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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?

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up vote 38 down vote accepted

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.

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

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It's not continuous at zero – Quark May 24 '13 at 17:14
It is continuous everywhere else, which I guess is the best you can do, given the argument above. – Nishant Jul 14 '14 at 21:09

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