Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
Solution 1:
For all $x\in\mathbb{R}$ and $n\in\mathbb{Z}$, $f(x)=f(x/2^n)$, so $f(x)=\lim_{n\to\infty}f(x/2^n)=f(0)$ by continuity.
Solution 2:
Counterexample: $f(x)=\sin(\log_a(x))$, select basis $a$ so that $\log_a(2)=2\pi$, that is, $a ^{2\pi}=2$.