Implicit Functions

Let F(x, y) be a function of two real variables x, y. When we consider the equation $$\displaystyle F(x, y)=0\, $$ and we fix a value of x, there could be either no value of y solving the equation, or exactly one y, or even several values of y satisfying (11.1). It is then of interest to establish under which conditions Eq. (11.1) allows to associate with x a unique value of y, i.e. when y can be expressed as a function of x, at least for some values of the variables. Said more precisely, we shall determine conditions for the existence of an interval $$I\subseteq {\mathbb {R}}$$ and of a unique function $$f:I \rightarrow {\mathbb {R}}$$ such that $$\displaystyle F(x, f(x)) = 0\,,\qquad \forall \,x \in I. $$