Understanding implicit function theorem Let's say we have $F(x, y) = x^2 +y^2 - 1$ and we want to apply the implicit function theorem. Then, after some calculations we see that for $y \neq 0$ we can indeed apply the theorem which means that there exist two open sets $U:=]-1, 1[$ of $x$,  and $V:=]0, 2[$ of $y$,  and a function $g: U \to V$ such that $F(x, g(x)) = 0$ $\forall$ $x \in U$. In this particular case we can explicitly write down $g$ by $g(x)=\sqrt{1-x^2}$ where $g(x)=y$ and $y>0$. 
What I don't understand is that even if  $y= 0$, $x=1$ and $x=-1$ are allowed $g$, still satisfies $F(x, g(x)) = 0$ $\forall$ $x \in U$. So is it right that if the conditions of the implicit function theorem are not satisfied this does not imply that there doesn't exist such a function at all?
Maybe I didn't understand the theorem correctly... any comments or help are welcome
 A: I'll make two remarks:


*

*It's true that that $F(x,g(x))=0$ for all $x\in [-1,1]$, but the idea of the implicit function theorem is that it gives sufficient conditions for a point in the level set of $F$ to have an open neighborhood over which the level set can be written locally as the graph of a function $g$. In this example, the points $(-1,0)$ and $(1,0)$ do not have any open neighborhoods in which the set $\{F(x,y)=0\}=\{x^2+y^2-1=0\}$ can be written as the graph of a function of $x$. However, $\frac{\partial F}{\partial x}\neq0$ at both of these points, which means there are functions $h$ such that $F(h(y),y)=0$ for neighborhoods of each of these points, respectively.

*You are correct that the converse of the implicit function theorem does not necessary hold. For an easy example, the derivative of the function $F(x,y)=x^3-y^3$ vanishes at $(0,0)$, but its zero level set can be written globally as the graph of the function $g(x)=x$ or $g(y)=y$.
In conclusion, you should understand the implicit function as providing a sufficient, but not necessary, condition to be able to express $\{F(x,y)=0\}$ locally as the graph of another smooth function.
