# What is the most surprising / interesting application of the inverse function theorem you have ever seen?

The inverse function theorem states that:

Suppose that $$f: U \subset \mathbb{R}^m \rightarrow \mathbb{R}^m$$ is a $$C^k$$-function and that there exists $$a \in U$$ such that $$f'(a): \mathbb{R}^m \rightarrow \mathbb{R}^m$$ is an isomorphism. Then, there exist $$\delta > 0$$ and an open ball $$B_{\delta} : = B(a, \delta) \subset U$$ such that $$f \mid_{B_{\delta}} : B_{\delta} \rightarrow V \ni f(a)$$ is a diffeomorphism, with $$V$$ being an open set.

There are two remarkable applications of this theorem:

1- Existence of matrices $$X$$ such that $$X^ k = Y$$ where $$Y$$ is a matrix sufficiently close to the identity;

2- Differentiable perturbation of the identity: Let $$U \subset \mathbb{R}^m$$ a convex and open set. If $$\varphi : U \rightarrow \mathbb{R}^m$$ is $$C^k$$, with $$|\varphi'(x)| \leq \lambda < 1$$ for all $$x \in U$$, then $$f : U \rightarrow \mathbb{R}^m$$ given by $$f(x) = x + \varphi(x)$$ is a diffeomorphism on $$U$$ onto its image $$f(U)$$.

My goal with this question is to broaden the knowledge about the application/importance of this theorem in other contexts in the areas of Analysis, Geometry, Differential Topology, etc...

• This is a subjective question and has no concrete, acceptable answer … – k.stm Nov 13 '19 at 23:21
• Dear @k.stm, from my view point your claim no souds good. In fact if someone give new/ interesting examples of application of this theorem will be a gain for community, such examples will be in fact concrete answers. – Marcelo Ng Nov 14 '19 at 15:09
• hint: dig in math.stackexchange.com/questions/tagged/… – janmarqz Dec 1 '19 at 0:44