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Problem 2

Discuss continuity and uniform continuity for the following functions. (a) \(f(z)=\frac{1}{1-z} \quad(|z|<1)\) (b) \(f(z)=\frac{1}{z} \quad(|z| \geq 1)\) (c) $f(z)=\left\\{\begin{array}{ll}\frac{|z|}{z} & \text { if } 0<|z| \leq 1 \\\ 0 & \text { if } z=0\end{array}\right.$ (d) $f(z)=\left\\{\begin{array}{ll}\frac{\operatorname{Re} z}{z} & \text { if } 0<|z|<1 \\ 1 & \text { if } z=0 .\end{array}\right.$

Expert verified

(a) \(f(z)=\frac{1}{1-z}\) is continuous but not uniformly continuous. (b) \(f(z)=\frac{1}{z}\) is continuous and uniformly continuous. (c) \(f(z)=\frac{\left|z\right|}{z}\) when \(0<\left|z\right|\leq 1\) and \(f(z) = 0\) when \(z=0\) is continuous but not uniformly continuous. (d) \(f(z)=\frac{Re\,z}{z}\) when \(0<\left|z\right|<1\) and \(f(z) = 1\) when \(z=0\) is neither continuous nor uniformly continuous.

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