Solutions 72: Continuity

Solutions to Odd-Numbered Exercises

1. Informally, if a function is continuous at x=c, then there is no break in the graph of the function at f(c), and f(c) is defined.

3. discontinuous at a=3 ; f(3) does not exist

5. removable discontinuity at a=4 ; f(4) is not defined

7. discontinuous at a=3 ; limx3f(x)=3, but f(3)=6, which is not equal to the limit.

9. limx2f(x) does not exist.

11. limx1f(x)=4;limx1+f(x)=1 . Therefore, limx1f(x) does not exist.

13. limx1f(x)=5limx1+f(x)=1 . Thus limx1f(x) does not exist.

15. limx3f(x)=6 , limx3+f(x)=13

Therefore, limx3f(x) does not exist.

17. f(2) is not defined.

19. f(3) is not defined.

21. f(0) is not defined.

23. Continuous on (,)

25. Continuous on (,)

27. Discontinuous at x=0 and x=2

29. Discontinuous at x=0

31. Continuous on (0,)

33. Continuous on [4,)

35. Continuous on (,) .

37. 1, but not 2 or 3

39. 1 and 2, but not 3

41. f(0) is undefined.

43. (,0)(0,)

45. At x=1, the limit does not exist. At x=1, f(1) does not exist.
At x=2, there appears to be a vertical asymptote, and the limit does not exist.

47. x3+6x27x(x+7)(x1)

49. fx={x2+4x12x=1