# Welcome Back!

armed with the known, we sail to explore the yet unknown

this document in PDF: WelcomeBack.pdf

The Known:

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 .

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Dror Bar-Natan 2003-01-07