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

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?

• Are there any assumptions on $f$? Is it continuous, differentiable, anything? Aug 22, 2018 at 10:01
• No there is no assumption, but if with an assumption we can get a partial result it would be nice. Aug 22, 2018 at 10:06
• Note that if $f$ has a fixed point, then it is $f(0)$. Aug 22, 2018 at 10:15

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 $f(f(x)-x)=f(1-x)=-1=f(1)=f(f(x))$.