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 $$x$$.