\begin{frame}
  \frametitle{Derivative as a Function}
  
  \begin{exampleblock}{}
    Where is $f(x) = |x|$ differentiable?
    \pause\medskip
    
    For $x = 0$ 
    \begin{talign}
      f'(0) = \lim_{h\to 0} \frac{f(0+h) - f(0)}{h}
       \mpause[1]{= \lim_{h\to 0} \frac{|h|}{h}}
    \end{talign}
    \pause\pause
    We need to look at the left and right limits:
    \begin{talign}
      \lim_{h\to 0^-} \frac{|h|}{h}
      &\mpause[1]{\quad\stackrel{\text{since $h < 0$}}{=}\quad    \lim_{h\to 0^-} \frac{-h}{h}}
      \mpause[2]{=  \lim_{h\to 0^-} -1} 
      \mpause[3]{=  -1} 
    \end{talign}
    \pause\pause\pause\pause
    and
    \begin{talign}
      \lim_{h\to 0^+} \frac{|h|}{h}
      &\mpause[1]{\quad\stackrel{\text{since $h > 0$}}{=}\quad    \lim_{h\to 0^+} \frac{h}{h}}
      \mpause[2]{=  \lim_{h\to 0^+} 1} 
      \mpause[3]{=  1} 
    \end{talign}
    \pause\pause\pause\pause
    
    The left and right limits are different.\pause\medskip
    
    Thus $f'(0)$ does not exist, and $f(x)$ is not differentiable at $0$.\pause\medskip
    
    Hence $f$ is differentiable at all numbers in $(-\infty,0) \cup (0,\infty)$.
  \end{exampleblock}
  \vspace{10cm}
\end{frame}