Let $g$ be defined on a set containing the range of a function $f$. Show that if $f$ is continuous at $z_0$ and $g$ is continuous at $f(z_0)$, then $g(f(z))$ is continuous at $z_0$.
Let $f$ be a continuous function on a domain $\mathcal{D}$. Suppose that $f$ has this property:
For each point $p \in\mathcal{D}$, there is a disc $\Delta$ centered at $p$ on which $f$ is constant.
Conclude that $f$ is constant throughout $\mathcal{D}$. You will have to use the fact that $\mathcal{D}$ is a domain. Which property is crucial?
Section 1.5 (functions)
Prove the following propertye of the $\cos(z)$ and $\sin(z)$:
Show that both $\cos (z)\to \infty$ and $\sin (z)\to \infty$ as
and $|\Im z| \to \infty$.
Is it true that $\cos (z)\to \infty $ and $\sin (z)\to \infty$ as $|z|\to \infty$? (it is not; explain why).
Is it true that $\cos (z) $ and $\sin (z)$ are bounded $|\Re z|\to \infty$ and $|\Im z|\le M$? (it is true; explain why).
Show that
\begin{align*}
&|\cos(z) | \le e^{|\Im z|} ;\\
&|\sin(z)|\le e^{|\Im z|} .
\end{align*}