Definition:Implicit Function

Definition
Consider a (real) function of two independent variables $z = \map f {x, y}$.

Let a relation between $x$ and $y$ be expressed in the form $\map f {x, y} = 0$ defined on some subset of $\R^2$.

If there exists a function:
 * $y = \map g x$

defined on some real interval $\mathbb I$ such that:
 * $\forall x \in \mathbb I: \map f {x, \map g x} = 0$

then the relation $\map f {x, y} = 0$ defines $y$ as an implicit function of $x$.

Also see
For sufficient conditions for the existence of such functions:
 * Implicitly Defined Real-Valued Function
 * Implicit Function Theorem