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The setting: bounded on , a partition of , , , , , , . Finally, if we say that `` is integrable on '' and set .
Theorem 1. For any two partitions , .
Theorem 2. is integrable iff for every there is a partition such that .
Theorem 3. If is continuous on then is integrable on .
Theorem 4. If then (in particular, the rhs makes sense iff the lhs does).
Theorem 5. If and are integrable on then so is , and .
Theorem 6. If is integrable on and is a constant, then is integrable on and .
Theorem . If on and both are integrable on , then .
Theorem 7. If on and is integrable on then .
Theorem 8. If is integrable on and is defined on by , then is continuous on .
Just for fun. Why did I put these boxed statements on this page? Can they both be true? Can they both be false? If just one is true, which one must it be?