Is ##J(f)## Non-Zero in ##U## for a Bijective ##C^m## 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.