1. 𝑓(−4)= 𝑓(−1)= 𝑓(1)= lim 𝑥→1 𝑓 𝑥 = lim 𝑥→2 𝑓(𝑥) = lim 𝑥→−3 𝑓(𝑥) =
Warm-Up
Find the indicated values
based on the graph below:
𝑓(−4)=
𝑓(−1)=
𝑓(1)=
lim 𝑥→1 𝑓 𝑥 =
lim 𝑥→2 𝑓(𝑥) =
lim 𝑥→−3 𝑓(𝑥) =
lim 𝑥→ −1 − 𝑓 𝑥 =
lim 𝑥→ −1 + 𝑓(𝑥) =
lim 𝑥→−1 𝑓(𝑥) =

2. F(x) J(x) K(x) G(x) H(x) Warm-Up Given the graph of f(x),
which is f(|x|) graph?
J(x)
K(x)
G(x)
H(x)

3. Warm-Up

4. 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 (

5. 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.

6. Examples
Continuous Functions

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

8. 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.

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

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

11. 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.

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

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

14. 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.

15. 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]

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

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

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

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

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

21. Important Idea
Theorem 1.10:
exists iff

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

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

24. 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.

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

26. 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.

27. 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.

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

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

30. Lesson Close
3 things must be true for a function to be continuous. What are they?

31. Assignment
Page 78 #1-6 all,7-19 odd,27-49 odd, 57, and 59