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 $f: \R^{n+1} \to \R, \left({x_1,x_2,\cdots,x_n,z}\right) \mapsto f\left({x_1,x_2,\cdots,x_n,z}\right)$, where $\left({x_1, x_2, \cdots, x_n}\right) \in \R^n, z \in \R$.

Let a relation between $x_1,x_2,\cdots,x_n$ and $z$ be expressed in the form $f \left({x_1,x_2,\cdots,x_n,z}\right) = 0$ defined on some subset of $S \subseteq \R^n$.

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

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

then the relation $f\left({x_1,x_2,\cdots,x_n,z}\right) = 0$ defines $z$ as an implicitly defined function of $x_1,x_2,\cdots,x_n$.

Sufficient conditions for the existence of such functions are given in Implicitly Defined Real-Valued Function and the Implicit Function Theorem.