1549901369-Elements_of_Real_Analysis__Denlinger_

5.3 Continuity on Compact Sets and Intervals 253

Proof. Suppose a :::; b, and f : [a, b] ___, [a, b] is continuous. Define the

function h on [a, b] by h( x) = f ( x) - x. By the algebra of continuous functions,

his continuous on [a, b]. Moreover,

h(a) = f(a) - a 2: 0 since f(a) E [a , b];

h(b) = f(b) - b:::; 0 since f(b) E [a, b].

If h(a) = 0 or h(b) = 0, then f(a) =a or f(b) = b, and we have found c as

desired. Thus, we suppose that h(a) # 0 and h(b) # 0. Then, 0 is between h(a)

and h(b), so by the intermediate value theorem, 3c E [a,b] such that h(c) = 0.

That is, f(c) - c = 0, i.e., f(c) = c. •

Corollary 5.3.14 Suppose f : I , JR is continuous, strictly monotone, and
bounded on I = (a, b), where a < b. Then f(I) is a bounded open interval. In
fact, f(I) = (c, d) where c =inf f(I) and d =sup f(I). Further, we can extend
f to a continuous, strictly monotone function f : [a, b] ,JR as follows:

(a) if f is strictly increasing on (a, b), define f(a) = c and f(b) = d.

(b) if f is strictly decreasing on (a, b), define f(a) = d and f(b) = c.

In either case, f is continuous and strictly monotone on [a, b], and f ([a, b]) =

[c,d].

Proof. Exercise 21. •

In Section 1.6 we proved that the completeness property guarantees that in
JR, every positive element has a square root. With the help of the intermediate
value theorem we can now go further and prove that \:In 2: 2 in N, every positive
real number has a unique positive nth root. First, we prove the following lemma.

Lemma 5.3.15 \:In EN, thefunctionf(x) = xn is 1-1 on the interval (0,+oo).

Proof. Since the case n = 1 is trivial, we assume n 2: 2 in N, and consider
the function f(x) = xn on (0 , + oo). Suppose a, b E (0, +oo) 3 f(a) = f(b).
Then an= bn. By factoring, this means

Since a, b > 0, the second factor cannot equal 0. Thus, a - b = 0, and so
a= b. That is , f is 1-1 on (0, +oo). •

Theorem 5.3.16 (Existence of Unique Positive nth Roots) \:In EN, and
Vx 0 > 0 in JR, 3 unique y > 0 such that yn = x 0. That is, every positive real
number x 0 has a unique positive nth root, y = ylxO.