Example of a non-trivial function such that $f(2x)=f(x)$ Could you give an example of a non-constant function $f$ such that 
$$
f(x) = f(2x).
$$ 
The one that I can think of is the trivial one, namely $\chi_{\mathbb{Q}}$, the characteristic function on the rationals.
I am wondering if there is any other such function other than this one. TQVM!

A less trivial example: define $f(0)$ to be any thing and for $x\neq 0$, define $f(x)=g(x/|x|)$ for any function $g:\mathbb{R}-\{0\}\to\mathbb{R}$.
