Continuity & One-Sided Limits

1. Continuity & One-Sided Limits Section 1.4

2. After this lesson, you will be able to: • determine continuity at a point and continuity on an open interval • determine one-sided limits and continuity on a closed interval • understand and use the Intermediate Value Theorem

3. f c Continuous Function Continuity at a Point fis continuous at c if the following three conditions are satisfied: f(c) 1) f(c) is defined Muy importante! Be able to recite by heart.

4. Some examples of functions that are NOT continuous at c. 1)f is not continuous at c because

5. Some examples of functions that are NOT continuous at c. 2)f is not continuous at c because

6. Some examples of functions that are NOT continuous at c. 3)f is not continuous at c because

7. f f(c) a c b Continuity on an Open Interval A function is continuous on an open interval (a, b) if it is continuous at each point in the interval. Continuous Function on the open interval (a, b). A function that is continuous over the set of real numbers is called everywhere continuous.

8. Discontinuity A function that is defined on the interval (a, b) (except possibly at c), and is not continuous at c is said to have a ___________________ at c. • There are two types of discontinuities: • Removable: If f can be made continuous by defining or redefining f(c)…this type would appear as a _________ in the graph. • Non-removable: f cannot be made continuous by changing only f(c)...e.g. a vertical asymptote at x = c.

9. Removable Discontinuity f f(c) Removable discontinuity at x=c, since f can be made continuous by just redefining f(c). c Discontinuity at x = c

10. Non-removable Discontinuity This function has a non-removable discontinuity at x = c. If only f(c) was redefined, the function would still be discontinuous. f f(c) c

11. Continuity Consider the graphs of the following functions and discuss the continuity of each.

12. One-sided Limits Limit from the right(right-hand limit) Limit from the left(left-hand limit)

13. One-sided Limits *One-sided limits are great for radical functions.* Example: We can’t take the limit of this function as x approaches 0 from the left side since negative numbers are not in the domain. We can only take the right limit.

14. One-sided Limits Example: Graph the function on the calculator. Then, determine the limit graphically. What is the domain of this function? ___________________ *One-sided limits are also great for functions with a closed interval as the domain.*

15. Examples 1 & 2

16. Example 3 is called the ______________ __________________ ______________.

17. Example: Find the one-sided limits at the endpoints of the function, At –2, we can only take a right limit: 2 -2 At 2, we can only take a left limit: Continuity on a Closed Interval A function is continuous on a closed interval if it is continuous everywhere inside the interval and has one-sided continuity at the endpoints.

18. Examples Discuss the continuity of the function on the closed interval.

19. Properties of Continuity If k is a real number and f and g are continuous at x = c, then f + g, f-g, f•g, kf, and f/g(provided g(c)  0) are also continuous. The following types of functions are continuous at every point within their domain: • Polynomial Functions • Rational Functions • Radical Functions • Trig Functions

20. We can also say, Continuity of a Composite Function If g is continuous at c and f is continuous at g(c), then the composite function given by is continuous at c.

21. Examples 1 & 2 Describe the intervals on which each function is continuous. Verify graphically.

22. Examples 3 & 4 Describe the intervals on which each function is continuous. Verify graphically.

23. Example 5 Describe the intervals on which each function is continuous. Verify graphically.

24. Example 6 Find the constant a such that f(x) is continuous on .

25. This theorem doesn’t tell you the value of c, but just tells you one exists. This is an existence theorem. f(b) k f(a) a c b f(c) = k Intermediate Value Theorem (IVT)“It ain’t a sandwich unless there’s something between the bread.” If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = k. A continuous function takes on all values between any two points that it assumes.

26. Weight (lbs) Day IVT Example Consider this example: A baby weighs 7.3 lbs on day 1 and weighs 8.9 lbs on day 24. There has to be a time between day 1 & 24 when the baby weighed exactly 8.0 lbs.

27. Finding Zeros So how do we use the IVT? We’ll use it primarily to locate zeros of a function that is continuous on an interval.

28. Finding Zeros Use the Intermediate Value Theorem to show that has a zero in the interval [0, 1]. Then use your calculator to find the zero accurate to four decimal places.

29. Homework Section 1.4:page 78 #1-17 odd, 25, 29 – 37 odd, 69, 71, 77, 79 Don’t Skip!!!!