Riemann integrable function

A bounded function $f:[a,b]\to\mathbb{R}$ is Riemann integrable on $[a,b]$ if

Equivalently, $f$ is Riemann integrable iff its upper integral $\inf_P U(f,P)$ (using upper sums ) equals its lower integral $\sup_P L(f,P)$ (using lower sums ), where the infimum/supremum are taken over all partitions $P$.

Riemann integrability is designed so that the area under the graph is well-defined and agrees with limits of Riemann sums .

Examples:

• Every continuous function on $[a,b]$ is Riemann integrable.
• The function $\mathbf{1}_{\mathbb{Q}}$ on $[0,1]$ is not Riemann integrable (upper sums are always $1$, lower sums always $0$).
• Any monotone function on $[a,b]$ is Riemann integrable.