Cauchy Criterion

Theorem

A bounded function is Riemann integrable iff for every there exists a partition of , such that:

Proof Hint

• To prove : consider and
• To prove : consider and

Note

is integrable on when:

• The set of points of discontinuity of a bounded function is finite.
• The set of points of discontinuity of a bounded function is finite number of limit points. (may have infinite number of discontinuities)

In these cases, the discontinuities don’t affect the result of the integration.