Density of zeroes of the Jacobian of an injective function

Question

I'm looking for a kind of reverse result to the Inverse Function Theorem. Let $f:\mathbb{R}^n\longrightarrow\mathbb{R}^n$ be a differentiable injective function. Is it true that points where the Jacobian of $f$ is zero are isolated? If so, then do you know any source for such a result? If it is false, then maybe you know of some other result that says something about zeroes of the Jacobian.