Prove or disprove $f(n) = O(f(2n))$ I wonder how to to prove or disprove that $f(n) = O(f(2n))$
I have tried many function, and think it is right, but still don't have any idea how to prove.
Could anyone give me a hint about it?

If
\begin{equation}
f(n)=\begin{cases}1 & n\text{ even,}\\ n& n\text{ odd,}\end{cases}
\end{equation}
then
\begin{equation}
\mathcal{O}(f(2n))=\mathcal{O}(1)\text{,}
\end{equation}
but $f(n)$ is not asymptotically constant.
