Question about inverse function

[facenian]

Let ##f:U\subset R^n\rightarrow V\subset R^n## be a biyective function of class ##C^m(m\geq 1)##, ##U## and ##V## are open sets in ##R^n##. I know from the inverse funtion theorem that when ##J(f)\neq 0## in a point of the the domain a local inverse exists, however, given the above conditions I'd like to know if it is true that ##J(f)\neq 0## in U.(kind of a reciprocal of the inverse function theorem)
For instance, if the inverse of a differentiable function is differentiable this would be easy to prove since
$$\frac{\partial(x_1\ldots x_n)}{\partial(y_1\ldots y_n)}\,\frac{\partial(y_1\ldots y_n)}{\partial(x_1\ldots x_n)}=1\neq 0$$
However I don't know whether this is correct or not.

 

[mathwonk]

x^3.

 

[math771]

It is not necessarily true that ##J(f)\neq 0## in ##U## for a bijective function of class ##C^m##. The inverse function theorem only guarantees the existence of a local inverse when ##J(f)\neq 0## at a point in the domain, but it does not necessarily hold for all points in the domain.

For example, consider the function ##f(x)=x^3##, which is a bijective function of class ##C^m## for any ##m\geq 1##. However, at ##x=0##, we have ##J(f)=0##, so the inverse function theorem does not apply. This means that there may not exist a local inverse at ##x=0##, even though the function is bijective and of class ##C^m##.

Therefore, it is not enough to just have a bijective function of class ##C^m## to guarantee that ##J(f)\neq 0## in ##U##. You would need additional conditions, such as the inverse of a differentiable function being differentiable, to prove this statement.