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August 10th, 2017, 08:22 AM   #1
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What is a "well-behaved" function?

Hello,

I am studying double integrals over general regions. My textbook was discussing the boundary curve of the general region you are integrating. Such as I have a general region D and I make a rectangle R around it. My textbook says

"if $f$ is continuous on D and the boundary curve is "well-behaved" (in a sense that is outside the scope of this book), then it can be shown that the double integral exists."

By any chance, could someone explain to me (a person with knowledge up to calculus 3) what a well-behaved function is and what qualities a well-behaved function has?

Thanks!

Jacob

Last edited by skipjack; August 11th, 2017 at 10:38 PM.
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August 10th, 2017, 01:54 PM   #2
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No explanation (and I am not familiar with "calculus 3") but this page might help.
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August 10th, 2017, 02:30 PM   #3
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Exactly what "well behaved curve" means depends on the context. Here it sounds like it means there are no "cusps"- the curve has a continuous derivative at every point. It probably also requires that the curve be a "Jordan curve"- that it does not cross itself.
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August 10th, 2017, 03:00 PM   #4
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What it means is that there are exceptions, but it would take 50 pages to explain them, and nothing we are going to work on is one of those exceptions, and nothing you are likely to run into is one of them.
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August 11th, 2017, 12:39 PM   #5
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Quote:
Originally Posted by JeffM1 View Post
What it means is that there are exceptions, but it would take 50 pages to explain them, and nothing we are going to work on is one of those exceptions, and nothing you are likely to run into is one of them.
Ha ha. Thank you for the response, Jeff.
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August 11th, 2017, 12:41 PM   #6
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Exactly what "well behaved curve" means depends on the context. Here it sounds like it means there are no "cusps"- the curve has a continuous derivative at every point. It probably also requires that the curve be a "Jordan curve"- that it does not cross itself.
Going to read up on Jordan Curve theorem. Thanks.
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August 11th, 2017, 01:26 PM   #7
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One very popular condition is that "the boundary has Jordan content 0". I'm not going to explain what this means, but this is likely the condition that your book wants. It is very restrictive however and it can be generalized a great deal.
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August 11th, 2017, 06:40 PM   #8
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For more information on Jordan content (Jordan measure), see here.
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August 12th, 2017, 03:49 AM   #9
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I notice, by the way, that while the title of this thread asks about a 'well behaved function', in the body of your post you ask about a 'well behaved curve'. Those are very different things.
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August 12th, 2017, 10:05 AM   #10
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What is a "well-behaved" function?
Jacob
A "well-behaved" function is a function that listens to his
parents and teachers, plus eats all his vegetables...
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