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Math Help - Continuous Functions

  1. #1
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    Continuous Functions

    How would I go about doing this question:

    Suppose\ that\ f\ and\ g are\ continuous\ functions\ on\ the\ closed\ interval [0,1]\ and\ that\ 0 \leq f(x) \leq 1\ for\ every\ x\ in\ [0,1]. Show\ that\ there\ exists\ a\ real\ number c \in [0,1]\ such\ that\ f(c)=c.

    Would you do this question using the intermediate value theorem to g(x)=f(x)-x
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  2. #2
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    Quote Originally Posted by acevipa View Post
    How would I go about doing this question:

    Suppose\ that\ f\ and\ g are\ continuous\ functions\ on\ the\ closed\ interval [0,1]\ and\ that\ 0 \leq f(x) \leq 1\ for\ every\ x\ in\ [0,1]. Show\ that\ there\ exists\ a\ real\ number c \in [0,1]\ such\ that\ f(c)=c.

    Would you do this question using the intermediate value theorem to g(x)=f(x)-x
    Yes, that's a good idea. I suggest you look at g(0) and g(1).
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    Yes, that's a good idea. I suggest you look at g(0) and g(1).
    Would you do it like this

    \frac{g(1)-g(0)}{1-0} = g'(c)

    \frac{f(1)-1-f(0)+0}{1-0}=f'(c)-1

    Given\ that\ f(1)=1\ and\ f(0)=0

    \frac{1-1-0}{1}=f'(c)-1

    \frac{0}{1}=f'(c)-1

    f'(c)-1=0

    f'(c)=1\ \forall c \in (0,1)

    What do I do from here?
    Last edited by acevipa; April 28th 2010 at 03:31 AM.
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