Riemann Integrability Criteria
A bounded is Riemann integrable iff for every some partition has ; equivalently, iff some sequence of partitions drives this gap to .
Proposition 4.8 (Riemann Integrability Criteria)
Let be bounded. Then is Riemann integrable iff
Sequential criterion
Proposition 4.9
Let be bounded. Then is Riemann integrable iff there exists a sequence of partitions of such that
Proof
Suppose the criterion holds. For every ,
so and is integrable.
Conversely, suppose . From the definition of and , there are partitions with
Since , . Take ; refining can only shrink the gap, so
Related
Stated in
- Proposition 4.8 (Riemann Integrability Criteria)§4.2 Integrability Criteria
- Proposition 4.9§4.2 Integrability Criteria