1. Continuity Lesson 1.1.14

2. Learning Objectives Given a function, determine if it is continuous at a certain point using the three criteria for continuity. Evaluate a one-sided limit. Determine if a function is continuous on a closed interval using one-sided limits. Apply the above objectives to the greatest integer function.

3. Continuity: The Big Picture Continuous functions are the “normal” functions. They are the ones that do not have any holes, asymptotes, jumps, or matter that break the smooth flow of the function. The functions on the right are continuous throughout y = x y = x 2

4. More Continuous Functions y = sin x y = x 3 y = 3y = e x

5. Which are continuous throughout?

6. Continuity at a Point The functions on the right are not continuous at x = 0, but they are continuous at all of the other points.

7. You can usually tell by looking if a function is continuous at a certain point. However, in calculus, we have a more formal definition of continuity.

8. Definition of Continuous A function f(x) is continuous at x = c if:

9. The functions on the right are y = 1/x and y = |x|/x. For both functions, f(0) does not exist, for it is undefined. Therefore, the functions are discontinuous at x = 0

10. Pre-Example 1 Observe the function on the right. Let’s see if it is continuous at x = 1. It certainly meets the first criteria, for f(1) exists. In fact, f(1) = 1. However, the limit as x  1 does not exist. Therefore, the function is not continuous at x = 1

11. Pre-Example 2 Observe the function on the right. Is it continuous at x = 1? First criteria: f(1) = 3. f(1) exists! The limit as x  1 is 1. The limit exists too! However, f(1) and the limit do not equal. Fails third criteria.

12. Example 1 Determine if the following function is continuous at h(1) and h(2). (Use your three criteria.)

13. One-sided Limits We’ve learned: The limit as x approaches c from the left. The limit as x approaches c from the right. The overall limit. This exists only if the left and right limits are equal.

14. In this lesson, you will get problems that ask you to evaluate one-sided limits.

15. Example 2 Evaluate:

16. The Greatest Integer Function This is also known as the step function. You can probably see why in the graph of y = ||x|| on the right. To evaluate the above function for a given x, determine the integer that is equal to or just below x.

17. For example: ||3.7|| = 3 ||4|| = 4 ||-5.9|| = 6

18. The greatest integer function is one in which one-sided limits are especially used. For example, what is:

19. Interval Review Remember: (a, b) is called an open interval. It is the range of every single number from a to b, not including a and b. [a, b] is called a closed interval. It is like an open interval except that it includes a and b.

20. Continuity on an Open Interval A function is continuous on the open interval (a, b) if it is continuous at every point in that interval.

21. Continuity on a Closed Interval A function is continuous on the closed interval [a, b] if: It is continuous on the open interval (a, b) The following one-sided limits exist:

22. Example 3 Determine if the function below is continuous on the interval [-1, 1]. Remember: First determine if it is continuous on (-1, 1) (You can do this by graphing.) Then evaluate the one-sided limits at -1 and 1.

23. Wrap-Up Know how to determine if a function is continuous at a point using the three criteria. Know one-sided limits. Know the greatest integer function. Know how to determine continuity on intervals.

24. Homework Reteaching