\begin{frame} \frametitle{How can a Function fail to be Derivable?} There are the following reasons for failure of being derivable: \begin{center} \scalebox{.7}{ \begin{tikzpicture}[default,baseline=1cm] \diagram{-.5}{3}{-.5}{3}{1} \diagramannotatez \begin{scope}[ultra thick] \draw[cred,dashed] (2,-.1) -- (2,3) node[at start,below] {$a$}; \draw[cgreen,ultra thick] plot[smooth,domain=-.5:2,samples=20] function{.5*x**2+.5}; \draw[cgreen,ultra thick] plot[smooth,domain=2:3,samples=20] function{.5*(x-4)**2+.5}; \end{scope} \end{tikzpicture}~\quad~% \begin{tikzpicture}[default,baseline=1cm] \diagram{-.5}{3}{-.5}{3}{1} \diagramannotatez \begin{scope}[ultra thick] \draw[cred,dashed] (1.5,-.1) -- (1.5,3) node[at start,below] {$a$}; \draw[cgreen,ultra thick] plot[smooth,domain=-.5:1.5,samples=20] function{sqrt(x+1)+.5}; \draw[cgreen,ultra thick] plot[smooth,domain=1.5:3,samples=20] function{.5+ sqrt(x-1)}; \node[include=cgreen] at (1.5,{sqrt(2.5)+.5}) {}; \node[exclude=cgreen] at (1.5,{.5+ sqrt(.5)}) {}; \end{scope} \end{tikzpicture}~\quad~% \begin{tikzpicture}[default,baseline=1cm] \diagram{-.5}{3}{-.5}{3}{1} \diagramannotatez \begin{scope}[ultra thick] \draw[cred,dashed] (1.5,-.1) -- (1.5,3) node[at start,below] {$a$}; \draw[cgreen,ultra thick] plot[smooth,domain=-.5:3,samples=200] function{1.5+sgn(x-1.5) * abs(x-1.5)**(1./3.)}; \end{scope} \end{tikzpicture} } \end{center} \begin{itemize} \pause \item graph changes direction abruptly (graph has a ``corner'') \pause \item the function is not continuous at $a$ \pause \item graph has a vertical tangent at $a$, that is: \begin{talign} \lim_{x\to a} |f'(x)| = \infty \end{talign} \pause Example for a vertical tangent is $f(x) = \sqrt[3]{x}$ at $0$. \end{itemize} \end{frame}