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.