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Math Help - Complex Analysis, uniformly continous function

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    Complex Analysis, uniformly continous function

    Prove that f(z)=z^3 is uniformly continuous in the region { {z\in C \mid \mid z \mid < 2 }}

    I do not understand what is the meaning of "uniformly continuous"... I read on wikipedia, but I still do not understand "uniformly continuous" in complex numbers. What theory should I use? How should I start in this case?
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    Quote Originally Posted by andreas View Post
    Prove that f(z)=z^3 is uniformly continuous in the region { {z\in C \mid \mid z \mid < 2 }}

    I do not understand what is the meaning of "uniformly continuous"... I read on wikipedia, but I still do not understand "uniformly continuous" in complex numbers. What theory should I use? How should I start in this case?
    A function f: D\to \mathbb{C} is uniformly continous on D if for every \epsilon > 0 there is \delta >0 so that if |x-y| < \delta \text{ and }x,y\in D \implies |f(x)-f(y)| < \epsilon. Note, if f is uniformly continous on D then certainly f is continous on D.

    An easy way to prove uniform continuity here is to note that f on |z|\leq 1 is a continous function. Continous functions on compact sets are uniformly continous on the set (this is a theorem). It follows that f is uniformly continous on |z| \leq 1. But if it is uniformly continous on |z|\leq 1 then it is surly uniformly continous on |z|<1 for it is a subset.
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    Quote Originally Posted by andreas View Post
    Prove that f(z)=z^3 is uniformly continuous in the region { {z\in C \mid \mid z \mid < 2 }}
    I do not understand what is the meaning of "uniformly continuous"
    The meaning of "uniformly continuous on C = \left\{ {z:\left| z \right| < 2} \right\} means that: \left( {\forall \varepsilon  > 0} \right)\left( {\exists \delta  > 0} \right)\left( {\forall z_0  \in C} \right)\left[ {\left| {z - z_0 } \right| < \delta \, \Rightarrow \,\left| {f(z) - f(z_0 )} \right| < \varepsilon } \right].
    In other words it is the usual \varepsilon \backslash \delta but it works for any point in the set.
    That is, we do not have to first pick a point.

    Note that z_0  \in C\, \Rightarrow \,\left| {z^3  - z_0 ^3 } \right| = \left| {z - z_0 } \right|\left| {z^2  + zz_0  + z_0 ^2 } \right| < \left| {z - z_0 } \right|(12).
    Thus if \varepsilon  > 0 \text{ let }\delta  = \frac{\varepsilon }{{12}}.
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