TITLE: Is it true that $f$ must be constant?
QUESTION [8 upvotes]: Let $f: \Bbb R \to \Bbb R$ be a function which satisfies $f(x) = f(2x-1)$ for all $x \in \Bbb R$ and $f$ is continuous at $x=1$. Is it true that $f$ must be constant?

REPLY [8 votes]: Remove a bit of obfuscation: Let $g(x)=f(1+x)$. Then $g$ is continuous at $x=0$ and $g(x)=g(2x)$. Now it is clearer how to proceed (though you can work with $f$ itself): By induction $g(2^nx)=g(x)$ for all $n\in\Bbb Z$. Then for any $x$, 
$$g(x)=\lim_{n\to\infty}g(x)= \lim_{n\to\infty}g(2^{-n}x)=g(0)$$