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 | 5x−2 |,a=25f(x)=ln | 5x−2 |,a=25
5. f(x)=x2−16x+4,a=−4f(x)=x2−16x+4,a=−4
6. f(x)=x2−16xx,a=0f(x)=x2−16xx,a=0
7. f(x)={x,x≠32x,x=3a=3f(x)={x,x≠32x,x=3a=3
8. f(x)={5,x≠03,x=0a=0f(x)={5,x≠03,x=0a=0
9. f(x)={12−x,x≠23,x=2a=2f(x)={12−x,x≠23,x=2a=2
10. f(x)={1x+6,x=−6x2,x≠−6a=−6f(x)={1x+6,x=−6x2,x≠−6a=−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)={3−x,x<1x,x=12x2,x>1a=1f(x)=⎧⎨⎩3−x,x<1x,x=12x2,x>1a=1
13. f(x)={3+2x,x<1x,x=1−x2,x>1a=1f(x)=⎧⎨⎩3+2x,x<1x,x=1−x2,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)={x2−9x+3,x<−3x−9,x=−31x,x>−3a=−3f(x)=⎧⎪ ⎪⎨⎪ ⎪⎩x2−9x+3,x<−3x−9,x=−31x,x>−3a=−3
16. f(x)={x2−9x+3,x<−3x−9,x=−3−6,x>−3a=3f(x)=⎧⎪⎨⎪⎩x2−9x+3,x<−3x−9,x=−3−6,x>−3a=3
17. f(x)=x2−4x−2, a=2f(x)=x2−4x−2, a=2
18. f(x)=25−x2x2−10x+25, a=5f(x)=25−x2x2−10x+25, a=5
19. f(x)=x3−9xx2+11x+24, a=−3f(x)=x3−9xx2+11x+24, a=−3
20. f(x)=x3−27x2−3x, 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)=x3−2x−15
24. f(x)=x2−2x−15x−5
25. f(x)=2⋅3x+4
26. f(x)=−sin(3x)
27. f(x)=|x−2|x2−2x
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)=√x−4
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,|x−2|≥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.

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].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3675/2018/09/27185344/CNX_Precalc_Figure_12_03_202F.jpg)
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.

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?

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)=x3−1x−1 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.

Figure 19
49. Find the limit limx→1f(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]](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3675/2018/09/27185352/CNX_Precalc_Figure_12_03_206.jpg)
Figure 20
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