Suppose $\displaystyle F(x,y)$ is a $\displaystyle C^{1}$ function such that $\displaystyle F(0,0) = 0$. What conditions on $\displaystyle F$ will guarantee that the equation $\displaystyle F(F(x,y),y) = 0$ can be solved for $\displaystyle y$ as a $\displaystyle C^{1}$ function of $\displaystyle x$ near $\displaystyle (0,0)$.

So obviously $\displaystyle \partial_{2}F(0,0) \ne 0$ is one condition just by what the hypothesis of the implicit function theorem uses (I think this part is obvious, but I could be wrong and if I am, please explain why this is part of a condition that will guarantee what we want). However, another condition that is needed that will guarantee what we want is $\displaystyle \partial_{1}F(0,0) \ne -1$, and I just don't see how to arrive at that conclusion. Any help would be appreciated.