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\noindent{\Large{\textsc{Complex Variables \hfill Assignment 8.2  \\ Fall 2024 \hfill Due October 23}}}

\large

\vspace{1cm}

\begin{exercise}
Our textbook defines a path $\gamma: [a,b] \to \C$ to be $C^1$ provided:
\begin{itemize}

\item[(1)]  $\gamma'(t)$ exists and is continuous on $(a,b)$;

\item[(2)]  $\displaystyle \lim_{t \to a^+} \gamma'(t)$ and $\displaystyle \lim_{t \to b^-} \gamma'(t)$ both exist.
\end{itemize}
Use the Mean Value Theorem from Calculus I to show that this definition is equivalent to: $\gamma'(t)$ exists and is continuous throughout $[a,b]$ (where at $a$ and $b$ we use {\em one-sided} derivatives, e.g.\
\[
\gamma'(a) = \lim_{t \to a^+} \frac{\gamma(t) - \gamma(a)}{t-a},
\]
and similarly for $\gamma'(b)$).

\end{exercise}

\begin{exercise}
Textbook exercise 1.5.28.
\end{exercise}

\begin{exercise}
Textbook exercise 1.R.18 (in part (b) use $\overline{f(\overline{z})}$ instead of what's stated).
\end{exercise}

\begin{exercise}
Textbook exercise 1.6.13.
\end{exercise}


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