2. 2.7: Continuity and the Intermediate Value Theorem Objectives: Define and explore properties of continuity Introduce Intermediate Value Theorem ©2002 Roy L. Gover (roygover@att.net)

3. Definition f(x) is continuous at x=c if and only if there are no holes, jumps, skips or gaps in the graph of f(x) at c.

4. Examples Continuous Functions

5. Examples Discontinuous Functions Removable discontinuity Jump Discontinuity (non- removable) Infinite discontinuity (non- removable)

6. Definition f(x) is continuous at x=c if and only if: 1. f (c) is defined …and 2. exists …and 3.

7. Examples x =2 Discontinuous at x =2 because f(2) is not defined

8. Examples x =2 Discontinuous at x =2 because, although f(2) is defined,

9. Definition f(x) is continuous on the open interval ( a,b ) if and only if f(x) is continuous at every point in the interval.

10. Try This Find the values of x (if any) where f is not continuous. Is the discontinuity removable? Continuous for all x

11. Try This Find the values of x (if any) where f is not continuous. Is the discontinuity removable? Discontinuous at x =o, not removable

12. Example Find the values of k, if possible, that will make the function continuous.

13. Definition f(x) is continuous on the closed interval [ a,b ] iff it is continuous on ( a,b ) and continuous from the right at a and continuous from the left at b.

14. Example a b f(x) f(x) is continuous on ( a,b ) f(x) is continuous from the right at a f(x) is continuous from the left at b f(x) is continuous on [ a,b ]

15. Graphing calculators can make non-continuous functions appear continuous. Graph: CATALOG F floor( Note resolution. The calculator “connects the dots” which covers up the discontinuities.

16. Graphing calculators can make non-continuous functions appear continuous. Graph: CATALOG F floor( GRAPH The open and closed circles do not show, but we can see the discontinuities.  If we change the plot style to “dot” and the resolution to 1, then we get a graph that is closer to the correct floor graph.

17. Intermediate Value Theorem If f is continuous on [ a,b ] and k is a number between f(a) & f( b), then there exists a number c between a & b such that f(c ) = k.

18. Intermediate Value Theorem a f(a) b f(b) k c

19. Intermediate Value Theorem an existence theorem; it guarantees a number exists but doesn’t give a method for finding the number. it says that a continuous function never takes on 2 values without taking on all the values between.

20. Example Kaley was 20 inches long when born. Let’s say that she will be 30 inches long when 15 months old. Since growth is continuous, there was a time between birth and 15 months when she was 25 inches long.

21. Try This Use the Intermediate Value Theorem to show that has a zero in the interval [-1,1].

22. Solution therefore, by the Intermediate Value Theorem, there must be a f (c)=0 where