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

1 Answer
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 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.