Inverse Function Theorem

The inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: its derivative is continuous and non-zero at the point.

The theorem also gives a formula for the derivative of the inverse function: In single variable calculus, the inverse rule states that .

In Multi-variate Context

A function has an inverse if and only if and for all and .

Computing inverse of a non-linear multi-variate functions is difficult. However, we can find a local inverse via Taylor expansion and linearization.

finding nonlinear inverses

Given

It is hard to find an inverse to this function. However, if we want to find an inverse to a particular point, say for close to zero, we can Taylor expand the function.

We will ignore the higher-order terms, and as a result we get

Now we get an linear function and the derivative is invertible.

The big idea: the derivative at a point tells you about the local nature of the function there, including invertibility.

The inverse rule in multi-variate context

The derivative of the inverse is the inverse of the derivative

assuming the function is differentiable

By definition and vice versa (via chain rule) thus

Note that the existence of an inverse should never be taken for granted.

Inverse function theorem

is locally invertible near if the derivative of at is invertible

In other word, is invertible near if (determinant of derivative)

The inverse function theorem implies that the linear data (derivatives) can control the existence of nonlinear inverse function.

Applications

References