Riemann integrability implies boundedness

Proposition: If $f:[a,b]\to\mathbb{R}$ is Riemann integrable on $[a,b]$, then $f$ is bounded on $[a,b]$.

In many texts, boundedness is built into the definition of Riemann integrability. This proposition records that boundedness is not optional: unbounded functions cannot have finite upper and lower sums .