10/98
\begin{frame}
\frametitle{Continuity}

\begin{block}{}
A function $f$ is \emph{continuous} at a number $a$ if
\begin{talign}
\lim_{x\to a} f(x) = f(a)
\end{talign}
\end{block}
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The definition implicitly requires that:
\begin{itemize}
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\item $f(a)$ is defined
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\item $\lim_{x\to a} f(x)$ exists
\end{itemize}
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Intuitive meaning of continuous:
\begin{itemize}
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\item gradual process without interruption or abrupt change
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\item small changes in $x$ produce only small change in $f(x)$
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\item graph of the function can be drawn without lifting the pen
\end{itemize}
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\begin{block}{}
A function $f$ is \emph{discontinuous} at a number $a$ if
\begin{itemize}
\item $f$ is defined near $a$ (except perhaps a), and
\item $f$ is not continuous at $a$
\end{itemize}
\end{block}
\end{frame}