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\begin{document}

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\noindent{\Large{\textsc{Complex Variables \hfill Assignment 5.2  \\ Fall 2024 \hfill Due October 2}}}

\large

\vspace{1cm}

\begin{exercise}
Let $f(z) = \frac{z \, \operatorname{Re}(z)}{|z|}$ for $z \ne 0$, and let $f(0)=0$.  Prove that $f$ is continuous throughout $\C$.
\end{exercise}

\begin{exercise}
For $z \ne 0$, let $f(z) = \frac{\overline{z}}{z}$.  Does $\displaystyle \lim_{z \to 0} f(z)$ exist?
\end{exercise}

\begin{exercise}
Let $f(z) = \frac{\left( \operatorname{Im}(z) \right)^2}{|z|}$ for $z \ne 0$, and let $f(0)=0$.  Is $f(z)$ continuous at $z=0$?
\end{exercise}



\end{document}
