\begin{frame} \frametitle{Continuity: Intermediate Value Theorem} \begin{block}{Intermediate Value Theorem} Suppose $f$ is continuous on the closed interval $[a,b]$ with $f(a) \ne f(b)$. If $N$ is strictly between $f(a)$ and $f(b)$. Then \begin{talign} f(c) = N &&\text{for some number $c$ in $(a,b)$} \end{talign} \end{block} \vspace{-2ex} \begin{center} \scalebox{.8}{ \begin{tikzpicture}[default] \diagram{-.5}{9}{-.5}{3.5}{1} \diagramannotatez % \diagramannotatex{1,2,3,4,5,6,7,8} % \diagramannotatey{1,2} \draw[cgreen] (0,2) to[out=-45,in=-150] (3,1); \draw[cgreen] (3,1) to[out=30,in=130,looseness=1.5] (5,1); \draw[cgreen] (5,1) to[out=-50,in=130,looseness=1.5] (8,3); \draw[cgreen] (8,3) to[out=-50,in=130,looseness=1.5] (9,3); \draw[cred,dashed] (3,-.25) -- node[at start,below] {$a$} (3,1) -- node[at end,left] {$f(a)$} (-0.2,1); \draw[cred,dashed] (8,-.25) -- node[at start,below] {$b$} (8,3) -- node[at end,left] {$f(b)$} (-0.2,3); \draw[cblue,dashed] (6.5,-.25) -- node[at start,below] {$c$} (6.5,2) -- node[at end,left] {$N$} (-0.2,2); \end{tikzpicture} } \end{center} Every $N$ between $f(a)$ and $f(b)$ occurs at least once on $(a,b)$.\\\pause Intuitively: the graph cannot jump over the line $y= N$. \bigskip \end{frame}