Inverse Function Theorem - Explanation

The inverse function theorem tells us that, under certain conditions, a differentiable function is invertible near a point a if its derivative at a is invertible. We offer some background which helps to appreciate the nature of this conclusion.

The derivative of f is a linear map from n-space to n-space. To say that it is an isomorphism, is to say that it is an invertible function.

Here is an example of the derivative f'(a) transforming the unit square of the plane.

The key now is that f itself transforms space near a like its derivative, just with some warping. Regard the left hand side copy of R² above as the tangent space to the left copy of R² at a below, and regard the right hand side copy of R² above as the tangent space to the right copy of R² at f(a) below.

Below is a picture of f transforming a piece of the plane near a.