1. 1-4: Continuity and One-Sided Limits
Objectives:
Define and explore properties of continuity
Discuss one-sided limits
Introduce Intermediate Value Theorem
©2002 Roy L. Gover

2. 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.

3. Examples
Continuous Functions

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

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

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

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

8. 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.

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

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

11. 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.

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

13. Definition
is a limit from the right which means x c from values greater than c

14. Definition
is a limit from the left which means x c from values less than c

15. Example
Find the limit of f(x) as x approaches 1 from the right:

16. Example
Find the limit of f(x) as x approaches 1 from the left:

17. Example
Find the limit of f(x) as x approaches 1:

18. Important Idea
Theorem 1.10:
exists iff

19. Try This
Use the graph to determine the limit, the limit from the right & the limit from the left as x0.

20. Try This
Use the graph to determine the limit, the limit from the right & the limit from the left as x1.
x=1

21. Intermediate Value Theorem
Theorem 1.13: 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.

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

23. 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.

24. Example
Ryan was 20 inches long when born and 30 inches long when 9 months old. Since growth is continuous, there was a time between birth and 9 months when he was 25 inches long.

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

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

27. Lesson Close
Tell me one thing you know about continuity and discontinuity.

28. Assignment
92/1-6 all,7-19 odd,27-49 odd,57,59,65, 81-84