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 interval $\mathbb I$.

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

defined on $\mathbb I$ 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 $x$.

More generally, let:
 * $f: \R^{n + 1} \to \R, \tuple {x_1, x_2, \ldots, x_n, z} \mapsto \map f {x_1, x_2, \ldots, x_n, z}$

where:
 * $\tuple {x_1, x_2, \ldots, x_n} \in \R^n, z \in \R$

Let a relation between $x_1, x_2, \ldots, x_n$ and $z$ be expressed in the form:
 * $\map f {x_1, x_2, \ldots, x_n, z} = 0$

defined on some subset $S \subseteq \R^n$.

If there exists a function $g: S \to \R$ such that:


 * $\forall \tuple {x_1, x_2, \ldots, x_n} \in S: z = \map g {x_1, x_2, \ldots, x_n} \iff \map f {x_1, x_2, \ldots, x_n, z} = 0$

then the relation $\map f {x_1, x_2, \ldots, x_n, z} = 0$ defines $z$ as an implicitly defined function of $x_1, x_2, \ldots, x_n$.

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