Example of $f(x) +f(2x) \neq 0$

Question

Given $\lim_{x\to 0}f(x)+f(2x)=0$ and $\lim_{x\to 0}f(x)$ not exist.

Does it imply that $f(x) +f(2x)=0$ and only that?

Can't find examples for the sum not equal to zero.

Thank you.

Answer 1

Try $$f(x) = x + \sin(\pi \log_2(|x|))$$

Answer 2

(Edited) Here is an example: $f(x) = x + \sin(\pi \log_2 |x|), $ for $x\neq 0$ and set $f(0)=1$. You can easily show that $f(x)+f(2x)\neq 0$ holds for all $x\in\mathbb{R}$. However, $\lim_{x\to 0} f(x)+f(2x) = 0,$ while $\lim_{x\to 0} f(x)$ does not exist.