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!

One more example
$$f(x) = \sin(2\pi\log_2x)$$
$$f(2x) = \sin(2\pi\log_2(2x)) = \sin(2\pi(1 + \log_2x)) = \sin(2\pi\log_2x) = f(x)$$
