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Unformatted text preview: 10. 11. 12. 13. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. Section 2.6 Continuity (a) Yes, f(l):1 (b) Yes, x1931] f(x)=2 (c) No (d) No (a) No (b) No [—170) U (0, 1) U (1.2) U (2,3) f(2) = 0, since X lin5_ f(x) : —2(2) + 4 = O = lim f(x) x——>2+ f(l) should be changed to 2 : lim1 f(x) x —) Nonremovable discontinuity at x = 1 because liml f(x) fails to exist (x lim_ f(x) : 1 and lim1+ f(x) : 0). x -—> —v x -—> Removable discontinuity at x : 0 by assigning the number lim0 f(x) = O to be the value of f(O) rather than X -—) f(0) = 1. lim+ f(x) : 1). x—r Nonremovable discontinuity at x : 1 because lim f(x) fails to exist ( lin}_ f(x) : 2 and X —) x—v Removable discontinuity at x : 2 by assigning the number lim2 f(x) = 1 to be the value of f(2) rather than x —> f(2) = 2. Discontinuous only when x — 2 = 0 :> x = 2 14. Discontinuous only when (x + 2)2 = O :> x = -2 Discontinuous only when x2 — 4x + 3 = 0 :> (x — 3)(x — 1) z 0 => x = 3 orx =1 Discontinuous only when x2 ~ 3x — 10 = 0 :> (x — 5)(x + 2): 0 => x = 5 or x = —2 Continuous everywhere. ( |x — 1| + sin x defined for all x; limits exist and are equal to function values.) Continuous everywhere. ( |x| + 1 76 0 for all x; limits exist and are equal to function values.) Discontinuous only at x = O Discontinuous at odd integer multiples of 1, i.e., x = (2n — 1) 1, n an integer, but continuous at all other x. M Discontinuous when 2x is an integer multiple of 7r, i.e., 2x 2 rm, [1 an integer :: x : 2 , n an integer, but continuous at all other x. Discontinuous when M is an odd inte er multi le of 5, i.e., "—X : 2n — 1) I, n an inte er :> x : 2n — 1, n an 2 g p 2 2 g integer (i.e., x is an odd integer). Continuous everywhere else. Discontinuous at odd inte er multi les of 1, i.e., x = (2n — 1) E. n an inte er, but continuous at all other x. g p 2 2 g Continuous everywhere since x4 + 1 2 1 and —1 s sin x g 1 :> 0 _<_ sin2x _<_ l :> 1 + sin2 x 2 1; limits exist and are equal to the function values. 3 Discontinuous when 2x + 3 < 0 or x < — 5 :> continuous on the interval [— g, 00) . 79 ...
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