Let $f:\mathbb{R}^2 \to \mathbb{R}$ be $C^1$ such that $f(0,0)=0$. Find conditions over $\frac{\partial f }{\partial x}(0,0)$ and $\frac{\partial f }{\partial y}(0,0)$ so that equation $f(x+f(x,y), y - f(x,y))= 0 $
Variable $x$ can be written like: $x=g(y)$ such that $0=g(0)$
I get that conditions should be (by differentiating x): $\frac{\partial f}{\partial x}(0,0)=0$ or $\frac{\partial f }{\partial y}(0,0)= 1+ \frac{\partial f}{\partial x}(0,0)$
The thing is that I don't know if that is correct, any idea?