Riemann integrable function

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

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

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 $[a,b]$.