1549901369-Elements_of_Real_Analysis__Denlinger_

278 Chapter 5 • Continuous Functions

exponent) to derive an important property of the function f(x) =ax (constant

base). Thus, the developments of these two functions are interrelated.

The disadvantage of this "early" approach to exponential and logarithm

functions is that it requires a rather difficult and tedious, "brute-force" ap-

proach. On the other hand, it puts to use many of the concepts and techniques

we have developed so far. You will see.

EXPONENTIAL FUNCTIONS

Let a be any positive real number. In Exercise 5.5.15(b ), we showed that

Vr E Q, ar is defined and positive. We now extend this definition to obtain a

function f(x) =ax defined for all x ER First, we need a few technical lemmas.

Lemma 5.6.1 For each a > 1 the exponential function f : Q __, JR given by

f(r) = ar (as defined in Exercise 5.5.15) is positive and strictly increasing on

Q. For each 0 <a< 1 the exponential function f: Q __,JR given by f(r) = ar

(as defined in Exercise 5.5.15) is positive and strictly decreasing on Q.

Proof. Suppose a > 1. Let r < sin Q. Then 3 m , n E Zand 3p EN 3

r = m/p, s = n/p and m < n. Then an-m ~ a > 1. Thus,

By Exercise 5.5.14, x^1 1P is strictly increasing on [O, +oo), so

(am)I/p < (an)I/p

ar = amfp < anfp =as.

Thus, f is strictly increasing on Q.

1

For 0 <a< 1, f(r) = ar = (~r, and the desired result follows from the

above argument, since ~ > 1. •

Lemma 5.6.2 Given any x E JR, there exists a monotone increasing sequence

{rn} of rational numbers converging to x.

Proof. Exercise 1. •

Lemma 5.6.3 Let a ~ 1 and x E R If {rn} is any monotone increasing

sequence of rational numbers converging to x, then { arn } converges.

Proof. Let a ~ 1 and x E R Suppose {rn} is any monotone increasing
sequence of rational numbers converging to x. Since the function f : Q __, JR
given by f ( r) = ar is strictly increasing, { arn } is monotone increasing. If r is