Open mapping theorem (complex analysis)

In complex analysis, the open mapping theorem states that if ${\displaystyle U}$ is a domain of the complex plane ${\displaystyle \mathbb {C} }$ and ${\displaystyle f:U\to \mathbb {C} }$ is a non-constant holomorphic function, then ${\displaystyle f}$ is an open map (i.e. it sends open subsets of ${\displaystyle U}$ to open subsets of ${\displaystyle \mathbb {C} }$, and we have invariance of domain.).

The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the real line, for example, the differentiable function ${\displaystyle f(x)=x^{2}}$ is not an open map, as the image of the open interval ${\displaystyle (-1,1)}$ is the half-open interval ${\displaystyle [0,1)}$.

The theorem for example implies that a non-constant holomorphic function cannot map an open disk onto a portion of any line embedded in the complex plane. Images of holomorphic functions can be of real dimension zero (if constant) or two (if non-constant) but never of dimension 1.