Let $f : \mathbb{R}^2 \to \mathbb{R}^2$ be a $C^1$ map such that $f^{-1}(y)$ is a finite set for all $y \in \mathbb{R}^2$. Show that the determinant det $df(x)$ of the Jacobi matrix of $f$ cannot vanish on an open subset of $\mathbb{R}^2$.
Does the statement in the question hold?
What about the identity map? Doesn't the determinant vanish on $\phi$, an open subset of $\mathbb{R}^2$?
For the identity map the Jacobi matrix is the identity matrix, so it is not singular.