Let $\displaystyle f \subseteq \mathbb{R}^2$ $\displaystyle \to \mathbb{R}^2$. Show that if the determinant of the Jacobian Matrix of $\displaystyle f$ is not equal to zero at $\displaystyle (x_o,y_o)\in D$, then $\displaystyle f$ is injective in some neighborhood of $\displaystyle (x_o,y_o)$.