# Math Help - Detrmine all solutions

1. ## Detrmine all solutions

Detrmine all solutions $f:R\to R$ of the functional equation:
$
f(2x+1)=f(x)
$

which are continuos at -1.

2. Originally Posted by dhiab
Detrmine all solutions $f:R\to R$ of the functional equation:
$
f(2x+1)=f(x)
$

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.