Definition:Implicit Function/General

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

be a real-valued function on $\R^{n + 1}$, 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 $x_1, x_2, \ldots, x_n$.