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Math Help - Application of Maximum Modulus Principle on the Boundary

  1. #1
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    Application of Maximum Modulus Principle on the Boundary

    Let U be a connected open set, and U^c its closure. Let f be a continuous function on U^c, analytic and nonconstant on U. If z_0 \in U^c is a maximum point for f, then z_0 lies on the boundary of U_c

    Now, the book says this comes from a direct application of the global maximum modulus principle, which states:

    If U is a connected open set, and f is analytic on U that has a maximum point in U. Then f is constant on U.

    I understand the proof of the principle, but how do I apply that to the first statement? Thanks.
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  2. #2
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    It's kind of trivial, if z_0 was in U then by the maximum modulus principle f would be constant on U and hence on U^c. So unless you were assuming f was constant this is a contradiction, and hence z_0 must belong to the boundary of U.
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