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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).
The theorem for example implies that a non-constant [[holomorphic function]] cannot map an open disk ''[[onto]]'' a portion of any real 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.
==Proof==
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