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Math Help - differentiable function and fixed point

  1. #1
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    Poland
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    differentiable function and fixed point

    HI! I've got some problem with this:

    Prove that if f: R->R is differentiable and |f '(x)|<a<1 then f(x) has a fixed point.
    Give and example of a function that |f '(x)|<1 doesn't guarantee the existance of fixed point.

    Thanks in advence for any kind of help.
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  2. #2
    Senior Member roninpro's Avatar
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    Are you familiar with the Contraction Mapping Theorem? If you can show that your map f:\mathbb{R}\to \mathbb{R} satisfies |f(x)-f(y)|\leq K|x-y| where K<1, then it is guaranteed to have a unique fixed point. To connect your derivative to this condition, I suggest using the Mean Value Theorem.

    For a counterexample, you need to find a function that never intersects g(x)=x but has a slope very close to 1 at all times.
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  3. #3
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    I figured it out that function with that conditions is Lipschitz continuous, just using the Mean Value Theorem, with K<1. But the problem is I can't use the Contraction Mapping Theorem, you've mentioned, because I haven't got it on my calculus lessons. Are there any other possible ways? Maybe the Intermediate value theorem can help?
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