664 Appendix C • Answers & Hints for Selected Exercises
- (a) Suppose f is uniformly continuous on (a, b) and (c, d), where a < b <
c < d. Let c > 0. Then 381 , 82 > 0 3
Vx, y E (a, b), Ix - YI < 81 =? lf(x) - f(y)I < c, and
Vx, y E (c, d), Ix - YI < 82 =? lf(x) - f(y)I < c.
Choose 8:::; min{8 1 , 8 2 , c - b}. Then Vx, y E (a, b) U (c, d),
Ix - YI < 8 =? x, y E (a, b) or x, y E (c, d), and Ix - YI < 81 and 82
=? lf(x) - f(y)I < c.
:. f is uniformly continuous on (a, b) U (c, d).
EXERCISE SET 5.5
- Suppose f is monotone increasing on A and x1, x2 E A. Then x2 :::; x1 =?
f(x2):::; f(x1), so f(x1) < f(x2) =? f(x2) i. f(x1) =? x2 i. X1 =? X1 < x2. - (a) x <yin I=? [f(x):::; f(y) and g(x):::; g(y)] =? f(x)+g(x):::; f(y)+g(y).
:. f + g is monotone increasing on I.
(b) Take f(x) = g(x) = X'On [-1, 1].
(c) Suppose f, g nonnegative on I. Then Vx, y EI,
x < y =? [f(x) :::; f(y) and g(x) :::; g(y)] =? f(x)g(x) < f(y)g(x) <
f(y)g(y).
:. f g is monotone increasing on I. - Take f(x) = x if xis rational, 1 - x if xis irrational.
- Redo the proof of Thm. 5.5.2, changing "increasing" to "decreasing" and
changing inequalities appropriately. - Suppose a< b < c in I and f(b) < min{f(a), f(c)}. Then 3r E JR. 3 f(b) <
y < min{f(a), J(c)}. Then f(b) < y < f(a), so by the intermediate value thm,
3x1 E (a, b) 3 f(xi) = y. Similarly, f(b) < y < f(c), so 3x2 E (a, b) 3 f(x2) =
y. Then we have x1 =/:-x2 in I 3 f (x1) = f (x2), contradicting the hypothesis
that f is 1-1 on I. :. in this case, f(b) is between f(a) and f(c). - Let x E JR.. Then 3 unique n E N 3 x E [n, n + 1), so 0 :::; x - n <
- Define <p(x) = cp(x - n) + n. Since cp : [O, 1) ---+ [O, 1) is continuous and
increasing, <j5: [n, n+ 1) ---+ [n, n+ 1) is continuous and increasing. Also, Vn EN,
lim <p(x) = lim cp(x - n) + n = cp(l) + n = 1+n=<p(n+1). Thus,
x->(n+1)- x->(n+l)-
lim <p(x)=<p(n)and lim <p(x)= lim cp(x-n)+n=cp(O)+n=n=<p(n),
x-+n- x-+n+ x-+n+
so <j5 is continuous on JR.. Moreover, <j5 is increasing on JR. since <p[n, n + 1) ~
[n,n+ 1).
EXERCISE SET 5.6
- Let x E JR.. By the density of Q in JR., 3 rational r 1 3 x - 1 < r 1 < x. Then,
3 rational r2 3 max{x - ~' ri} < r 2 < x. Continuing inductively, 3 rational