|Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:||(147)||
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Setting. bounded on , a partition of , , , , , , . Finally, if we say that `` is integrable on '' and set .
Theorem 13-1. For any two partitions , .
Theorem 13-2. is integrable iff for every there is a partition such that .
Theorem 13-3. If is continuous on then is integrable on .
Theorem 14-2. (The Second Fundamental Theorem of Calculus) If is integrable on , where is some differentiable function, then .
Theorem 13-4. If then (in particular, the rhs makes sense iff the lhs does).
The Yet Unknown:
Convention. and if we set .
Theorem 13-4'. so long as all integrals exist, no matter how , and are ordered.
Theorem 13-5. If and are integrable on then so is , and .
Theorem 13-6. If is integrable on and is a constant, then is integrable on and .
Theorem 13-7a. If on and both are integrable on , then .
Theorem 13-7. If on and is integrable on then .
Theorem 13-8. If is integrable on and is defined on by , then is continuous on .
Theorem 14-1. (The First Fundamental Theorem of Calculus) Let be integrable on , and define on by . If is continuous at , then is differentiable at and .