Open mapping theorem (complex analysis)
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 (i.e. it sends open subsets of U to open subsets of 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 f(x) = x2 is not an open map, as the image of the open interval (−1, 1) is the half-open interval [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.
Assume f : U → C is a non-constant holomorphic function and U is a domain of the complex plane. We have to show that every point in f(U) is an interior point of f(U), i.e. that every point in f(U) has a neighborhood (open disk) which is also in f(U).
Consider an arbitrary w0 in f(U). Then there exists a point z0 in U such that w0 = f(z0). Since U is open, we can find d > 0 such that the closed disk B around z0 with radius d is fully contained in U. Consider the function g(z) = f(z)−w0. Note that z0 is a root of the function.
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