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this document in PDF: Integration.pdf

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?

The generation of this document was assisted by
L^{A}TEX2`HTML`.

Dror Bar-Natan 2005-01-05