Open mapping theorem (complex analysis)

In <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, the open mapping theorem states that if 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is a <a href="/facts/Domain_(mathematical_analysis)/xT44oI3E">domain</a> of the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 and 
 
 
 
 f
 :
 U
 →
 
 C
 
 
 
 {\displaystyle f:U\to \mathbb {C} }
 
 is a non-constant <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic function</a>, then 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is an <a href="/facts/Open_map/abHUKX7l">open map</a> (i.e. it sends open subsets of 
 
 
 
 U
 
 
 {\displaystyle U}
 
 to open subsets of 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
, and we have <a href="/facts/Invariance_of_domain/n1uFhWRY">invariance of domain</a>.).
The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the <a href="/facts/Real_line/6eq42xX5">real line</a>, for example, the differentiable function 
 
 
 
 f
 (
 x
 )
 =
 
 x
 
 2
 
 
 
 
 {\displaystyle f(x)=x^{2}}
 
 is not an open map, as the image of the <a href="/facts/Open_interval/e9se3vTe">open interval</a> 
 
 
 
 (
 −
 1
 ,
 1
 )
 
 
 {\displaystyle (-1,1)}
 
 is the half-open interval 
 
 
 
 [
 0
 ,
 1
 )
 
 
 {\displaystyle [0,1)}
 
. 
The theorem for example implies that a non-constant <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic function</a> cannot map an open disk <a href="/facts/Onto/KezhLpCb">onto</a> 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.

Open mapping theorem (complex analysis) open-in-new

Open mapping theorem (complex analysis)