreasing), then $f$ is Riemann integrable on $[a,b]$.

This is a key example showing that Riemann integrability does not require continuity; controlled discontinuities are allowed.