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...

This preview shows page 1 - 2 out of 2 pages.

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
Image of page 1
Image of page 2

You've reached the end of your free preview.

Want to read both pages?

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

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes
A+ icon
Ask Expert Tutors