\begin{frame} \frametitle{Continuity: Examples} \begin{block}{} A function $f$ is \emph{continuous form the right} at a number $a$ if \begin{talign} \lim_{x\to a^+} f(x) = f(a) \end{talign} \end{block} \pause \begin{block}{} A function $f$ is \emph{continuous form the left} at a number $a$ if \begin{talign} \lim_{x\to a^-} f(x) = f(a) \end{talign} \end{block} \pause\smallskip \begin{minipage}{.58\textwidth} \begin{exampleblock}{} Where is $\lfloor x \rfloor$ (dis)continuous? \begin{talign} \lfloor x \rfloor = \text{`\;the largest integer $\le x$\;'} \end{talign}\vspace{-2ex} \begin{itemize} \item<4-> discontinuous at all integers \item<5-> left-discontinuous at all integers\\ $\lim_{x\to n^-} \lfloor x \rfloor = n-1 \ne n = f(n)$ \item<6-> \alert{but} right-continuous everywhere\\ $\lim_{x\to n^+} \lfloor x \rfloor = n = f(n)$ \end{itemize} \end{exampleblock} \end{minipage}~\quad~ \begin{minipage}{.39\textwidth} \scalebox{.75}{ \begin{tikzpicture}[default] \diagram{-.5}{4}{-.5}{3.2}{1} \diagramannotatez \diagramannotatex{1,2,3} \diagramannotatey{1,2} \foreach \x in {0,1,2,3} { \draw[cred,ultra thick] (\x,\x) -- ++(1,0); \node[include={cred}] at (\x,\x) {}; \node[exclude={cred}] at (\x+1,\x) {}; } \end{tikzpicture} } \end{minipage} \end{frame}