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Math Help - Function

  1. #1
    Junior Member
    Joined
    Aug 2009
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    Function

    Hi everybody,

    f is a function defined from [a;b] to [a;b], such as:

    (\forall (x;t)\in[a;b]^2) |f(x)-f(t)|<|x-t|

    1)- i must show that f is continuous on [a;b]

    2)-And i must show also that f accepts a fixed point on [a;b]

    Can you help me please?

    And thank you anyway.
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  2. #2
    MHF Contributor red_dog's Avatar
    Joined
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    Medgidia, Romania
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    1) Let x_0\in[a,b]. Then |f(x)-f(x_0)|<|x-x_0|

    0\leq\lim_{x\to x_0}|f(x)-f(x_0)|\leq\lim_{x\to x_0}|x-x_0|=0\Rightarrow

    \Rightarrow\lim_{x\to x_0}(f(x)-f(x_0))=0\Rightarrow\lim_{x\to x_0}f(x)=f(x_0)

    Then the function is continuous.

    2) Let g(x)=f(x)-x, \ g:[a,b]\to\mathbb{R}

    g(a)=f(a)-a\geq 0, \ g(b)=f(b)-b\leq 0

    But g is continuous, so exists c\in[a,b] such as g(c)=0\Rightarrow f(c)=c.
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