EXAMPLE 5 Where is the function

f(x)=|x|

differentiable?\ SOLUTION If

x>0

, then

|x|=

and we can choose

h

small enough that

x+h>0

and hence\

|x+h|=

\ Therefore, for

x>0

we have\

f^(')(x)=\\\\lim_(h->0)(|x+h|-|x|)/(h)\ =\\\\lim_(h->0)((x+h)-x)/(h)\ =\\\\lim_(h->0)()/(h)\ =\\\\lim_(h->0)\ =

\ and so

f

is differentiable for any

x>

\ Similarly, for

x

we have

|x|=

\

|x+h|=

\

- Therefore for x

\

f^(')(x)=\\\\lim_(h->0)(|x+h|-|x|)/(h)\ =\\\\lim_(h->0)(-(x+h)-(-x))/(h)\ =\\\\lim_(h->0)()/(h)\ =\\\\lim_(h->0)\ =