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Math Help - Intermediate Value Theorem

  1. #1
    Junior Member
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    Intermediate Value Theorem

    Hi,for this question I can visualize what it is asking but I'm having troubles writing a formal proof for it.

    Suppose that f is continuous on the closed interval [0,1] and that 0 \leq f(x) \leq1 for every x in [0,1]. Show that there exists a real number c \in [0,1] such that  f(c) = c.
    I was thinking of applying the intermediate value theorem to g(x) = f(x) - x, but I just keep getting stuck.

    Any help will be highly appreciated
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  2. #2
    MHF Contributor red_dog's Avatar
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    g(0)=f(0)\geq 0

    g(1)=f(1)-1\leq 0

    g is also continuous and g(0)g(1)\leq 0

    so \exists c\in[0,1] such as g(c)=0\Rightarrow f(c)=c
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