Prove that if $F:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ is continuously differentiable, the set of points for which $\nabla F$ is nonsingular is open. I want to write the set $\{(x,y)\mid\det \nabla F\neq 0\}$ as the preimage of an open set under a continuous function.
Can I define a function $G:\mathbb{R}^2 \rightarrow \mathbb{R}$ given by $$G(x,y)=\frac{\partial F_1}{\partial x}(x,y)\frac{\partial F_2}{\partial y}(x,y)-\frac{\partial F_1}{\partial y}(x,y)\frac{\partial F_2}{\partial x}(x,y)$$ and then $G^{-1}(\mathbb{R}-\{0\})=\{(x,y) \mid \det \nabla F \neq 0\}$?