1. Chapter 3 Limits and the Derivative
Section 3
Continuity
(Part 1)

2. Learning Objectives for Section 3.3 Continuity
The student will understand the concept of continuity
The student will be able to apply the continuity properties
The student will be able to solve word problems.
The student will be able to solve inequalities
Barnett/Ziegler/Byleen Business Calculus 12e

3. Continuity
In this lesson, we’ll take a closer look at graphs that are discontinuous due to:
Holes
Gaps
Asymptotes
Barnett/Ziegler/Byleen Business Calculus 12e

4. Definition of Continuity
A function f is continuous at a point x = c if it meets these three criteria: 1.
2. f (c)  undefined 3.
lim 𝑥→𝑐 𝑓(𝑥)≠𝐷𝑁𝐸
Barnett/Ziegler/Byleen Business Calculus 12e

5. Example 1 Is f(x) continuous at x = 2 ? ? lim 𝑥→2 𝑓 𝑥 =𝑓(2) 𝐷𝑁𝐸
𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
f(x) is not continuous at x = 2
Is f(x) continuous at x = 3 ? ?
lim 𝑥→3 𝑓 𝑥 =𝑓(3) 1
𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
f(x) is not continuous at x = 3
Continuous over the interval:
−∞, 2 ∪(2, 3)∪(3,∞)
Barnett/Ziegler/Byleen Business Calculus 12e

6. Example 2 Is f(x) continuous at x = -2 ? ? lim 𝑥→−2 𝑓 𝑥 =𝑓(−2) 3 −1
f(x) is not continuous at x = -2
Continuous over the interval:
−∞, −2 ∪(−2,∞)
Barnett/Ziegler/Byleen Business Calculus 12e

7. Example 3 𝑓 𝑥 = 2𝑥 2 +𝑥−1 Is f(x) continuous at x = 3? ?
lim 𝑥→3 𝑓 𝑥 =𝑓(3) 20 20
f(x) is continuous at x = 3
Barnett/Ziegler/Byleen Business Calculus 12e

8. Example 4 𝑓 𝑥 = 𝑥 2 −4 𝑥+2 Is f(x) continuous at x = -2? ?
𝑓 𝑥 = 𝑥 2 −4 𝑥+2
Is f(x) continuous at x = -2?
lim 𝑥→−2 𝑥 2 −4 𝑥+2 =𝑓(−2) ?
1. lim 𝑥→−2 𝑥 2 −4 𝑥+2 =
lim 𝑥→−2 (𝑥+2)(𝑥−2) 𝑥+2
= lim 𝑥→−2 (𝑥−2) = −4
2. 𝑓 −2 =
𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
f(x) is NOT continuous at x = -2
Barnett/Ziegler/Byleen Business Calculus 12e

9. Example 5 𝑓 𝑥 = 𝑥−5 𝑥−5 Is f(x) continuous at x = 5? Explain.
𝑓 𝑥 = 𝑥−5 𝑥−5
Is f(x) continuous at x = 5? Explain.
lim 𝑥→5 𝑥−5 𝑥−5 =𝑓(5) ?
lim 𝑥→ 5 − 𝑥−5 𝑥−5 =
lim 𝑥→ 5 − −(𝑥−5) 𝑥−5 = −1
1. lim 𝑥→5 𝑥−5 𝑥−5
lim 𝑥→ 𝑥−5 𝑥−5 =
lim 𝑥→ (𝑥−5) 𝑥−5 = 1
lim 𝑥→5 𝑓 𝑥 =𝐷𝑁𝐸
2. 𝑓 5 =
𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
f(x) is NOT continuous at x = 5
Barnett/Ziegler/Byleen Business Calculus 12e

10. Example 6 𝑓 𝑥 = 𝑥−5 𝑥−5 Is f(x) continuous at x = 2?
𝑓 𝑥 = 𝑥−5 𝑥−5
Is f(x) continuous at x = 2?
lim 𝑥→2 𝑥−5 𝑥−5 =𝑓(2) ?
1. lim 𝑥→2 𝑥−5 𝑥−5 =
2−5 2−5 = −1
2. 𝑓 2 = −1
f(x) is continuous at x = 2
Barnett/Ziegler/Byleen Business Calculus 12e

11. Continuity Where is a graph continuous?
Where there are no asymptotes or holes.
Where the function is defined.
Barnett/Ziegler/Byleen Business Calculus 12e

12. Example 7 Where is 𝑓 𝑥 = 2𝑥+6 continuous? 2𝑥+6≥0 𝑥≥−3
𝑓(𝑥) is continuous over the interval: [−3 ,∞)
Barnett/Ziegler/Byleen Business Calculus 12e

13. Example 8 Where is 𝑓 𝑥 = (𝑥+1)(𝑥−4) (𝑥+1)(𝑥−2) continuous? x≠−1, 2
𝑓(𝑥) is continuous over the interval:
(−∞,−1)∪(−1,2)∪(2 ,∞)
Barnett/Ziegler/Byleen Business Calculus 12e

14. Example 9 Where is 𝑓 𝑥 continuous?
𝑓(𝑥) is continuous over the interval:
(−∞,−2)∪(−2,1)∪(1,2)∪(2 ,∞)
Barnett/Ziegler/Byleen Business Calculus 12e

