Inverse Function Theorem

If Df(a) is invertible, f itself is locally invertible near a -- and the inverse is just as smooth.

The Inverse Function Theorem says that if $f$ is $C^1$ near $\mathbf{a}$ and $Df(\mathbf{a})$ is an invertible matrix, then $f$ has a local inverse near $\mathbf{a}$ , and that inverse is itself $C^1$ . It converts a linear-algebra fact about one matrix into a genuinely nonlinear conclusion about $f$ itself.

Examples

Polar coordinates $(r,\theta)\mapsto(r\cos\theta,r\sin\theta)$ are locally invertible everywhere except $r=0$ , exactly where the Jacobian determinant vanishes.