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Intermediate value theorem problem

  1. Nov 23, 2008 #1
    1. How can you prove if F is continuous, then there exists a fixed point of F in [0,1]?



    I know F:[0,1] ---> [0,1] an bijective, but what is f(c)=c mean?
     
  2. jcsd
  3. Nov 23, 2008 #2

    morphism

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    Re: Stuck

    Have you tried drawing a picture? It might be enlightening.
     
  4. Nov 23, 2008 #3

    Dick

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    Re: Stuck

    I think it means more or less exactly what it says. There is a c in [0,1] such that F(c)=c. Use the intermediate value theorem. If F is bijective there is an a such that F(a)=0 and a b such that F(b)=1. What happens in between? Apply the IVT to F(x)-x on the interval [a,b].
     
  5. Nov 23, 2008 #4
    Re: Stuck

    Does that mean I need to pick a point like .5, which is between [0,1].

    Also, would there be any point where F isn't continuous and a fixed point may not exist.

    Thanks guys
     
  6. Nov 23, 2008 #5

    Dick

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    Re: Stuck

    I don't think you really understood what I wrote. As morphism said, draw a picture. Look up the IVT.
     
  7. Nov 24, 2008 #6

    HallsofIvy

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    Re: Stuck

    No, you can't "pick" a point. And you are told that the function is continuous. Why are you asking if it isn't?

    If F is from [0, 1] what can you say about F(0)? What can you say about F(1)?

    What can you say about 0- F(0) and 1- F(1)?
     
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