If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant.

Question

How do I prove that if $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $$f(x)=f(2x)$$ for each $x\in\Bbb{R}$, then $f$ is constant?

Answer 1

If $f(a)\neq f(b)$, then $f(a\,2^{-n})=f(a)$ and $f(b\,2^{-n})=f(b)$ for each $n$. By continuity, $f(0)=f(a)=f(b)$ which is a contradiction.

It is usually appreciated if you also write down your ideas or trials.

Answer 2

The functional equation implies $f\Bigl(\dfrac{x}{2}\Bigr)=f(x)$, hence $f(x)=f\Bigl(\dfrac{x}{2^n}\Bigr)$ for all $n$.