Prove/Disprove for $f:\mathbb N\to\mathbb N,$ if $f\circ f$ is periodic then $f$ is periodic

Question

We define periodic according to the standard definition where $f(x)=f(x+p)$, this means that there exists p so that $f(f(x)) = f(f(x)+p)$.

However I am unable to find a counter example or a constructing a formal proof. Any insights are welcome

Answer 1

Hint: Start with

$$f(x)=\begin{cases} 0, & \text {if $x$ is even}\\ 1, & \text {if $x$ is odd} \end{cases} $$

This has $f\circ f = f$ periodic. Now find a small change $f'$ to $f$ such that $f'\circ f' = f$ (so periodic), but $f'$ becomes "slightly " non-periodic.