Def 28.5 pg. 228 Darboux integrable
For and be a bounded function. Then is Darboux integrable if
Def 28.1 Riemann integrable
where is a partition, is the max mesh size, and is a set with points from every sub-interval in , is Riemann Sum.
Thm 28.4
Darboux integrable imply Riemann integrable
Thm 29.5
continue imply integrable
Lemma 10.4
let be Lipschitz and integrable, we have integrable.
Also
Thm 31.1 Lebesgueβs Criterion for Riemann Integrability
is bounded and , the following are equivalent:
- is Riemann integrable on
- the set of discontinuities of on has Lebesgue measure zero
Basically, it means, is only discontinue in measure zero set, and continue otherwise.
Thm 31.2
is continue, integrable integrable
OH 35.1
Thm A
Lipschitz and integrable integrable
Thm B
continue and integrable integrable
Proof
By Thm C, we can say there exists has Lebesgue Zero Measure.