Inverse Function Theorem - Statement of The Theorem

Statement of The Theorem

For functions of a single variable, the theorem states that if ƒ is a continuously differentiable function with nonzero derivative at the point a, then ƒ is invertible in a neighborhood of a, the inverse is continuously differentiable, and

where b = ƒ(a).

For functions of more than one variable, the theorem states that if the total derivative of a continuously differentiable function F defined from an open set U of Rn into Rn is invertible at a point p (i.e., the Jacobian determinant of F at p is non-zero), then F is an invertible function near p. That is, an inverse function to F exists in some neighborhood of F(p). Moreover, the inverse function is also continuously differentiable. In the infinite dimensional case it is required that the Fréchet derivative have a bounded inverse at p.

Finally, the theorem says that

where denotes matrix inverse and is the Jacobian matrix of the function G at the point q.

This formula can also be derived from the chain rule. The chain rule states that for functions G and H which have total derivatives at H(p) and p respectively,

Letting G be F -1 and H be F, is the identity function, whose Jacobian matrix is also the identity. In this special case, the formula above can be solved for . Note that the chain rule assumes the existence of total derivative of the inside function H, while the inverse function theorem proves that F -1 has a total derivative at p.

The existence of an inverse function to F is equivalent to saying that the system of n equations y_i = F_j(x₁,...,x_n) can be solved for x₁,...,x_n in terms of y₁,...,y_n if we restrict x and y to small enough neighborhoods of p and F(p), respectively.