TITLE: Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
QUESTION [29 upvotes]: If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples?

REPLY [14 votes]: Counterexample: $f(x)=\sin(\log_a(x))$, select basis $a$ so that $\log_a(2)=2\pi$, that is, $a ^{2\pi}=2$.