Question About Implicit Function Therorem

Question

I have a function $F(x,y)=0$. Based on the implicit function theorem, suppose I can derive $y=f_{1}(x)$ in a neighborhood of the point $x_{1}$, and I can also derive $y=f_{2}(x)$ in a neighborhood of the point $x_{2}$. Now I am wondering whether $f_{1}(x)=f_{2}(x)$ ?

Answer 1

You can write $y=f_1(x)$ in a neighborhood $U_1$ of $x_1$ only if $$\frac{\partial F(x_1,y_1)}{\partial y}\neq 0,$$ where $y_1$ is a quantity satisfying $F(x_1,y_1)=0$. You can do the same for a point $x_2$ if the condition holds there too. If the neighborhoods of corresponding to $x_1$ and $x_2$ overlap, you have $f_1(x)=f_2(x)$ is the overlap, as long as $f(U_1)$ contains $y_2$ and $f(U_2)$ contains $y_1$.