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