# Show that the function |x| is not differentiable at all points?

Jun 22, 2017

graph{|x| [-10, 10, -5, 5]}

For the derivative to exist the limit definition of the derivative must exist, and that limit requires a consistent result as you approach $0$ from the left and the right.

However,

${\lim}_{x \rightarrow {0}^{+}} \frac{f \left(x\right) - f \left(0\right)}{x - 0} = 1$
${\lim}_{x \rightarrow {0}^{-}} \frac{f \left(x\right) - f \left(0\right)}{x - 0} = - 1$

So as we do not have a consistent result then in general

${\lim}_{x \rightarrow 0} \frac{f \left(x\right) - f \left(0\right)}{x - 0}$

is not defined, and thus the derivative at $x = 0$ does not exist.

What you are suggesting is taking an average value, but that approach does not hold up to vigour.