EXAMPLE 5 Video Example Where is the function f(x)=(x) differentiable? SOLUTION If x > 0, then [x]- F(x) = lim x+hi-x h (x+1) h - lim lim lim and so is differentiable for any x >> Similarly, for x < and F(x) = m 1x + h-x h -(x+h)-(-x) - lim lim - lim and sof is differentiable for any x < 0. For x = 0 we have to investigate lim lim we have |x|-| lim f(0+h)-f(0) h 10+h-101 h Let's compute the left and right hand limits separately. lim 10+h-10 h and we can chooseh small enough that x+h> 0 and hence (x + h) - 10+h|-|0| h lim lim h-a lim h-a h 1 h h (if it exists). h lim (-1) and h can be chosen small enough that x + he Since these limits are different, f'(0) does not exist. Thus, f is differentiable at all x except 0. A formula for fis given by f(x) = . Therefore, for x > 0 we have and so (x + h) - . Therefore for x < 0,

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