Find all solutions to the functional equation F:R->R $$f(x+f(x))=f(x)+x$$ I have no idea how to solve this. I can substitute $x=0$ to obtain $f(f(0))=f(0)$. But other than that I can't make any progress. There are two obvious solutions, namely $f(x)=x$ and $f(x)=-x$. But are there any other solutions?

Thanks in advance.

Find all solutions to the functional equation $f:\mathbb{R}\rightarrow \mathbb{R}$ $$f(x+f(x))=f(x)+x$$ I have no idea how to solve this. I can substitute $x=0$ to obtain $f(f(0))=f(0)$. But other than that I can't make any progress. There are two obvious solutions, namely $f(x)=x$ and $f(x)=-x$. But are there any other solutions?

Thanks in advance.