Open Mapping Theorem from $C^n$ to $C^n$

Question

I am looking for an Open Mapping Theorem for a holomorphic function $f: U \subset \mathbb{C}^n \to \mathbb{C}^n$ where $U$ is a domain. I believe the following is true:

Let $f: U \subset \mathbb{C}^n \to \mathbb{C}^n$ be holomorphic, where $U$ is a domain. Suppose the determinant of the Jacobian of $f$ is not identically zero on $U$. Then $f(U)$ is open.

References and thoughts are welcome.

Answer 1

Not true: $(x,y)\mapsto(x, xy)$.