SECTION 1.6 1ccw/calculus/Chapter_01/Page61-71.pdf · Before looking at this definition, consider...

Continuity

In mathematics, the term “continuous” has much the same meaning as it does ineveryday use. To say that a function is continuous at means that there is nointerruption in the graph of f at c. The graph of f is unbroken at and there areno holes, jumps, or gaps. As simple as this concept may seem, its precise defini-tion eluded mathematicians for many years. In fact, it was not until the early1800s that a precise definition was finally developed.

Before looking at this definition, consider the function whose graph is shownin Figure 1.60. This figure identifies three values of x at which the function f isnot continuous.

1. At is not defined.

2. At does not exist.

3. At

At all other points in the interval the graph of f is uninterrupted, whichimplies that the function f is continuous at all other points in the interval

Roughly, you can say that a function is continuous on an interval if its graphon the interval can be traced using a pencil and paper without lifting the pencilfrom the paper, as shown in Figure 1.61.

�a, b�.�a, b�,

f�c3� � limx→c3

f�x�.x � c3,

limx→c2

f�x�x � c2,

f�c1�x � c1,

c,x � c

SECT ION 1.6 Continuity 61

1.6 C O N T I N U I T Y

■ Determine the continuity of functions.■ Determine the continuity of functions on a closed interval.■ Use the greatest integer function to model and solve real-life problems.■ Use compound interest models to solve real-life problems.

Definition of Continuity

Let c be a number in the interval and let f be a function whosedomain contains the interval The function f is continuous at thepoint c if the following conditions are true.

1. is defined.

2. exists.

3.

If f is continuous at every point in the interval then it is continuouson an open interval �a, b�.

�a, b�,

limx→c

f�x� � f�c�.

limx→c

f�x�

f�c�

�a, b�.�a, b�,

xa c2 c3 bc1

(c3, f (c3))

(c2, f (c2))

y

F I G U R E 1 . 6 0 is not continuouswhen x � c1, c2, c3.

f

x

y = f (x)

ba

y

F I G U R E 1 . 6 1 On the intervalthe graph of can be traced

with a pencil.f�a, b�,

360040_0106.qxd 1/5/05 1:04 PM Page 61

In Section 1.5, you studied several types of functions that meet the three con-ditions for continuity. Specifically, if direct substitution can be used to evaluatethe limit of a function at c, then the function is continuous at c. Two types of functions that have this property are polynomial functions and rational functions.

Polynomial functions are one of the most important types of functions usedin calculus. Be sure you see from Example 1 that the graph of a polynomial func-tion is continuous on the entire real line, and therefore has no holes, jumps, orgaps. Rational functions, on the other hand, need not be continuous on the entirereal line, as shown in Example 2.

62 CHAPTER 1 Functions, Graphs, and Limits

Continuity of Polynomial and Rational Functions

1. A polynomial function is continuous at every real number.

2. A rational function is continuous at every number in its domain.

E X A M P L E 1 Determining Continuity of a Polynomial Function

Discuss the continuity of each function.

(a)

(b)

SOLUTION Each of these functions is a polynomial function. So, each is contin-uous on the entire real line, as indicated in Figure 1.62.

f�x� � x3 � x

f�x� � x2 � 2x � 3

31 2

4

3

1

2

−1x

f (x) = x2 − 2x + 3

y

(a)

F I G U R E 1 . 6 2 Both functions are continuous on ��, ��.

−2

−1

2

1

1 2−2x

f (x) = x3 − x

y

(b)

T R Y I T 1


(a) (b) f �x� � x3 � xf �x� � x2 � x � 1

S T U D Y T I P

A graphing utility can give misleading information aboutthe continuity of a function.Graph the function

in the standard viewing window.Does the graph appear to becontinuous? For what values ofx is the function continuous?

f�x� �x3 � 8x � 2

Most graphing utilities candraw graphs in two different

modes: connected mode and dotmode. The connected mode workswell as long as the function is continuous on the entire intervalrepresented by the viewing win-dow. If, however, the function isnot continuous at one or more x-values in the viewing window,then the connected mode may tryto “connect” parts of the graphsthat should not be connected. Forinstance, try graphing the function

on the viewing window and Do you noticeany problems?

�6 ≤ y ≤ 6.�8 ≤ x ≤ 8

y1 � �x � 3��x � 2�

T E C H N O L O G Y

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Consider an open interval I that contains a real number c. If a function fis defined on I (except possibly at c), and f is not continuous at c, then f is said to have a discontinuity at c. Discontinuities fall into two categories: removableand nonremovable. A discontinuity at c is called removable if f can be made continuous by appropriately defining (or redefining) For instance, the function in Example 2(b) has a removable discontinuity at To remove thediscontinuity, all you need to do is redefine the function so that

A discontinuity at is nonremovable if the function cannot be madecontinuous at by defining or redefining the function at For instance,the function in Example 2(a) has a nonremovable discontinuity at x � 0.

x � c.x � cx � c

f �1� � 2.�1, 2�.

f�c�.


