Calculus function create

Question

Create an example of a function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(f(\mathbb{R}))) = f(f(\mathbb{R})) \neq f(\mathbb{R})$.

Answer 1

Lots of functions work.
Figure out what the conditions imply. Try to modify when you get stuck.
This is a good question to play around with, so I don't want you to just see the answer. (But, given how this site works, just saying this will lead to negative comments, so I gave an explicit construction hidden below)

$f (x) = \begin{cases} 0 & x \text{ rational} \\ 1 & x \text{ irrational} \end{cases} $