1549901369-Elements_of_Real_Analysis__Denlinger_
Answers & Hints for Selected Exercises 651 ( {:::) Suppose every real number is a cluster point of A. Let a < b. Let c = ...
652 Appendix C • Answers & Hints for Selected Exercises EXERCISE SET 3.4 C ~ [O, 1], so it is bounded. C is the intersectio ...
Answers & Hints for Selected Exercises 653 (c) If B ~A, then An B = B , and by (μ1) and (μ5), 0::; μ(A - B) =μ(A) - μ(An B) ...
654 Appendix C • Answers & Hints for Selected Exercises By Parts (a) and (b), lim [f(x)-g(x)] = lim [f(x)+(-l)g(x)] = lim f ...
Answers & Hints for Selected Exercises 655 (e) f(x 0 ) = 2 and f(xci) = 3, so lim f(x) does not exist. X-tXQ (g) f(x 0 ) = 0 ...
656 Appendix C • Answers & Hints for Selected Exercises and 3 02 > 0 3 0 <Ix - xol < 02 => g(x) > VM. Then 0 ...
Answers & Hints for Selected Exercises 657 (b) \Ix :f: 0, 0:::; j si~x j :::; - 11 x 1 --> 0, so by squeeze principle, x- ...
658 Appendix C • Answers & Hints for Selected Exercises :::::} l4x + 31 Ix - 21 < 15 1 e 5 :::::} l(4x^2 - 5x - 3) - 31 & ...
Answers & Hints for Selected Exercises 659 Suppose I is continuous at x 0. Then 38 > 0 3 '<Ix E V(f), Ix - xol < ...
660 Appendix C • Answers & Hints for Selected Exercises f has a removable discontinuity at x 0 ::::} lim f(x) exists, but e ...
Answers & Hints for Selected Exercises 661 (b) Suppose A not open. Let g : JR ---+ JR denote the characteristic function of ...
662 Appendix C • Answers & Hints for Selected Exercises Suppose 1 : I ~ JR is continuous, strictly monotone, and bounded on ...
Answers & Hints for Selected Exercises 663 EXERCISE SET 5.4 For [-3, 1], choose 6 = c:/17. For (-2, 2), choose 6 = c:/ 13. ...
664 Appendix C • Answers & Hints for Selected Exercises (a) Suppose f is uniformly continuous on (a, b) and (c, d), where a ...
Answers & Hints for Selected Exercises 665 rk+l 3 max{x - k!l' rk} < rk+l < x. Then, 'Vn EN, x - ~ < rn < rn+l & ...
666 Appendix C • Answers & Hints for Selected Exercises 1 1 (f) lim .;x - vxo = lim Vx - ftO = lim -l X->Xo X - XQ X-> ...
Answers & Hints for Selected Exercises 667 . f(x) - f(O).. (b) hm 0 = hm xr-l sm ~. By the above argument, x->O+ X - x-&g ...
668 Appendix C 11 Answers & Hints for Selected Exercises d~fo(goh) = J'[(goh)(x)]·(goh)'(x) = J'[(goh)(x)]· g'(h(x))·h'(x) ...
Answers & Hints for Selected Exercises 669 Revise the proof of Part (a), making appropriate changes. (a) Increasing on (-2, ...
670 Appendix C • Answers & Hints for Selected Exercises Suppose f"(x) = 0 for all x ER By Thm. 6.4.4, f'(x) must be constan ...
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