Definition:Implicit Function

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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: