Thread: Injective

Let \subseteq \mathbb{R}^2" alt="f \subseteq \mathbb{R}^2" /> . Show that if the determinant of the Jacobian Matrix of is not equal to zero at , then is injective in some neighborhood of .

First, the Jacobian is continuous in (x, y) (you may need to show that) so if it is not 0 at it is not 0 in some neighborhood of that point. Now use the "mean value theorem". If then for some point (x, y) on the line between and