1549901369-Elements_of_Real_Analysis__Denlinger_

6.1 The Derivative and Differentiability 299

Case 2 (xo < 0): Then

1

. f(x) - f(xo)
lm 1
. im---lxl - lxol
x-+xo x - Xo x-+xo x - Xo

• x-(-xo)
  = lim since x < 0 as x ---+ xo < 0
  x-+Xo x - Xo
  = lim -(x - xo)
  x-+xo x - x 0
  = -1.

Thus, when xo < 0, f'(xo) = -1.

Case 3 (xo = 0): Then

lim _f (_x_) -_f (_x_o) = lim lxl - 0

x-+O X - Xo x-+0 X - 0

= lim l:'.l, which does not exist.

x-+O X

(See Exercise 4.3.l (a).)

Thus, the function f(x) = lxl is not differentiable at 0. D

Example 6.1.5 The function f(x) = ..jX is differentiable on (0, +oo), and

't:/xo E (0, +oo), f'(xo) = ~·
2yXO

Proof. Let f(x) = ..jX on (0, +oo), and let xo E (0, +oo). Then,

lim _f (_x_) _-_!_( x_o_) = lim ..jX -ftO

x-+xo x - Xo x-+xo x - Xo

1

. ..jX - ftO
= im
X-+Xo ( ..jX - ftQ) ( ..jX + ftQ)
. 1
= hm
X-+Xo ..jX + ftO

(since ..jX - ftO -::/-0)

1 1

since xo -::/-0. D.

2ft0

Sometimes Definition 6.1.1 is less convenient to use than another, equiva-
lent, definition that is based on the following observation:
If g is any function, and xo E JR is any cluster point of 'D(g), then

lim g(x) = L ~ lim g(x 0 + h) = L.

X-+Xo h-+0