Problem Set 72: Continuity

1. State in your own words what it means for a function ff to be continuous at x=cx=c.

2. State in your own words what it means for a function to be continuous on the interval (a,b)(a,b).

For the following exercises, determine why the function ff is discontinuous at a given point aa on the graph. State which condition fails.

3. f(x)=ln | x+3 |,a=3f(x)=ln | x+3 |,a=3

4. f(x)=ln | 5x2 |,a=25f(x)=ln | 5x2 |,a=25

5. f(x)=x216x+4,a=4f(x)=x216x+4,a=4

6. f(x)=x216xx,a=0f(x)=x216xx,a=0

7. f(x)={x,x32x,x=3a=3f(x)={x,x32x,x=3a=3

8. f(x)={5,x03,x=0a=0f(x)={5,x03,x=0a=0

9. f(x)={12x,x23,x=2a=2f(x)={12x,x23,x=2a=2

10. f(x)={1x+6,x=6x2,x6a=6f(x)={1x+6,x=6x2,x6a=6

11. f(x)={3+x,x<1x,x=1x2,x>1a=1f(x)=3+x,x<1x,x=1x2,x>1a=1

12. f(x)={3x,x<1x,x=12x2,x>1a=1f(x)=3x,x<1x,x=12x2,x>1a=1

13. f(x)={3+2x,x<1x,x=1x2,x>1a=1f(x)=3+2x,x<1x,x=1x2,x>1a=1

14. f(x)={x2,x<22x+1,x=2x3,x>2a=2f(x)=x2,x<22x+1,x=2x3,x>2a=2

15. f(x)={x29x+3,x<3x9,x=31x,x>3a=3f(x)=⎪ ⎪⎪ ⎪x29x+3,x<3x9,x=31x,x>3a=3

16. f(x)={x29x+3,x<3x9,x=36,x>3a=3f(x)=x29x+3,x<3x9,x=36,x>3a=3

17. f(x)=x24x2, a=2f(x)=x24x2, a=2

18. f(x)=25x2x210x+25, a=5f(x)=25x2x210x+25, a=5

19. f(x)=x39xx2+11x+24, a=3f(x)=x39xx2+11x+24, a=3

20. f(x)=x327x23x, a=3f(x)=x327x23x, a=3

21. f(x)=x|x|, a=0f(x)=x|x|, a=0

22. f(x)=2|x+2|x+2, a=2f(x)=2|x+2|x+2, a=2

For the following exercises, determine whether or not the given function ff is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

23. f(x)=x32x15f(x)=x32x15

24. f(x)=x22x15x5f(x)=x22x15x5

25. f(x)=23x+4f(x)=23x+4

26. f(x)=sin(3x)f(x)=sin(3x)

27. f(x)=|x2|x22xf(x)=|x2|x22x

28. f(x)=tan(x)+2f(x)=tan(x)+2

29. f(x)=2x+5xf(x)=2x+5x

30. f(x)=log2(x)f(x)=log2(x)

31. f(x)=ln x2f(x)=ln x2

32. f(x)=e2xf(x)=e2x

33. f(x)=x4f(x)=x4

34. f(x)=sec(x)3f(x)=sec(x)3 .

35. f(x)=x2+sin(x)f(x)=x2+sin(x)

36. Determine the values of bb and cc such that the following function is continuous on the entire real number line.

f(x)={x+1,1<x<3x2+bx+c,|x2|1f(x)={x+1,1<x<3x2+bx+c,|x2|1

For the following exercises, refer to Figure 15. Each square represents one square unit. For each value of a, determine which of the three conditions of continuity are satisfied at x=a and which are not.

Graph of a piecewise function where at x = -3 the line is disconnected, at x = 2 there is a removable discontinuity, and at x = 4 there is a removable discontinuity and f(4) exists.

Figure 15

37. x=3

38. x=2

39. x=4

For the following exercises, use a graphing utility to graph the function f(x)=sin(12πx) as in Figure 16. Set the x-axis a short distance before and after 0 to illustrate the point of discontinuity.

Graph of the sinusodial function with a viewing window of [-10, 10] by [-1, 1].

Figure 16

40. Which conditions for continuity fail at the point of discontinuity?

41. Evaluate f(0).

42. Solve for x if f(x)=0.

43. What is the domain of f(x)?

For the following exercises, consider the function shown in Figure 17.

Graph of a piecewise function where at x = -1 the line is disconnected and at x = 1 there is a removable discontinuity.

Figure 17

44. At what x-coordinates is the function discontinuous?

45. What condition of continuity is violated at these points?

46. Consider the function shown in Figure 18. At what x-coordinates is the function discontinuous? What condition(s) of continuity were violated?

Graph of a piecewise function where at x = -1 the line is disconnected and where at x = 1 and x = 2 there are a removable discontinuities.

Figure 18

47. Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at x=7 and x=1.

48. The function f(x)=x31x1 is graphed in Figure 19. It appears to be continuous on the interval [3,3], but there is an x-value on that interval at which the function is discontinuous. Determine the value of x at which the function is discontinuous, and explain the pitfall of utilizing technology when considering continuity of a function by examining its graph.

Graph of the function f(x) = (x^3 - 1)/(x-1).

Figure 19

49. Find the limit limx1f(x) and determine if the following function is continuous at x=1:

50. The function is discontinuous at x=1 because the limit as x approaches 1 is 5 and f(1)=2.

51. The graph of f(x)=sin(2x)x is shown in Figure 20. Is the function f(x) continuous at x=0? Why or why not?

Graph of the function f(x) = sin(2x)/x with a viewing window of [-4.5, 4.5] by [-1, 2.5]

Figure 20