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Thread: 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 $\displaystyle f$ is continuous on the closed interval [0,1] and that $\displaystyle 0 \leq f(x) \leq1$ for every $\displaystyle x$ in [0,1]. Show that there exists a real number $\displaystyle c \in [0,1]$ such that $\displaystyle f(c) = c.$
    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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    $\displaystyle g(0)=f(0)\geq 0$

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

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

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