How do you find the non differentiable points for a function?

Apr 30, 2015

A function is non-differentiable at any point at which

a) it is discontinuous,

b) it has a corner point or a cusp

$\textcolor{w h i t e}{\text{sssss}}$ This happens at $a$ if

$\textcolor{w h i t e}{\text{sssss}}$ ${\lim}_{h \rightarrow {0}^{-}} \frac{f \left(a + h\right) - f \left(a\right)}{h} \ne {\lim}_{h \rightarrow {0}^{+}} \frac{f \left(a + h\right) - f \left(a\right)}{h}$

c) It has a vertical tangent line

$\textcolor{w h i t e}{\text{sssss}}$ This happens at $a$ if

$\textcolor{w h i t e}{\text{sssss}}$ ${\lim}_{x \rightarrow {a}^{-}} \left\mid f ' \left(x\right) \right\mid = \infty$ or ${\lim}_{x \rightarrow {a}^{+}} \left\mid f ' \left(x\right) \right\mid = \infty$