Proving the determinant of the derivative of a function can not be zero on any open set.

Question

I'm working on the following problem for my multivariable calculus course:

Let $f: R^2 \to R^2 $ be a $C^1$ function such that for each $y \in R^2$, the set $f^{-1}(y)$ is finite. Show that $\det Df(x)$ cannot vanish identically on any open subset of $R^2$

I'm assuming I need to use the inverse function theorem somewhere, but I cannot find a place to apply it.