Inverse function in singular points

Question

Hi there! First of all i'm not a matematician, i'm just mechanical engineer who interested in some math.

Probably trivial question. Suppose I have a mapping $F: \mathbb{R}^n \to \mathbb{R}^n$ with Jacobian $J(x)$. And at some point $x_0$ Jacobian degenerate $\det J(x_0) = 0$. Is there general condition, for the existence of continuous inverse function near $x_0$, which is weaker than inverse function theorem. For example $y=x^3$ has continuous inverse $x = \sqrt[3]{y}$ near $x = 0$, but this fact is not provided by IFT.

Some usefull thinks I found in book "The implicit function theorem: history, theory, and applications" by Steven George Krantz, Harold R. Parks, but there are only special cases.

Thanks for all inputs.

Answer 1

• One result is that small Lipschitz perturbations of the identity on a Banach space are homeomorphisms onto an open set: see here for a more precise statement. Note that the classic Inverse Mapping Theorem is usually proved as a consequence of this.

• Another one deals with a strongly monotone map on a real Hilbert space, $F:H\to H$, that is, a map satisfying $(F(x)-F(y),x-y)\ge c |x-y|^2$ for all $x$ and $y$. A continuous strongly monotone map on $H$ is a homeomorphism onto $H$ (see e.g.Thm 11.2 in Klaus Deimling's book Nonlinear functional analysis, where you can also find more advanced inversion theorems). For one variable, real-valued functions this reduces to the well-known criterium.

• In the case of $\mathbb{R}^n$ let's recall the theorem of Invariance of Domain. Injectivity is usually simpler to check than surjectivity, and in the case of a continuous map $f:U\to \mathbb{R}^n$ on an open set $U$ of $\mathbb{R}^n$ guarantees that $f$ is a homeomorphism onto an open set $f(U)$.