Well, similar questions have already been asked. But they are not identical and the solution methods offered there are not the same one as here. Anyway, I want to solve functional equation $f(m + f(n)) = f(m) - n$ where $f:\mathbb{Z}\rightarrow \mathbb{Z}$. Somehow I showed that such function doesn't exist and will be very grateful if someone checks my proof. Thanks in advance.
1) $f(f(n)) = f(0 + f(n)) = f(0) - n$. From this we get that $f$ is injective since if $f(m) = f(n)$ then $f(0)-m = f(0)-n$ and thus $m=n$.
2) $f(n+f(0))=f(n)-0=f(n)$ and by injectivity we have $n + f(0) = n$ and $f(0) = 0$.
3) $f(f(n))=f(0)-n=0-n=-n$ and then $f(m-n) = f(m+f(f(n))) = f(m)-f(n)$.
4) $f(-n)=f(0-n)=f(0)-f(n)=0-f(n)=-f(n)$ and then $f(m+n)=f(m-(-n))=f(m)-f(-n)=f(m)-(-f(n))=f(m)+f(n)$.
5) Now it is a standard result that any function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $f(m+n) = f(m) + f(n)$ must be of the form $f(n) = kn$ for some $k \in \mathbb{Z}$. Then on one side $f(f(1))=k^2$ and from above $f(f(1))=-1$. Thus $k^2 = -1$ which doesn't hold for any integer and so no such function $f$ exists.
EDIT. Why duplicate? My question was asked in 2015, and the one you mention is from 2016!