\begin{definition}[Definition:Implicit Function/General]
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 $x_1, x_2, \ldots, x_n$.
\end{definition}