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In [[complex analysis]], the '''open mapping theorem''' states that if ''U'' is a [[Domain (mathematical analysis)|___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''').
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'') = ''x''<sup>2</sup> is not an open map, as the image of the [[open interval]] (−1, 1) is the half-open interval [0, 1).
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[[Image:OpenMapping1.png|frame|right|Blue dots represent zeros of ''g''(''z''). Black spikes represent poles. The boundary of the open set ''U'' is given by the dashed line. Note that all poles are exterior to the open set. The smaller red circle is the set ''B'' constructed in the proof.]]
Assume ''f'' : ''U'' → '''C''' is a non-constant holomorphic function and ''U'' is a [[Domain (mathematical analysis)|___domain]] of the complex plane. We have to show that every [[Point (geometry)|point]] in ''f''(''U'') is an [[interior point]] of ''f''(''U''), i.e. that every point in ''f''(''U'') is contained in a disk which is contained in ''f''(''U'').
Consider an arbitrary ''w''<sub>0</sub> in ''f''(''U''). Then there exists a point ''z''<sub>0</sub> in ''U'' such that ''w''<sub>0</sub> = ''f''(''z''<sub>0</sub>). Since ''U'' is open, we can find ''d'' > 0 such that the closed disk ''B'' around ''z''<sub>0</sub> with radius ''d'' is fully contained in ''U''. Consider the function ''g''(''z'') = ''f''(''z'')−''w''<sub>0</sub>. Note that ''z''<sub>0</sub> is a [[root of a function|root]] of the function.
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