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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2010 • Question 8

Question

Math 6210 • Real Analysis Spring 2010 • Question 8

Let \(U\) be an open connected domain in the complex plane, and let \(f\) be analytic on \(U\).

(a) Prove or disprove: if \(f\) is injective on \(U\), then the derivative \(df/dz\) is never zero on \(U\).

(b) Prove or disprove: if the derivative \(df/dz\) is never zero on \(U\), then \(f\) is injective on \(U\).

Raw LaTeX

Let \(U\) be an open connected domain in the complex plane, and let
\(f\) be analytic on \(U\).

(a) Prove or disprove: if \(f\) is injective on \(U\), then the derivative
\(df/dz\) is never zero on \(U\).

(b) Prove or disprove: if the derivative \(df/dz\) is never zero on \(U\), then
\(f\) is injective on \(U\).

Topic Tags

Analytic Continuation and Monodromy Automorphism Groups and Biholomorphisms Riemann Mapping Theorem and Simply Connected Domains Zeros of Holomorphic Functions and the Identity Theorem