Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $D$ be a region in $\hat{\mathbb C}$, the Riemann surface, $I=[T_0,T_1]$ be a bounded closed interval in $\mathbb R$ and $\phi: I\rightarrow D$ be a function whose real part and imaginary part are continuous as real functions. I want to show that there exists a partition $T_0=t_0<t_1<\dots<t_n=T_1$ of $I$ such that for every $1\le j\le n$ there exists an open disk $\Delta_j\subset D$ that contains $\phi([t_{j-1},t_j])$, and if $\infty\in\Delta_j$ either $\phi(t_{j-1})$ or $\phi(t_j)$ is $\infty$.

How do you prove this?

share|cite|improve this question
up vote 1 down vote accepted

As the real and the imaginary part of $\phi$ are continuous on the compact interval $I$ the point $\infty$ is not part of the problem.

The essential ingredient in the following proof is the compactness of $I$ resp. its image in $D$.

For $t\in I$ put $\rho(t):=\sup\{r>0\ |\ D_r\bigl(\phi(t)\bigr)\subset D\}$. We may assume $\rho(t)<\infty$ for all $t\in I$, or there would be nothing to prove. The function $t\mapsto \rho(t)$ is positive and continuous on $I$. To prove the latter assume that an $\epsilon>0$ is given. Then there is a $\delta>0$ with $|\phi(t)-\phi(t')|<{\epsilon\over 2}$ as soon as $|t-t'|<\delta$. It follows that $\rho(t')\geq \rho(t)-{\epsilon\over 2}$ when $|t-t'|<\delta$, whence by symmetry $|\rho(t')- \rho(t)|\leq{\epsilon\over 2}$ when $|t-t'|<\delta$.

Therefore we may conclude that there is a $\delta>0$ with $\rho(t)>2\delta$ for all $t\in I$. Choose $N$ so large that $|\phi(t)-\phi(t')|<\delta$ when $|t-t'|<h:=(T_1-T_0)/N$, and put $t_j:=T_0+ jh$ $\ (0\leq j\leq N)$. Then the open disks $\Delta_j$ $\ (1\leq j\leq N)$ with center $\phi(t_j)$ and radius $\delta$ are subsets of $D$ and contain the respective arcs $\phi\bigl([t_{j-1},t_j]\bigr)$.

share|cite|improve this answer
Before I asked this, I thought something like your $\rho$ should be used to prove this, but I couldn't prove its continuity. How do you do this? – Pteromys Apr 3 '12 at 0:05
@Pteromys: See my edit. – Christian Blatter Apr 3 '12 at 8:13

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.