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Math Help - Continuity of complex function

  1. #1
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    Continuity of complex function

    Note: Sorry if this is the wrong section, the course this is from is 'Complex Analysis', so I assumed it belonged here.

    The question
    f(z) = im(z)

    Where is f continuous?

    My attempt
    Unless I'm mistaken, we need to show that
    \lim_{z \to z_0} f(z) = f(z_0)

    The limit is y_0, and f(z) is also y_0. So is the function continuous everywhere, or am I doing this incorrectly?

    Thanks.
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  2. #2
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    Quote Originally Posted by Glitch View Post
    The question
    f(z) = im(z) Where is f continuous?
    My attempt
    Unless I'm mistaken, we need to show that
    \lim_{z \to z_0} f(z) = f(z_0)
    The answer depends level of rigor required.
    If you need a \varepsilon /\delta proof here is a suggestion.
    If |z-z_0|<\delta then |y-y_0|\le\sqrt{(x-x_0)^2+(y-y_0)^2}=|z-z_0|.
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  3. #3
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    I don't think we have to use epsilon delta proofs. I'm quite sure my lecturer mentioned that it was non-examinable.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    An alternative is to use the property


    \displaystyle\lim_{z\to z_0}f(z)=w_0\Leftrightarrow \displaystyle\lim_{z\to z_0}\textrm{Re}(f(z))=\textrm{Re}(w_0)\;\wedge\;  \displaystyle\lim_{z\to z_0}\textrm{Im}(f(z))=\textrm{Im}(w_0)
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