Riemann integrability implies boundedness

Riemann integrability implies boundedness: Let $a<b$ and let $f:[a,b]\to\mathbb{R}$ be a Riemann integrable function . Then $f$ is bounded on $[a,b]$ (equivalently, $f$ is both bounded above and bounded below ).

This is a necessary condition for the Riemann integral to be well-defined, since upper sums and lower sums take suprema and infima of $f$ on subintervals of a partition .