 Implicit function theorem

If $F(x_0,y_0)=0$ and the partial derivative with respect to $y$ is invertible at $(x_0,y_0)$, then the nearby zero set is the graph $y=g(x)$ of a continuously differentiable function.

The Implicit Function Theorem is a fundamental result in calculus and differential topology that provides conditions under which a relation defines a function implicitly.