Definition:Implicit Function

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 $$x$$.