• I
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.