I'm working on a proof that is not a homework assignment - that's why I'm posting it here. My question is simple.(adsbygoogle = window.adsbygoogle || []).push({});

The epsilon-delta definition of continuity at a point a in an open subset E of the Complex plane is:

[tex] \forall a \in E, \ \forall \ \varepsilon > 0 \ , \exists \ \delta > 0 \ [/tex] such that [tex] \ \forall \ z \in E \ [ \ |z - a| < \delta \ ] \Rightarrow [ \ |f(z) - f(a)| < \varepsilon \ ] [/tex]

This definition has a sequential equivalent as follows:

[tex] \forall \{ z_n \} \in E \ , [ \ z_n \rightarrow a \ ] \Rightarrow [ \ f(z_n) \rightarrow f(a) \ ] [/tex]

My question deals with uniform continuity at a point a in an open subset E of the Complex plane. I know the epsilon-delta definition is:

[tex] \forall \ \varepsilon > 0 \ , \exists \ \delta > 0 \ [/tex] such that [tex] \ \forall \ z, a \in E \ [ \ |z - a| < \delta \ ] \Rightarrow [ \ |f(z) - f(a)| < \varepsilon \ ] [/tex]

What I am looking for is a sequential equivalent for uniform continuity. It seems that the only difference is that now a is not fixed, it varies as does z. Any ideas? I have to put my kids to bed, but I'll be back...

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# Uniform Continuity

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