Implicit function theorems and differentiable maps

Abstract

Suppose we are given a relation in ℝ₂ of the form

$$F\left( {x,{\rm{ }}y} \right){\rm{ }} = {\rm{ }}0$$

((14.1))

Then to each value of x there may correspond one or more values of y which satisfy (14.1)—or there may be no values of y which do so. If I = {x: x ₀ — h < x < x ₀ + h} is an interval such that for each x ∈ I there is exactly one value of y satisfying (14.1), then we say that F(x, y) = 0 defines y as a function of x implicitly on I. Denoting this function by f, we have F[x, f(x)] = 0 for x on I.