\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} \pause\medskip The definition implicitly requires that: \begin{itemize} \pause \item $f(a)$ is defined \pause \item $\lim_{x\to a} f(x)$ exists \end{itemize} \pause\medskip Intuitive meaning of continuous: \begin{itemize} \pause \item gradual process without interruption or abrupt change \pause \item small changes in $x$ produce only small change in $f(x)$ \pause \item graph of the function can be drawn without lifting the pen \end{itemize} \pause\medskip \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}