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Math Help - complex analysis analytic functions

  1. #1
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    complex analysis analytic functions

    Prove or disprove: Suppose f(z) and g(z) are analytic functions on an open and connected region \omega and f(z)g(z)=0 \in \omega, then either f(z)=0 or g(z)=0 in \omega.
    Can I get some help please?
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  2. #2
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    otherwise choose any point p that f(p) is not 0. For any natural number n, there must be a point p_n in the open ball B(p,1/n) that f(p_n) is 0, otherwise g will be indentically zero in this open ball which leads to a global zero of g.
    So we get a series of points p_n \rightarrow p with f(p_n)=0 which implies f(p)=0.
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