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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2003 • Question 6

Question

Math 6210 • Real Analysis 2003 • Question 6

Let \(f\) be holomorphic on the unit disc \(U\). Let \(h(z,w)\) be the function on \(U\times U\) defined by \[ h(z,w)= \begin {cases} \dfrac {f(z)-f(w)}{z-w} & \text {if } z\ne w,\\[0.6ex] f'(z) & \text {if } z=w \end {cases} \] Prove that \(f\) is continuous as a function of two variables on \(U\times U\). You need only check the continuity at points of the diagonal. At other points it is obvious.

Raw LaTeX

Let \(f\) be holomorphic on the unit disc \(U\). Let \(h(z,w)\) be the
function on \(U\times U\) defined by
\[
h(z,w)=
\begin{cases}
\dfrac{f(z)-f(w)}{z-w} & \text{if } z\ne w,\\[0.6ex]
f'(z) & \text{if } z=w
\end{cases}
\]
Prove that \(f\) is continuous as a function of two variables on \(U\times U\). You
need only check the continuity at points of the diagonal. At other points it is
obvious.

Topic Tags

Absolute Continuity and Singular Measures Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Zeros of Holomorphic Functions and the Identity Theorem