
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
September 28th, 2011, 09:40 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, 09:26 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, 03:15 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, 02:43 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, 08:08 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, 08:25 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, 09:22 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, 09:27 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, 09:37 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 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
integrable functions  kapital  Calculus  10  July 26th, 2012 08:36 AM 
Differentiability of Riemann Integrable Functions  veronicak5678  Real Analysis  1  April 15th, 2012 02:12 PM 
continuous integrable function  alexolympia  Real Analysis  0  October 13th, 2009 07:44 AM 
more continuous functions  julian21  Real Analysis  2  May 9th, 2009 02:45 AM 
Give an example of a sequence of integrable functions that..  nedaiii  Real Analysis  2  February 9th, 2009 04:32 PM 