Give an example of a function $f$ satisfying $\lim_{x\to 0}(f(x)f(2x))=0$,but $\lim_{x\to 0}f(x)$ does not exists

Question

Question:

Give an example of a function $f$ satisfying the condition $$\lim_{x\to 0}(f(x)f(2x))=0$$ and such that $$\lim_{x\to 0}f(x)$$ does not exists.

I think this question have many example. But now I can't to find any example.
Thank you

Answer 1

We put $f(x)=\begin{cases} 1 &\mbox{if } x=\frac{1}{3^n},\quad n\in\mathbb{N} \\ 0 & \mbox{otherwise } . \end{cases} $

We have $f(x)f(2x)=0$ for every $x\in\mathbb{R}$ and $f$ does not have a limit at $0$.

Answer 2

Take $$f=\chi_{A}$$ where $$A=\bigcup_{k=1}^{\infty}A_k,\qquad A_k=\left(\frac{-1}{2^{2k}},\frac{-1}{2^{2k+1}}\right)\cup\left(\frac{1}{2^{2k+1}},\frac{1}{2^{2k}}\right).$$ Then $$\lim_{x\to0}f(x)$$ does not exist since $f(x)=0$ for some $x$ arbitrarily close to $0$ and $f(x)=1$ for some $x$ arbitrarily close to $0$. However, if $x\in A$ then $2x\in A^{c}$ so

$$\lim_{x\to0}f(x)f(2x)=0$$