Proving that $f$ is inegrable over rectangular $R=R' \cup R''$, union of disjoint rectangulars iff it's integrable over $R'$ and $R''$

Asked May 09 '14 at 22:16

Active May 09 '14 at 22:58

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Prove that $f : \mathbb{R}^n \to \mathbb{R}$ is inegrable over rectangular $R=R' \cup R''$, union of disjoint rectangulars iff it's integrable over $R'$ and $R''$.

My definition of "integrable": for any $\epsilon > 0$ there is a partition $p$ such that $U(f,p) - L(f,p) < \epsilon$.

I've been trying to prove it for lots of time. I tried proving the $\rightarrow$ direction by assuming $f$ isn't integrable at $R'$, but I wasn't able using it to my advantage.

You have my gratitude for any assistance you may provide!

edited May 09 '14 at 22:58

Caleb Stanford

45,895

asked May 09 '14 at 22:16

Choko

1

What does "integrable" mean to you? – May 09 '14 at 22:17
I tried using, that for every $\epsilon >0$, $f$ is integrable iff there is a partition $p$ s.t.: $U(f,p)-L(f,p)>\epsilon$ – Choko May 09 '14 at 22:19
@Choko I'm assuming these are rectangles in $\mathbb{R}^n$ given the [tag:multivariable-calculus] tag, but you should really give more details in asking a question. – Caleb Stanford May 09 '14 at 23:00
@Choko I edited your question. Did I get everything right? – Caleb Stanford May 09 '14 at 23:02
@Choko How do you define a partition of a non-rectangular region? – Caleb Stanford May 09 '14 at 23:04

Proving that $f$ is inegrable over rectangular $R=R' \cup R''$, union of disjoint rectangulars iff it's integrable over $R'$ and $R''$

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