Math 185 Homework 2 Solutions - Math 185 Spring 2015 Homework 2 Solution sketches Problem 1 Show that if f is holomorphic at z C then f is continuous at

Math 185 Homework 2 Solutions - Math 185 Spring 2015...

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Math 185 - Spring 2015 - Homework 2 - Solution sketches Problem 1. Show that if f is holomorphic at z C then f is continuous at z . For any w 6 = z we have f ( z ) - f ( w ) = ( z - w ) f ( z ) - f ( w ) z - w 0 · f 0 ( z ) = 0 as w z. Problem 2. Let Ω C be open. Show that Ω is connected if and only if it is path connected. Suppose Ω is connected. To show Ω is path connected, it suffices to show that for any z 0 Ω, the set A := { z Ω : there exists a curve in Ω joining z 0 to z } is equal to Ω. So fix z 0 Ω and define A as above. As Ω is connected, to prove A = Ω we can show A is non-empty, closed in Ω, and open in Ω. The set A is clearly non-empty, since z 0 A . Next, suppose { z n } ⊂ A converges to some z Ω. Since Ω is open, we may find r > 0 such that B r ( z ) Ω. As z n z we may find z n B r ( z ). Since z n A we may join z 0 to z n with a curve in Ω . We can then concatenate this curve with the line segment joining z n to z , thus giving a curve in Ω joining z 0 to z . In particular z A , so that A is closed in Ω. Finally suppose z A . Then since Ω is open we can find some r > 0 such that B r ( z ) Ω. We can now find curves in Ω joining z 0 to any w B r ( z ) by first joining z 0 to

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• Fall '07
• Lim
• Math, Topology, Topological space, Ω, open connected subset

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