Suppose there is a holomorphic function $\mathbf{f}:U\rightarrow V$ with several complex variables, where $U$ and $V$ are both open sets in $\mathbb{C}^n$. If the Jacobian matrix of $\mathbf{f}$ on $x_0$, $\mathbf{Df}(x_0)$ is invertible, then we can conclude that $\mathbf{f}$ has an inverse function in a neighborhood $U’$ of $x_0$. Can I immediately conclude that $\mathbf{f}(U’)$ is open?
This question is not the same as Open Mapping Theorem from $C^n$ to $C^n$. The difference is the condition about Jacobian matrix. Mine is that Jacobian matrix on a point is invertible, but his/hers is that Jacobian matrix is not identically zero on $U$.
Thank you very much for your answers.