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Math Help - Help with Continuity Proof

  1. #1
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    Help with Continuity Proof

    I think I kinda know the direction this proof needs to go but I'm not sure how to word it all. Any help would be appreciated.

    Let f:[0,1]\rightarrow [0,1] be continuous. Prove that there is a c\in [0,1], such that f(c)=c. [Hint: consider g(x)=x-g(x)

    I'm thinking a proof by contradiction might be apprpriate here. Assume that there is no c that works and then find a contradiction by using the hint somehow????
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    For which x in [0,1] g(x)>0, for which x in [0,1] g(x)<0 ?

    Now use Intermediate value theorem - Wikipedia, the free encyclopedia
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Suppose that f(x)\ne x for all x\in[0,1] and consider \displaystyle g:[0,1]\to\{-1,1\}:\frac{f(x)-x}{|f(x)-x|}. Since the denominator is non-zero this is continuous and surjective. But, this implies that \{-1,1\} is connected....
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