\begin{frame} \frametitle{Continuity: Examples} \begin{center} \scalebox{.8}{ \begin{tikzpicture}[default] \diagram{-.5}{9}{-.5}{3.2}{1} \diagramannotatez \diagramannotatex{1,2,3,4,5,6,7,8} \diagramannotatey{1,2} \draw[cred] (-.5,0) to[out=51,in=180] (3,3); \draw[cred] (3,2) to[out=-45,in=180] (5,1); \draw[cred] (5,1) to[out=0,in=150] (9,3); \node[exclude={cred}] at (1,2) {}; \node[include={cred}] at (3,2) {}; \node[exclude={cred}] at (3,3) {}; \node[exclude={cred}] at (5,1) {}; \node[include={cred}] at (5,2.5) {}; \end{tikzpicture} } \end{center} Where is this graph continuous/discontinuous? \begin{itemize} \pause \item discontinuous at $x = 1$ since $f(1)$ is not defined \pause \item discontinuous at $x = 3$ since $\lim_{x\to 3} f(x)$ does not exist \pause \item discontinuous at $x = 5$ since $\lim_{x\to 5} f(x) \ne f(5)$ \end{itemize} \pause Everywhere else it is continuous. \end{frame}