3.22 Implicit functions

For $f(x,y)$ a differentiable function of two variables, the formula $f(x,y)=0$ defines a curve $C$ in the plane; often we say that $y$ is an implicit function of $x$. More precisely, given $(a,b)$ such that $f(a,b)=0$ and $f_{y}(a,b)\neq 0$, the set of $(x,y)$ near to $(a,b)$ and such that $f(x,y)=0$ will generally form a graph of a function $y(x)$. Sometimes we can solve $f(x,y)=0$ for $y$, but usually we cannot.