A continuous function that when iterated, becomes eventually constant Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function, and let $c$ be a number. Suppose that for all $x \in \mathbb{R}$, there exists $N_x > 0$ such that $f^n(x) = c$ for all $n \geq N_x$. Is it possible that $f^n$ (this is $f$ composed with itself $n$ times) is not constant for any $n$?
I don't know if this is true or false. So far I haven't made much progress besides trivial stuff (in an example $c$ is the unique fixed point of $f$, we have $f(x) > x$ for all $x > c$ or $f(x) < x$ for all $x > c$, etc..)

Take for example
$$f_k(x) = \begin{cases}0 &\text{ for } x \geq -k \\ x+k & \text{ otherwise}\end{cases}$$
which are all continuous and we have $f_1^n = f_n$ which is not constant.
I hope this helps $\ddot\smile$
