Functional equation $f(f(x)-x)=f(f(x))$

Question

Consider the following functional equation:

$$f(f(x)-x)=f(f(x))$$

where $f: \mathbb{R} \rightarrow \mathbb{R}$.

Obviously $f(x)$ cannot be inverted.

One solution is $f(x) = k$ where $k \in \mathbb{R}$ is a constant.

Are there any other solution?

Answer 1

Define $f(x)$ by

$$ f(x)= \begin{cases} \ \ \,1&\text{if $x<0$} \\ \ \ \,0&\text{if $x=0$} \\ -1&\text{if $x>0$} \\ \end{cases} $$ The equation holds for $x=0$ as $f(f(0)-0)=f(f(0))$.

The equation holds for $x>0$ as $f(f(x)-x)=f(-1-x)=1=f(-1)=f(f(x))$.

The equation hold for $x<0$ as $f(f(x)-x)=f(1-x)=-1=f(1)=f(f(x))$.