Let $f:\mathbb{R}^3 \to \mathbb{R}^3, f(x,y,z) = (x,xy,z^2)$. Determine where $f$ is a local diffeomorphism.
Our lecturer gave us this definition
Let $U \subset R^n$ be an open set and $x_0 \in U$. If $$f \in C^1(U, \mathbb{R}^n) \text{ such that } J_f(x_0) \ne 0$$ then $f$ is a local diffeomorphism at $x_0$.
So for the problem, I can just find the Jacobian determinant and restrict it so that the Jacobian matrix is invertible.
I have that $$Df(x,y,z) = \begin{bmatrix} 1 && 0 && 0 \\ y && x && 0 \\ 0 && 0 && 2z \end{bmatrix}$$
and $$\det(Df(x,y,z)) = 2xz.$$
Thus $f$ is locally invertible for all $(x,y,z) \in \mathbb{R}^3$ for which $xz \ne 0.$
However, they did not provide any proof for this and I'm left wondering why is this the case? This seems to be related to the inverse function theorem?