1. State in your own words what it means for a function to be continuous at .
2. State in your own words what it means for a function to be continuous on the interval .
For the following exercises, determine why the function is discontinuous at a given point on the graph. State which condition fails.
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For the following exercises, determine whether or not the given function 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.
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36. Determine the values of and such that the following function is continuous on the entire real number line.
For the following exercises, refer to Figure 15. Each square represents one square unit. For each value of , determine which of the three conditions of continuity are satisfied at and which are not.

Figure 15
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For the following exercises, use a graphing utility to graph the function 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 .
42. Solve for if .
43. What is the domain of
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 and .
48. The function is graphed in Figure 19. It appears to be continuous on the interval , but there is an x-value on that interval at which the function is discontinuous. Determine the value of 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 and determine if the following function is continuous at
50. The function is discontinuous at because the limit as approaches 1 is 5 and .
51. The graph of is shown in Figure 20. Is the function continuous at 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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- Precalculus. Authored by: Jay Abramson, et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution. License Terms: Download for free at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface