Problem Set 72: Continuity

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

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

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

3. f(x)=ln | x+3 |,a=3

4. f(x)=ln | 5x2 |,a=25

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

6. f(x)=x216xx,a=0

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

8. f(x)={5,x03,x=0a=0

9. f(x)={12x,x23,x=2a=2

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

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

12. f(x)={3x,x<1x,x=12x2,x>1a=1

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

14. f(x)={x2,x<22x+1,x=2x3,x>2a=2

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

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

17. f(x)=x24x2, a=2

18. f(x)=25x2x210x+25, a=5

19. f(x)=x39xx2+11x+24, a=3

20. f(x)=x327x23x, a=3

21. f(x)=x|x|, a=0

22. f(x)=2|x+2|x+2, a=2

For the following exercises, determine whether or not the given function f 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)=x32x15

24. f(x)=x22x15x5

25. f(x)=23x+4

26. f(x)=sin(3x)

27. f(x)=|x2|x22x

28. f(x)=tan(x)+2

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

30. f(x)=log2(x)

31. f(x)=ln x2

32. f(x)=e2x

33. f(x)=x4

34. f(x)=sec(x)3 .

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

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

f(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