R-integrable function (Riemann integrable)

See Riemann integral#Integrability

Boundedness requirement: By definition, an R-integrable function must be bounded on the interval of integration. This is a necessary condition for Riemann integrability.

Examples of R-integrable functions:

• Continuous functions on closed intervals
• Monotonic functions on closed intervals
• Step functions
• Functions with finitely many discontinuities

Relationship with Lebesgue integrability

Every R-integrable function is L-integrable, and the two integrals coincide. The converse fails: ${\chi }_{\mathbb{Q}\cap [0,1]}$ is L-integrable (integral $=0$) but not R-integrable. See L-integrable for the comparison.