Find a subset of $\mathbb{R}^2$ on which the mapping $F(x,y) = (x^2-y^2,2xy)$ is stable

Question

Find a subset of $\mathbb{R}^2$ on which the mapping $F(x,y) = (x^2-y^2,2xy)$ is stable.
We can show that there is no neighborhood of a point for which a mapping is stable by taking the derivative function. Does the logic also work backwards?
So, can I say that $u = x^2-y^2, v = 2xy$, and so $$DF = \begin{bmatrix}u_{x}&u_{y}\\v_{x}&v_{y}\end{bmatrix} = \begin{bmatrix}2x&2y\\2y&2x\end{bmatrix}$$ So, det(DF) = $4x^2 - 4y^2$. Now pick a subset in $\mathbb{R}^2$ that won't make the determinant singular? I don't think this is right, but it's the only idea I have. Can anyone help? Also, how can I denote a subset of a function in two dimensions?