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Math Help - proof question 4

  1. #1
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    proof question 4

    Suppose that f : R->R is conitnuous and that its image f(R) is bounded. Prove that there is a solution for the equation f(x)=x for x in R
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let m=\inf f(\mathbf{R}), \ M=\sup f(\mathbf{R})\Rightarrow m\leq f(x)\leq M, \ \forall x\in\mathbf{R}

    Let g:\mathbf{R}\to\mathbf{R}, \ g(x)=f(x)-x

    g(m)=f(m)-m\geq 0, \ g(M)=f(M)-M\leq 0

    g continuous \Rightarrow \exists c\in[m,M] such as g(c)=0\Rightarrow f(c)=c
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