I am studying the implicit function theorem. I have a question about the condition $F_y(x_0,y_0)\neq 0.$
More precisely, let $w=F(x,y)$ be a $C^1$ function on an open rectangle $R=(a,b) \times (c,d)$. Consider $F(x,y)=C,$ where $C \in \mathbb{R}$ is fixed, and $F(x_0,y_0)=C$ for some $(x_0,y_0) \in R.$
By implicit function theorem, if $F_y(x_0,y_0)\neq 0,$ $\exists$ a $C^1$ function $y=y(x)$ near $(x_0,y_0)$ satisfying $F(x,y(x))=C.$
If $\exists$ a differential function $y=y(x)$ near $(x_0,y_0)$ satisfying $F(x,y(x))=C,$ then, by chain rule, $F_x(x_0,y_0)+F_y(x_0,y_0)y'(x_0)=0,$ which implies that the case $F_x(x_0,y_0)\neq 0$ and $F_y(x_0,y_0)=0$ is impossible.
My question: Is it possible that $F_x(x_0,y_0)= F_y(x_0,y_0)=0$?
I would be grateful if you give any comment for my question. Thanks in advance.