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In complex analysis, the open mapping theorem states that if U is a domain of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map.
The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the real line, for example, the differentiable function f = x is not an open map, as the image of the open interval is the half-open interval.
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 or two but never of dimension 1.
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