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I wonder if this is correct: there is a holomorphic function on an open connected subset $G$ of $\mathbb{C}$ which maps $G$ onto a subset of a straight line, and I have to show that the function is constant.

I thought I can suppose that the straight line is the real axis (otherwise I can find a rotation and a traslation that will do so) and so using the Cauchy-Riemann equations I find that the function is constant since its imaginary part is zero. Is that correct? Thank you

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    Yes, that is correct and a much better argument (as it is much more elementary) than appealing to the open mapping theorem as suggested in the answers.2012-08-11

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