If $f: U \to \mathbb{R}^{n}$ is a $C^{1}$ function, $U \subset \mathbb{R}^{n}$ open, show that $f$ is an open map.

Question

Let $f: U \to \mathbb{R}^{n}$ be a $C^{1}$ function with $U \subset \mathbb{R}^{n}$ open. Show that $f$ is an open map.

If $\det Df \neq 0$ (maybe it's not so direct), I think I can use the Inverse Function Theorem for show that $f$ is a local diffeomorphism and so, write $f(O)$ as union of open sets for each $O \subset U$ open. But without this hypothesis, how can I ensure the same conclusion? I really dont see the way.

Answer 1

You can't prove it, since it is false. Take any constant function, for instance.