Let, $ f(x,y)=(x^3+3xy^2-15x-12y,x+y). $ And also let S $=\{ (x,y) \in \mathbf{R}^2: f$ is locally invertible at $(x,y)\}.$ Then which of the following is true,
1.$ S= \mathbf{R}^2 -\{(0,0)\} $
2.S in open in $ \mathbf{R}^2 $
3.S in dense in $ \mathbf{R}^2 $
- $ \mathbf{R}^2 -S$ in countable.
By the inverse function theorem I see that $ T=\{ (x,y) : x-y =-1~ ~or,~ ~ x-y =1\} \subset S$. Because $Df$ is invertible on the whole $ \mathbf{R}^2 $ except on the two lines $x-y=-1$ and $x-y=1$.
Now T satisfies option 2 and 3 and in the answer key the given correct options are 2 and 3. But How to find S? Since Inverse Function theorem is sufficient but not necessary, I can't use it find S. By nature of T can we conclude nature of S?