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August 10th, 2017, 07:22 AM  #1 
Senior Member Joined: Nov 2015 From: Alabama Posts: 140 Thanks: 17 Math Focus: Geometry, Trigonometry, Calculus  What is a "wellbehaved" 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 "wellbehaved" (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 wellbehaved function is and what qualities a wellbehaved function has? Thanks! Jacob Last edited by skipjack; August 11th, 2017 at 09:38 PM. 
August 10th, 2017, 01:30 PM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,649 Thanks: 681 
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.

August 10th, 2017, 02:00 PM  #4 
Senior Member Joined: May 2016 From: USA Posts: 785 Thanks: 312 
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.

August 11th, 2017, 11:39 AM  #5 
Senior Member Joined: Nov 2015 From: Alabama Posts: 140 Thanks: 17 Math Focus: Geometry, Trigonometry, Calculus  
August 11th, 2017, 11:41 AM  #6  
Senior Member Joined: Nov 2015 From: Alabama Posts: 140 Thanks: 17 Math Focus: Geometry, Trigonometry, Calculus  Quote:
 
August 11th, 2017, 12:26 PM  #7 
Senior Member Joined: Oct 2009 Posts: 141 Thanks: 59 
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.

August 12th, 2017, 02:49 AM  #9 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,649 Thanks: 681 
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, 09:05 AM  #10 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 10,472 Thanks: 693  

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