\begin{frame}
  \frametitle{Continuity: Composition of Functions}
  
  \begin{block}{}
    If $f$ and $g$ are continuous at $a$ and $c$ is a constant, 
    then the following functions are continuous at $a$:
    \begin{enumerate}
    \pause
      \item $f+g$
    \pause
      \item $f-g$
    \pause
      \item $c\cdot f$
    \pause
      \item $f\cdot g$
    \pause
      \item $\frac{f}{g}$ if $g(a) \ne 0$
    \end{enumerate}
  \end{block}
  \pause\bigskip
  
  All of these can be proven from the limit laws!
  \pause\bigskip
  
  For example, (1) can be proven as follows:
  \begin{talign}
    \lim_{x\to a}(f+g)(x) 
    &\mpause[1]{= \lim_{x\to a}[f(x) + g(x)]}
    \mpause[2]{= \lim_{x\to a} f(x) + \lim_{x\to a} g(x)}\\
    &\mpause[3]{= f(a) + g(a)}
    \mpause[4]{= (f+g)(a)}
  \end{talign}
  \pause\pause\pause\pause\pause
  Thus $f+g$ is continuous at $a$.
  \vspace{10cm}
\end{frame}