Definition:Implicit Function

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

Let a relation between $x$ and $y$ be expressed in the form $f \left({x, y}\right) = 0$ defined on some interval $\mathbb I$.

If there exists a function:
 * $y = g \left({x}\right)$

defined on $\mathbb I$ such that:
 * $\forall x \in \mathbb I: f \left({x, g \left({x}\right)}\right) = 0$

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

More generally, let:
 * $f: \R^{n + 1} \to \R, \left({x_1, x_2, \ldots, x_n, z}\right) \mapsto f \left({x_1, x_2, \ldots, x_n, z}\right)$

where:
 * $\left({x_1, x_2, \ldots, x_n}\right) \in \R^n, z \in \R$

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

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

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


 * $\forall \left({x_1, x_2, \ldots, x_n}\right) \in S: z = g \left({x_1, x_2, \ldots, x_n}\right) \iff f \left({x_1, x_2, \ldots, x_n, z}\right) = 0$

then the relation $f \left({x_1, x_2, \ldots, x_n, z}\right) = 0$ 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