Detrmine all solutions

**dhiab** · February 2nd 2010, 06:52 AM

Detrmine all solutions $f:R\to R$ of the functional equation:
$<br /> f(2x+1)=f(x)<br />$
which are continuos at -1.

**Drexel28** · February 2nd 2010, 01:32 PM

Originally Posted by dhiab

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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