Countable Discontinuities implies Integrability
If and is finite or countable, then is Riemann integrable.
If , and , then
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is finite implies that is Riemann integrable.
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is countable implies that is Riemann integrable.
Strength and limits of the criterion
Part (1) is stronger than the piecewise-continuity criterion: no one-sided limits are required at the discontinuities, so oscillating functions like near are covered.
The countability hypothesis cannot be dropped. If is not Riemann integrable then cannot be countable; but uncountably many discontinuities do not by themselves obstruct integrability either, since the indicator function of the Cantor set has uncountably many discontinuities and is still Riemann integrable.
The Dirichlet function and the Thomae function show that infinitely many discontinuities alone do not decide the question: both are discontinuous at infinitely many points, yet one fails to be integrable while the other is integrable.
Related
Stated in
- Proposition 4.17 (Countable Discontinuities Implies Integrability)§4.2 Integrability Criteria