Riemann integrable function - Imported Knowlpedia Content

A Riemann integrable function on $[a,b]$ is a bounded function $f:[a,b]\to\mathbb R$ such that for every $\varepsilon>0$ there exists a partition $P$ with

U(f,P)-L(f,P)<\varepsilon,

where $U(f,P)$ is the upper sum and $L(f,P)$ is the lower sum. Equivalently,

\sup_P L(f,P)=\inf_P U(f,P).

Remarks

When $f$ is Riemann integrable, the common value is the Riemann integral of $f$ . Integrability is closely tied to the set of discontinuities (see the Lebesgue criterion for Riemann integrability).

Examples

Any function that is continuous on $[a,b]$ is Riemann integrable.
The indicator function of $\mathbb Q\cap[a,b]$ is not Riemann integrable on $[a,b]$ .