matrix of it at is:
:with the determinant:
:

The determinant is nonzero everywhere. Thus the theorem guarantees that, for every point in , there exists a neighborhood about over which is invertible. This does not mean is invertible over its entire domain: in this case is not even injective since it is periodic: .

Counter-example

If one drops the assumption that the derivative is continuous, the function is no longer necessarily locally injective. For example and

1.Previous
3.Next