5.6 When a Function Is Not Differentiable at a Point

Case 2

When $f_{+}^{\prime }(x_0)$ and $f_{-}^{\prime }(x_0)$ both exist but $f_{+}^{\prime }(x_0)\ne f_{-}^{\prime }(x_0)$. In this case, there is a corner in the graph of $f$. For instance, see Figure 2.

Case 3

When the tangent line is vertical. In this case, $$\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x_0+\mathrm{\Delta }x)-f(x_0)}{\mathrm{\Delta }x}=+\mathrm{\infty }\text{ or }-\mathrm{\infty }.$$ For example, consider $f(x)=x^{1/3}.$ As you can see in Figure 3, the tangent to its graph at $(0,0)$ is vertical. This  function does not have a derivative at $x=0$, for $$\begin{array}{rl}{\frac{d}{dx}\sqrt[3]{x}|}_{x=0}& =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\sqrt[3]{0+\mathrm{\Delta }x}-\sqrt[3]{0}}{\mathrm{\Delta }x}\\ & =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\sqrt[3]{\mathrm{\Delta }x}}{\mathrm{\Delta }x}\\ & =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{1}{\mathrm{\Delta }{x}^{1-1/3}}\\ & =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{1}{\sqrt[3]{(\mathrm{\Delta }x{)}^2}}\\ & =+\mathrm{\infty }.\end{array}$$

Case 4

When the increment quotient ${\displaystyle \frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}}$ does not tend neither to any number nor to $\pm \mathrm{\infty }$ as $\mathrm{\Delta }x\to 0$.
For example, consider $$f(x)=\{\begin{array}{ll}x\sin \frac{1}{x}& \text{if }x\ne 0\\ 0& \text{if }x=0\end{array}.$$ This function is not differentiable (although it is continuous) at $x=0$, because $$\begin{array}{rl}{f}^{\prime }(0)& =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(0+\mathrm{\Delta }x)-f(0)}{\mathrm{\Delta }x}\\ & =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }x\sin \frac{1}{\mathrm{\Delta }x}-0}{\mathrm{\Delta }x}\\ & =\underset{\mathrm{\Delta }x\to 0}{lim}\sin \frac{1}{\mathrm{\Delta }x}\end{array}$$ does not exist. The graph of $f(x)$ is shown in Figure 4.