Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$:$f(f(f(x)+y)+y)=x+y+f(y)$

Question

Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$: $f(f(f(x)+y)+y)=x+y+f(y)$

I got the following: (1)$f$ is injective (2) $f(0)=0$

(3)$f(f(f(x)))=x$

But then how to proceed?

Answer 1

If you set $y=-f(x)$, then you get

$$f(f(0)-f(x))=x-f(x)+f(-f(x))$$

Now I'm not sure how you got $f(0)=0$, but if indeed you did, then we have

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

which implies $f(x)=x$ for all $x$.