15. #3-3A Pg 161 (7-14, 23, 27, 29, 31, 49-59 odd) Homework
Barnett/Ziegler/Byleen Business Calculus 12e

16. Chapter 3 Limits and the Derivative
Section 3
Continuity
(Part 2)

17. Learning Objectives for Section 3.3 Continuity
The student will understand the concept of continuity
The student will be able to apply the continuity properties
The student will be able to solve word problems.
The student will be able to solve inequalities
Barnett/Ziegler/Byleen Business Calculus 12e

18. Application: Media
A music website called MyTunes charges $0.99 per song if you download less than 100 songs per month and $0.89 per song if you download 100 or more songs per month.
Write a piecewise function f(x) for the cost of downloading x songs per month.
Graph the function.
Is f(x) continuous at x = 100? Explain.
Barnett/Ziegler/Byleen Business Calculus 12e

19. Solution 𝑓 𝑥 = 0.99𝑥 0≤𝑥<100 0.89𝑥 𝑥≥100 ? lim 𝑥→100 𝑓 𝑥 =𝑓(100)
𝑓 𝑥 = 0.99𝑥 0≤𝑥< 𝑥 𝑥≥100 ?
$99
$89
$79
100
110
lim 𝑥→100 𝑓 𝑥 =𝑓(100)
𝐷𝑁𝐸 89
f(x) is NOT continuous at x=100
Barnett/Ziegler/Byleen Business Calculus 12e

20. Application: Natural Gas Rates
The table shows the monthly rates for natural gas charged by MyGas Company. The charge is based on the number of therms used per month. (1 therm = 100,000 Btu)
Write a piecewise function f(x) of the monthly charge for x therms.
Graph f(x).
Is f(x) continuous at x = 40? Give a mathematical reason.
Amount
Cost
Base Charge
$8.00
First 40 therms
$0.60 per therm
Over 40 therms
$0.35 per therm
Barnett/Ziegler/Byleen Business Calculus 12e

21. Solution 𝑓 𝑥 = 8+0.60𝑥 0≤𝑥≤40 8+.60 40 +.35(𝑥−40) 𝑥>40
Amount
Cost
Base Charge
$8.00
First 40 therms
$0.60 per therm
Over 40 therms
$0.35 per therm
𝑓 𝑥 = 𝑥 ≤𝑥≤ (𝑥−40) 𝑥>40
= 𝑥 ≤𝑥≤40 .35𝑥 𝑥>40
Barnett/Ziegler/Byleen Business Calculus 12e

22. Solution 𝑓(𝑥)= 8+0.60𝑥 0≤𝑥≤40 .35𝑥+18 𝑥>40 ? lim 𝑥→40 𝑓 𝑥 =𝑓(40) 32
𝑓(𝑥)= 𝑥 ≤𝑥≤40 .35𝑥 𝑥>40
$90
$60
$30 40 80 ?
lim 𝑥→40 𝑓 𝑥 =𝑓(40) 32 32
f(x) IS continuous at x=40
Barnett/Ziegler/Byleen Business Calculus 12e

23. Solving Inequalities
Up until now, we have solved inequalities using a graphical approach.
𝑓 𝑥 <0  Where is the graph below the x-axis?
𝑓 𝑥 >0  Where is the graph above the x-axis?
Now we will learn an algebraic approach that is based on continuity properties.
Barnett/Ziegler/Byleen Business Calculus 12e

24. Constructing Sign Charts
Find all numbers which are:
a. Holes or vertical asymptotes.
Plot these as open circles on the number line.
b. x-intercepts
Plot these according to the inequality symbol.
Select a test number in each interval and determine if f (x) is positive (+) or negative (–) in the interval.
Determine your answer using the signs and the inequality symbol and write it using interval notation.
Barnett/Ziegler/Byleen Business Calculus 12e

25. Polynomial Inequalities
Ex 3: Solve and write your answer in interval notation.
𝑥 2 −4𝑥−12>0
Factor f(x) and solve for zeros.
(x−6)(x+2)>0
Graph the zeros on a number line. Use open circles for < or >, use closed circles for  or . + - + 6 −2
Test numbers on all sides of the zeros by plugging them into the inequality.
𝐴𝑛𝑠𝑤𝑒𝑟:(−∞,−2)∪(6,∞)
Since f(x) > 0 we want the positive intervals.
Barnett/Ziegler/Byleen Business Calculus 12e

26. Rational Inequalities
Ex 4: Solve and write your answer in interval notation.
𝑥 2 −4 𝑥+4 ≤0
Factor the top and bottom.
Graph the holes and vertical asymptotes as open circles.
(𝑥+2)(𝑥−2) 𝑥+4 ≤0
Graph the x-intercepts. Use open circles for < or >, use closed circles for  or . − + − +
Test numbers on all sides of the points by plugging them into the reduced inequality. 2 −2 −4
𝐴𝑛𝑠𝑤𝑒𝑟: −∞,−4 ∪[−2,2]
Since f(x)  0, we want the negative intervals.
Barnett/Ziegler/Byleen Business Calculus 12e

27. Homework
#3-3B
Pg. 162
(36-43, 45, 81, 85, 86)
Barnett/Ziegler/Byleen Business Calculus 12e