Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
add comment

2 Answers

up vote 32 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.

share|improve this answer
add comment

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

share|improve this answer
1  
It's not continuous at zero –  Quark May 24 '13 at 17:14
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.