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Math Help - Continuity of Complex numbers

  1. #1
    Member aldrincabrera's Avatar
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    Unhappy Continuity of Complex numbers

    hello there,. i need your help with this continuity in complex plane problem.

    Show that if a function f is continuous at a point z0 in some domain and f(z0) not equal to 0, then there exists a neighborhood of z0 throughout which f(z) not equal to 0.

    thnx
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Continuity of Complex numbers

    Quote Originally Posted by aldrincabrera View Post
    hello there,. i need your help with this continuity in complex plane problem. Show that if a function f is continuous at a point z0 in some domain and f(z0) not equal to 0, then there exists a neighborhood of z0 throughout which f(z) not equal to 0.
    Choose \epsilon=|f(z_0)| .
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  3. #3
    Member aldrincabrera's Avatar
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    Re: Continuity of Complex numbers

    the book suggested me with this,.
    write the continuity as |f(z0) - f(z)|/2 where epsilon = |f(z0)|/2. then note the contradiction if f(z)=0 at some point z in every neighborhood of z0.

    now i am confused,.
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    Re: Continuity of Complex numbers

    Quote Originally Posted by aldrincabrera View Post
    the book suggested me with this,.
    write the continuity as |f(z0) - f(z)|/2 where epsilon = |f(z0)|/2. then note the contradiction if f(z)=0 at some point z in every neighborhood of z0.
    Suppose that |f(w)-f(z_0)|<\frac{|f(z_0)|}{2}.

    Then it follows that 0<\frac{|f(z_0)|}{2}<|f(w)|.

    Can we have 0<|f(w)|=0~?.
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  5. #5
    Member aldrincabrera's Avatar
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    Re: Continuity of Complex numbers

    no??what should i do next??my mind is messed up
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    Re: Continuity of Complex numbers

    Quote Originally Posted by aldrincabrera View Post
    no??what should i do next??my mind is messed up
    You are done, at no w\in \mathcal{B}_{(f(z_0)/2)} can f(w)=0.
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  7. #7
    Member aldrincabrera's Avatar
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    Re: Continuity of Complex numbers

    ,.sir,.what will happen if f(w) = 0??will there have any contradiction??
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    Re: Continuity of Complex numbers

    Quote Originally Posted by aldrincabrera View Post
    ,.sir,.what will happen if f(w) = 0??will there have any contradiction??
    If |f(w)|>0 then f(w)\ne 0.
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  9. #9
    MHF Contributor FernandoRevilla's Avatar
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    Re: Continuity of Complex numbers

    Quote Originally Posted by aldrincabrera View Post
    the book suggested me with this,. write the continuity as |f(z0) - f(z)|/2 where epsilon = |f(z0)|/2. then note the contradiction if f(z)=0 at some point z in every neighborhood of z0. now i am confused,.
    Of course we can choose \epsilon=|f(z_0)|/2 but it is also valid if \epsilon=|f(z_0)| . As f is continuous at z_0 there exists \delta >0 such that |f(z)-f(z_0)|<|f(z_0)| if |z-z_0|<\delta .

    If f(z)=0 then, |f(z)-f(z_0)|=|f(z_0)|<|f(z_0)| (contradiction).
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