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 August 10th, 2017, 08:22 AM #1 Senior Member     Joined: Nov 2015 From: United States of America Posts: 162 Thanks: 21 Math Focus: Calculus and Physics 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.
 August 10th, 2017, 01:54 PM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,640 Thanks: 959 Math Focus: Elementary mathematics and beyond No explanation (and I am not familiar with "calculus 3") but this page might help. Thanks from SenatorArmstrong
 August 10th, 2017, 02:30 PM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 2,820 Thanks: 750 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. Thanks from SenatorArmstrong
 August 10th, 2017, 03:00 PM #4 Senior Member   Joined: May 2016 From: USA Posts: 823 Thanks: 335 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. Thanks from SenatorArmstrong
August 11th, 2017, 12:39 PM   #5
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 Originally Posted by JeffM1 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.

August 11th, 2017, 12:41 PM   #6
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 Originally Posted by Country Boy 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.

 August 11th, 2017, 01:26 PM #7 Senior Member   Joined: Oct 2009 Posts: 142 Thanks: 60 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. Thanks from greg1313 and SenatorArmstrong
 August 11th, 2017, 06:40 PM #8 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,640 Thanks: 959 Math Focus: Elementary mathematics and beyond For more information on Jordan content (Jordan measure), see here. Thanks from SenatorArmstrong
 August 12th, 2017, 03:49 AM #9 Math Team   Joined: Jan 2015 From: Alabama Posts: 2,820 Thanks: 750 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.
August 12th, 2017, 10:05 AM   #10
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Quote:
 Originally Posted by SenatorArmstrong 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...

,

### well behaved function examples

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