\begin{frame}
  \frametitle{Continuity: Function Composition}
  
  The composite function \alert{$f \circ g$} is defined by 
  \begin{talign}
    (f \circ g)(x) = f(g(x))
  \end{talign}\vspace{-2ex}
  \pause
  \begin{block}{}
    If
    \begin{itemize}
      \item $g$ is continuous at $a$, and
      \item $f$ is continuous at $g(a)$,
    \end{itemize} 
    then the composite function $f \circ g$ is continuous at $a$.
  \end{block}
  \pause
  
  A continuous function of a continuous function is continuous.
  \pause\bigskip
  
  \begin{exampleblock}{}
    Where is $h(x) = \sin x^2$ continuous? \\\pause 
    Both $x^2$ and $\sin$ are continuous everywhere (on $(-\infty,\infty)$).\\\pause
    Thus $h(x)$ is continuous everywhere.
  \end{exampleblock}
  \pause
  
  \begin{exampleblock}{}
    Where is $h(x) = \ln (1+\cos x)$ continuous? \\\pause 
    The functions $1$, $\cos$ (and their sum) and $\ln$ are on their domain.\\\pause 
    Thus $h(x)$ is continuous on its domain: $\mathbb{R} \setminus \{\pm\pi, \pm 3\pi,\pm 5\pi,\ldots\}$.
  \end{exampleblock}
\end{frame}