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Math Help - Continuity problem

  1. #1
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    Continuity problem

    Hello MHF, please help on this problem; (and by the way, we only did the continuity lesson )
    We are given :
    f:[0,1]\rightarrow [0,1]
    g:[0,1]\rightarrow [0,1]
    both f and g are continuious on  [0,1]
    f(g(x)) = g(f(x))
    If f(a) = a then f(g(a)) = g(a)

    Show that \exists \alpha \in [0,1]  f(\alpha) = g(\alpha)

    P.S : use proof by contradiction
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  2. #2
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    Re: Continuity problem

    I tried to do this
    let h(x) = f(x) - g(x)
    for every x in [0,1] h(x)≠0
    then for every x in [0,1] h(x) > 0 ( or < 0)
    but after this i could't find a contradiction
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  3. #3
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    Re: Continuity problem

    Please I need to solve it before Monday morning.
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  4. #4
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    Re: Continuity problem

    Hi,
    Since you need this by Monday morning, I can only assume it's either an exam question or homework for credit. So here are some hints:
    Let a be a fixed point of f (f(a)=a) -- you need to prove why a exists. Now consider the sequence x1 = a and xn+1 = g(xn).

    By the way, the above leads to a direct proof, not a proof by contradiction.
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  5. #5
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    Re: Continuity problem

    To start you off, suppose without loss of generality
     f(x)>g(x) \quad \text{ for an open interval in } [0,1]
    You'll get a contradiction along the way.

    Note: Using the auxiliary function h you defined, could be used for a direct proof, but at that point you guys are assumed to know Intermediate Value Theorem, etc.

    Disclaimer: If my line of thought is wrong or it has missing holes, please correct and/or inform me.
    Last edited by chen09; September 28th 2013 at 09:21 PM.
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  6. #6
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    Re: Continuity problem

    Hi again,
    If you're still interested after your deadline, here's a complete solution:

    Continuity problem-mhfcalc18.png

    Chen, I'm interested in seeing a proof along the lines suggested by your post. Could you provide such?
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