Suppose a function $f: \mathbb{R}\rightarrow\mathbb{R}$ with the property
$$ \lim_{n\rightarrow\infty}\; \underbrace{f\circ f\circ \dots \circ f}_{\mathrm{n \;times}}(x) = c = \mathrm{const.} $$
i.e. it converges in the limit to a finite number when repeatedly applied to itself. Also $\exists\, a, b \in \mathbb{R}, s.t. \; f(a)\neq f(b)$ should hold (i.e. only non-constant functions).
As pointed out in the comments $f(x) = \sqrt{|x|}$ is not a valid example. So I don't have any example function with that property but $f(x) = \frac{1}{x}$ on the other hand is bound though it doesn't converge.
Now I am interested in the following aspects:
- Can one identify a subset of all functions for which this property is present (for any value of $c$)?
- Do such functions have other specific properties that are perhaps common among all of them (and related to the value of $c$)?
- Do functions exist for which $c = 0$ (perhaps $f(x) = \sin(x)$ but I'm not sure how to approach this)?
- Do functions exist for which $c \neq 0$?
I am particularly interested in the case $c = 0$.