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 $x$.