\begin{frame}
  \frametitle{Derivatives and Continuity}
  
  \begin{block}{}
    If $f$ is differentiable at $a$, then $f$ is continuous at $a$.
  \end{block}
  \pause
  The proof is in the book. Intuitively it holds because\ldots\hspace{-2ex}
  \pause\bigskip
  
  Differentiable at $a$ means:
  \begin{talign}
    f'(a) = \lim_{h\to 0} \frac{f(a+h) - f(a)}{h} \quad \text{exists}
  \end{talign}
  \pause
%   
  Continuous at $a$ means:
  \begin{talign}
    \lim_{x\to a} f(x) = f(a) 
    &\mpause[1]{\quad\iff\quad \lim_{x\to a} (f(x) - f(a)) = 0 }\\
    &\mpause[2]{\quad\iff\quad \lim_{h\to 0} (f(a+h) - f(a)) = 0 }
  \end{talign}
  \pause\pause\pause
  If the latter limit would not be $0$ (or not exist), \\
  then $\frac{f(a+h) - f(a)}{h}$ would get arbitrarily large for small $h$.
  \pause
  
  \begin{alertblock}{}
    If $f$ is continuous at $a$, then $f$ is \alert{not always} differentiable at $a$.
  \end{alertblock}
  \pause
  E.g. $|x|$ is continuous at $0$ but not differentiable at $0$.
  \vspace{10cm}
\end{frame}