1549901369-Elements_of_Real_Analysis__Denlinger_

6.2 Rules for Differentiation 305

16. (a) Prove that if f is differentiable at x 0 , then lim f(xo + h) ~f(xo - h)

h->O 2

exists and equals f' ( xo).

(b) Find an example of a function f and a point x 0 such that

. f (xo + h) - f (xo - h)....
hm
2

h exists but f is not d1fferent1able at x 0.

h->O

17. A function f defined on an open interval I is said to "satisfy a Lipschitz

condition^3 of order o:" on I if :JM > 0 3 Yx, y E J, lf(x) - f(y)I ~

Mlx-yl"·

(a ) Prove that if f satisfies a Lipschitz condition of order o: on I , for some

real number o: > 1, then f is differentiable on I, and f'(x) = 0 on I.

(b) Find an example of a function f that satisfies a Lipschitz condition

of order o: = 1 on an interval I but f is not differentiable on I.

6.2 Rules for Differentiation

We now get down to the business of proving the familiar differentiation formu-
las.

Theorem 6.2.1 (Power Rule) For a given natural number n, the function
f(x) = xn is differentiable everywhere, and Yxo E JR., f'(xo) = nx 0 -^1.

Proof. The case n = 1 is covered by Theorem 6.1.2. Hence, assume n 2: 2.
Let f(x) = xn. Then, Yxo E JR.,

lim f(x) - f(xo) = lim xn - x'Q

x -+xo x - xo x-+xo x - x o

. (x - x o)(xn-1 + xn-2xo + ... + xxn-2 + xn-1)
= hm o o
x-+xo x - Xo
= lim (xn-1 + Xn-2Xo + ... + XXo-2 + Xo-1)
X-+Xo

= x0-1 + x0-1 + ... + x0-1 + x0-1 ( n terms)

= nx0-^1 (since there are n terms)

Therefore, f(x ) = xn is differentiable everywhere, and f'(xo) = nx 0 -^1. •

3. In Exercise 5.4.10 we proved that if f satisfies a Lipschitz condition of order 1 on I then
   f is uniformly continuous on I.