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Math Help - Detrmine all solutions

  1. #1
    Super Member dhiab's Avatar
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    Detrmine all solutions

    Detrmine all solutions  f:R\to R of the functional equation:
     <br />
f(2x+1)=f(x)<br />
    which are continuos at -1.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by dhiab View Post
    Detrmine all solutions  f:R\to R of the functional equation:
     <br />
f(2x+1)=f(x)<br />
    which are continuos at -1.
    Define a sequence \left\{x_n\right\}_{n\in\mathbb{N}} by x_1=x,\text{ }2x_{n+1}+1=x_1. It is relatively easy to prove that \lim\text{ }x_n=-1. Also, consider that f(x_n)=f(2x_{n+1}+1)=f(x_{n+1}) from where by induction we may conclude that f(x)=f(x_n). Also, since x_n\to -1 and f is continuous at -1 we see that f(-1)=\lim\text{ }f(x_n)=f(x). Thus, f is constant.
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