1549901369-Elements_of_Real_Analysis__Denlinger_

242 Chapter 5 11 Continuous Functions

Theorem 5.2.1 7 Suppose f is monotone increasing and bounded on an open

interval I= (a, b), where a< b. Then

(a) Ve E (a, b], lim f(x) exists and equals sup{f(x) : a< x < c};

x-.c-

(b) Ve E [a, b), lim f(x) exists and equals inf{j(x): c < x < b};

x--+c+

(c) Ve E (a, b), lim f(x) ::; f(c)::; lim f(x); i.e., f(c-)::; f(c)::; f(c+);

x--+c- x-.c+

(d) Ve< din (a, b), lim f(x)::; lim f(x); i.e., f(c+)::; f(d-).

x--+c+ x-.d-

Proof. Suppose f is monotone increasing and bounded on the open interval
I= (a, b), where a< b.
(a) Suppose c E (a,b], and let A = {f(x) : a < x < c}. Then A is
nonempty since f(x 1 ) EA for any a< x 1 < c. Also, A is bounded above since
f is bounded on I. Thus, by the completeness property, :Ju= sup A. We shall
prove that lim f(x) = u.
x-+c-
Let c > 0. By the c criterion for supremum (Theorem 1.6.6), :J y E A 3
u-E: < y. Since y EA, y = f(xo) for some xo E (a, c). So we have u -E: < f(xo).
Let o = c - xo. Then o > 0 and
c - o < x < c =? xo < x < c by definition of O
=? f(xo) ::; f(x) ::; u since f is monotone on (a, b)
=? u - c < f(x) < u + c
=? lf(x) - ul < c.
Therefore, lim f ( x) = u.
(b) Exercise 8.
( c) Exercise 9.
(d) Exercise 10. •

Theorem 5.2.1 8 Suppose f is monotone decreasing and bounded on an open

interval I= (a, b), where a< b. Then

(a) Ve E (a,b], lim f(x) exists and equals inf{J(x): a< x < c};

x--+c-

(b) Ve E [a,b), lim f(x) exists and equals sup{f(x): c < x < b};

x-+c+

(c) Ve E (a, b), lim f(x) ~ f(c) ~ lim f(x); i.e., f(c) ~ f(c) ~ f(c+);

x_.c- x--+c+

(d) Ve< din (a, b), lim f(x) ~ lim f(x); i.e., f(c+) ~ f(d-).

x-+c+ x--+d-

Proof. Exercise 11. •