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.

Proof sketch: Assume $f$ is nondecreasing. For a partition $P$ with mesh $\|P\|$, one can estimate $U(f,P)-L(f,P)\le (f(b)-f(a))\,\|P\|.$ Choosing $\|P\|<\varepsilon/(f(b)-f(a))$ (or handling the constant case separately) forces $U-L<\varepsilon$, proving integrability.