TITLE: If $\mathrm{Int}(\mathrm{Im}(f(\mathbb {R^n})))=\varnothing$ then the determinant of the Jacobian matrix is zero.
QUESTION [9 upvotes]: I need some light in this exercise:
"If $f:\mathbb {R^n}\rightarrow \mathbb {R^n}$ is continuous differentiable and the image of $f(\mathbb {R^n})$ has $\text{int} =\varnothing$ then the determinant of the Jacobian matrix is zero."
I really don't know where to start... I appreciate any help that helps me to get started.

REPLY [10 votes]: If the determinant of the Jacobian was non-zero at some point, then the inverse function theorem would show that it is locally invertible, and hence maps a small open set into an open set, which would contradict the image having empty interior.

REPLY [4 votes]: Suppose there is a point $x$ such that $\operatorname{det}(f'(x))\neq 0$. What happens  if you use the inverse function theorem?