Analytic Function Mapping to a Line: Constant Throughout Domain?

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If an analytic function w = f(z) maps a domain D onto a portion of a line, it must be constant throughout D. The discussion revolves around the application of the open mapping theorem, which indicates that non-constant analytic functions cannot map to lower-dimensional sets like lines. A participant questions whether they can express w as w = u(x,y)(a + bi), where a, b, and u(x,y) are real, to support their argument. They have demonstrated that the derivatives are zero, reinforcing the conclusion that f is constant. Thus, the analytic function must indeed be constant across the domain.
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Homework Statement


Show that if the analytic function w= f(z) maps a domain D onto a portion of a line, then f must be constant throughout D.


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The Attempt at a Solution


I just have one question, can I write w = u(x,y) (a+bi) since it maps to a portion of a line? Where a,b, and u(x,y) are reals. From there I have shown that the derivatives are 0.
 
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