Question

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

Answer 1

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 rarr 0^+) (f(x)-f(0))/(x-0) = 1`
`lim_(x rarr 0^-) (f(x)-f(0))/(x-0) = -1`

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

`lim_(x rarr 0) (f(x)-f(0))/(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.