Page 72 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 3] FUNCTIONS, LIMITS, AND CONTINUITY 63

3.72. Prove that lim f ðxÞ¼ l if and only if lim f ðxÞ¼ lim f ðxÞ¼ l.
x!x 0 x!x 0 þ x!x 0
CONTINUITY
In the following problems assume the largest possible domain unless otherwise stated.
2
3.73. Prove that f ðxÞ¼ x 3x þ 2iscontinuous at x ¼ 4.
3.74. Prove that f ðxÞ¼ 1=x is continuous (a)at x ¼ 2, (b)in 1 @ x @ 3.
3.75. Investigate the continuity of each of the following functions at the indicated points:
3
sin x x 8
; x 6¼ 0; f ð0Þ¼ 0; x ¼ 0 ; x 6¼ 2; f ð2Þ¼ 3; x ¼ 2
x x 4
ðaÞ f ðxÞ¼ ðcÞ f ðxÞ¼ 2
sin x; 0 < x < 1

f ðxÞ¼ x jxj; x ¼ 0 ; x ¼ 1:
ðbÞ ðdÞ f ðxÞ¼
ln x 1 < x < 2
Ans. (a)discontinuous, (b)continuous, (c)continuous, (d)discontinuous
3.76. If ½xŠ¼ greatest integer @ x,investigate the continuity of f ðxÞ¼ x ½xŠ in the interval (a)1 < x < 2,
(b)1 @ x @ 2.
3
3.77. Prove that f ðxÞ¼ x is continuous in every finite interval.
3.78. If f ðxÞ=gðxÞ and gðxÞ are continuous at x ¼ x 0 ,prove that f ðxÞ must be continuous at x ¼ x 0 .

3.79. Prove that f ðxÞ¼ ðtan 1 xÞ=x, f ð0Þ¼ 1iscontinuous at x ¼ 0.

3.80. Prove that a polynomial is continuous in every finite interval.
3.81. If f ðxÞ and gðxÞ are polynomials, prove that f ðxÞ=gðxÞ is continuous at each point x ¼ x 0 for which gðx 0 Þ 6¼ 0.

3.82. Give the points of discontinuity of each of the following functions.
x p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðx 3Þð6 xÞ; 3 @ x @ 6
ðaÞ f ðxÞ¼ ðcÞ f ðxÞ¼
ðx 2Þðx 4Þ
2
f ðxÞ¼ x sin 1=x; x 6¼ 0; f ð0Þ¼ 0 1 :
ðbÞ ðdÞ f ðxÞ¼
1 þ 2 sin x
Ans. (a) x ¼ 2; 4, (b) none, (c) none, (d) x ¼ 7 =6 2m ; 11 =6 2m ; m ¼ 0; 1; 2; ...
UNIFORM CONTINUITY
3
3.83. Prove that f ðxÞ¼ x is uniformly continuous in (a)0 < x < 2, (b)0 @ x @ 2, (c)any finite interval.
2
3.84. Prove that f ðxÞ¼ x is not uniformly continuous in 0 < x < 1.
2
3.85. If a is a constant, prove that f ðxÞ¼ 1=x is (a)continuous in a < x < 1 if a A 0, (b) uniformly
continuous in a < x < 1 if a > 0, (c) not uniformly continuous in 0 < x < 1.
3.86. If f ðxÞ and gðxÞ are uniformly continuous in the same interval, prove that (a) f ðxÞ gðxÞ and (b) f ðxÞgðxÞ
are uniformly continuous in the interval. State and prove an analogous theorem for f ðxÞ=gðxÞ.

MISCELLANEOUS PROBLEMS
3.87. Give an ‘‘ ; ’’ proof of the theorem of Problem 3.31.
3.88. (a)Prove that the equation tan x ¼ x has a real positive root in each of the intervals =2 < x < 3 =2,
3 =2 < x < 5 =2, 5 =2 < x < 7 =2; ... .