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Problem:

I am stuck on the following problem and I appreciate if someone is willing to help solving it. The problem is as follows:

I am given a uniformly continuous function : $f:\mathbb{R}^2 \rightarrow [0,\infty )$ such that the following condition is satisfied: $$\iint_{ R^2} f(x,y)\,dx\,dy< \infty .$$ The question is to prove that:$$\lim_{| (x,y)| \to \infty}f(x,y)=0$$

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You have the letter $r$ for two different dummy variables. –  anon Apr 26 '12 at 3:54
    
Hi, your last edit removed most of the problem, rendering the queston incomprehensible. –  Johannes Kloos Apr 27 '12 at 12:12
    
@BoyanKlo: You are the one being disrespectful. Zarrax, nor anyone else, is under any obligation to help you, and when they choose to help you they can do so in any manner they want. I have deleted your comments. Any further such comments will result in suspension. –  Zev Chonoles Apr 27 '12 at 15:54
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Sadly, this problem is from a 48 hour exam given at some university in the US. A friend of mine who is a first year PHD student there shared with me his homework and exam problems because there were some interesting problems there. Many of the questions of Boyan Klo are from those homekorks and the last two are for the exam. Maybe I am wrong, but it is very unlikely that someone will post exactly the right problems at exactly the time they were given as homework/exam. I hope I am wrong... –  Beni Bogosel Apr 27 '12 at 16:13
    
@Beni: If you have your friend get in contact with their professor, and the professor asks the moderators to delete the questions until the exam is over, we can do so. –  Zev Chonoles Apr 27 '12 at 16:17
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1 Answer

up vote 3 down vote accepted

Hint: Try a contrapositive proof. If the limit were not true, you'd have an $\epsilon > 0$ and $(x_n,y_n)$ of magnitude at least $n$ such that $f(x_n,y_n) > \epsilon$. Use the uniform continuity to show there's a little disk $B_n$ centered at $(x_n,y_n)$ such that the integral of $f$ over the disk $B_n$ is at least $\epsilon '$ for some fixed $\epsilon '$....

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There are going to be infinitely many $B_n$'s.. so the total integral is infinite. –  Zarrax Apr 26 '12 at 5:13
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You'll have to wait a little longer, I'd prefer to not give away too much of a homework problem. –  Zarrax Apr 26 '12 at 17:00
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@BoyanKlo No one is under any obligation to assist you in your homework. If your mathematical skill level is not enough to comprehend my hint, either find your own solution or seek help elsewhere. People here are volunteering. This means they can help you as little as they want, or not at all. In your case clearly I should have ignored you, which I will do in the future. –  Zarrax Apr 27 '12 at 13:38
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