Find all functions $f$ and $g$ for which $f(x+y) = g(xy)$.
Is there anything wrong with this?
We see that $f(1) =g(0)$ and $f(0) = g(0)$ so $f(1) = f(0)$. Also, $f(x) = g(0)$ and therefore $f(x) = f(1)$ and so $f$ must be constant? Similarly $g(x) = f(x+1) = f(1)$?
Nothing is wrong with your reasoning. Your presentation is somewhat unclear though. For instance, I assume your reasoning is to say $$f(1+0)=g(1\cdot 0)$$ but you only write $f(1)=g(0)$, which is not as clear. One might also note that one can combine the steps by saying $$f(x+0)=g(x\cdot 0)=g(0)=g(y\cdot 0)=f(y+0)$$ which gives $f$ to be constant. Then $g(x\cdot 1)=f(x+1)$ gives $g$ to be constant.
