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Math Help - Continuous function

  1. #1
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    Continuous function

    f is a continuous function from R->R and f satisfies f(2x+1)=f(x), then prove that f is constant.
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  2. #2
    A Plied Mathematician
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    Is f also differentiable?
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  3. #3
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    Hi--

    No f is not differentiable. Ok, suppose you have an counterexample then please do provide me with one or else, prove the result with differentiability holding true. As far as i know, i dont think f is differntiable.
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    It doesn't need to be differentiable!

    Hint : show that f is constant on a dense subset of \mathbb{R}.
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  5. #5
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    If it's of any help, f is differentiable at x = 1 with derivative 0, and f(1) = f(0).

     \frac{f(1+2h) - f(1)}{2h} = 0 for all non-zero h.

    And  f(0) = f(1 + 2*0) = f(1) .
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  6. #6
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    hi

    Hi--

    How do we prove it for dense subsets. The only way i can think of is prove this true for rationals since they are dense in R.

    But i have an other idea. Let g(x)=f(x-1)=> g(2x)=f(2x-1)=f(x-1)=g(x). But by doing this repeatedly we can have g(x)->g(0). Does this work.
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