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September 28th, 2011, 10:40 PM  #1 
Member Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0  On Continuous, Integrable Functions on [0,1]
Let be fixed. Determine all nonnegative continuous functions which satisfy the following three conditions: 1.) , 2.) , 3.) . First of all, I have no idea how to begin this problem. We were told that we can use general facts from Calculus without rigor to prove this, but even with that I cannot seen to find a place to begin. 
September 29th, 2011, 10:26 AM  #2 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  Re: On Continuous, Integrable Functions on [0,1]
You would agree, I hope, that if is nonnegative then the function is nonnegative everywhere. What happens when you evaluate its integral over [0,1], and what does this tell you about 
September 29th, 2011, 04:15 PM  #3 
Member Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0  Re: On Continuous, Integrable Functions on [0,1]
I believe it is nonegative, but I am not really sure I follow you on the rest. The trouble I seem to be having is integrating an undefined function f(x) over a definite integral. 
September 30th, 2011, 03:43 AM  #4 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  Re: On Continuous, Integrable Functions on [0,1]
If we expand we get You know what you get when you integrate and over the interval [0,1], so you should be able to work out what the integral of the above function is over the same interval without knowing anything more explicit about f.

September 30th, 2011, 09:08 AM  #5 
Newbie Joined: Dec 2008 From: Copenhagen, Denmark Posts: 29 Thanks: 0  Re: On Continuous, Integrable Functions on [0,1]
Multiply 1) with a squared and 2) with a. Then set them equal and you reach the point mattpi is making.

September 30th, 2011, 09:25 AM  #6 
Member Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0  Re: On Continuous, Integrable Functions on [0,1]
Oh, I see what you are saying. Are we looking for a contradiction? 
September 30th, 2011, 10:22 AM  #7 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  Re: On Continuous, Integrable Functions on [0,1]
What do we know about a continuous function if it integrates to zero over an interval, and is nonnegative in that same interval?

September 30th, 2011, 10:27 AM  #8 
Newbie Joined: Dec 2008 From: Copenhagen, Denmark Posts: 29 Thanks: 0  Re: On Continuous, Integrable Functions on [0,1]
I wouldn't call it a contradiction, but an implication. Let's see, we have thus and which means that Knowing that is nonnegative for all , this last integral equation implies something about . What does it imply? 
September 30th, 2011, 10:37 AM  #9 
Member Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0  Re: On Continuous, Integrable Functions on [0,1]
It must be that f is the zero function. But that contradicts 1) So no such functions exist. 

Tags 
continuous, functions, integrable 
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