1549901369-Elements_of_Real_Analysis__Denlinger_

298 Chapter 6 • Differentiable Functions

Proof. Let f(x) =ax + b. For arbitrary Xo E IR,

lim _f-'---(x--'--)_-_f-'(_xo-'-) = lim (ax+ b) - (ax^0 + b)

x-->xo x - Xo X-->Xo x - Xo

1

. ax - axo
= lffi
x-->xo x - x 0
= lim a(x - xo) =a.
X-->Xo x - Xo

Therefore, f is differentiable at xo and f' ( x) = a. •

Corollary 6.1. 3 (a) The derivative of a constant function is 0. More precisely,

if f is constant on a neighborhood of xo E IR, then f' (xo) = 0.

(b) The function f ( x) = x is difjerentiable everywhere, and f' ( x) = 1.

Example 6.1.4 The function f(x) = lxl is differentiable at every xo except 0.

Proof. Consider the function f(x) = lxl. Let xo ER

Case 1 (xo > 0): Then

lim f(x) - f(xo) = lim lxl - lxol

x-->xo x - xo x-->xo x - x 0

X-Xo

= lim --since x > 0 as x ____, xo > 0

x-->xo x - Xo

= 1.

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

y

x

Figure 6.1