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September 28th, 2011, 09:40 PM   #1
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On Continuous, Integrable Functions on [0,1]

Let be fixed.
Determine all non-negative 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.
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September 29th, 2011, 09:26 AM   #2
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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
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September 29th, 2011, 03:15 PM   #3
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Re: On Continuous, Integrable Functions on [0,1]

I believe it is no-negative, 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.
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September 30th, 2011, 02:43 AM   #4
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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.
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September 30th, 2011, 08:08 AM   #5
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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.
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September 30th, 2011, 08:25 AM   #6
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Re: On Continuous, Integrable Functions on [0,1]

Oh, I see what you are saying.

Are we looking for a contradiction?
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September 30th, 2011, 09:22 AM   #7
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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 non-negative in that same interval?
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September 30th, 2011, 09:27 AM   #8
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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 non-negative for all , this last integral equation implies something about . What does it imply?
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September 30th, 2011, 09:37 AM   #9
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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.
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