E X A M P L E 2 Determining Continuity of a Rational Function


(a) (b) (c)

SOLUTION Each of these functions is a rational function and is therefore continuous at every number in its domain.

(a) The domain of consists of all real numbers except So, thisfunction is continuous on the intervals and [See Figure1.63(a).]

(b) The domain of consists of all real numbers exceptSo, this function is continuous on the intervals and

[See Figure 1.63(b).]

(c) The domain of consists of all real numbers. So, this func-tion is continuous on the entire real line. [See Figure 1.63(c).]

f�x� � 1��x2 � 1�

�1, ��.��, 1�x � 1.f�x� � �x2 � 1��x � 1�

�0, ��.��, 0�x � 0.f�x� � 1�x

f�x� � 1��x2 � 1� f�x� � �x2 � 1��x � 1� f�x� � 1�x

3

3

2

21

1

−1

−1

x

f (x) = x1

y

(a) Continuous on and (b) Continuous on and (c) Continuous on

F I G U R E 1 . 6 3

��, ��1, ��, 1��0, ��, 0�

321

−2

−1

−2

1

2

3

x

(1, 2)

x2 − 1f (x) = x − 1

y

−2

−1

1 2−1−2−3

3

2

x

y

x2 + 11f (x) =

T R Y I T 2


(a) (b) (c) f �x� �1

x2 � 2f �x� �

x2 � 4x � 2

f �x� �1

x � 1

360040_0106.qxd 1/5/05 1:04 PM Page 63

Continuity on a Closed Interval

The intervals discussed in Examples 1 and 2 are open. To discuss continuity on a closed interval, you can use the concept of one-sided limits, as defined inSection 1.5.

Similar definitions can be made to cover continuity on intervals of the form and or on infinite intervals. For example, the function

is continuous on the infinite interval �0, ��.

f�x� � �x

�a, b�,�a, b�


Definition of Continuity on a Closed Interval

Let f be defined on a closed interval If f is continuous on the openinterval and

and

then f is continuous on the closed interval Moreover, f is contin-uous from the right at a and continuous from the left at b.

[a, b].

limx→b�

f�x� � f�b�limx→a�

f�x� � f�a�

�a, b��a, b�.

E X A M P L E 3 Examining Continuity at an Endpoint

Discuss the continuity of

SOLUTION Notice that the domain of f is the set Moreover, f is continuous from the left at because

For all the function f satisfies the three conditions for continuity. So,you can conclude that f is continuous on the interval as shown in Figure 1.64.

��, 3�,x < 3,

� f�3�. � 0

limx→3�

f�x� � limx→3�

�3 � x

x � 3��, 3�.

f�x� � �3 � x.

T R Y I T 3

Discuss the continuity of f �x� � �x � 2.

S T U D Y T I P

When working with radical functions of the form

remember that the domain of f coincides with the solution of g�x� ≥ 0.

f�x� � �g�x�

1 2 3−1

1

2

3

4

x

f (x) = 3 − x

y

F I G U R E 1 . 6 4

360040_0106.qxd 1/5/05 1:04 PM Page 64

The Greatest Integer Function

Many functions that are used in business applications are step functions.For instance, the function in Example 9 in Section 1.5 is a step function. Thegreatest integer function is another example of a step function. This function isdenoted by

greatest integer less than or equal to x.

For example,

greatest integer less than or equal to



Note that the graph of the greatest integer function (Figure 1.66) jumps up oneunit at each integer. This implies that the function is not continuous at each integer.

In real-life applications, the domain of the greatest integer function is oftenrestricted to nonnegative values of x. In such cases this function serves the purpose of truncating the decimal portion of x. For example, 1.345 is truncatedto 1 and 3.57 is truncated to 3. That is,

and 3.57 � 3.1.345 � 1

1.5 � 1. 1.5 �

�2 � �2 �2 �

�2.1 � �3 �2.1 �

x �


E X A M P L E 4 Examining Continuity on a Closed Interval

Discuss the continuity of .

SOLUTION The polynomial functions and are continuous on theintervals and respectively. So, to conclude that g is continuous onthe entire interval you only need to check the behavior of g when You can do this by taking the one-sided limits when

Limit from the left

and

Limit from the right

Because these two limits are equal,

So, g is continuous at and, consequently, it is continuous on the entireinterval The graph of g is shown in Figure 1.65.��1, 3�.

x � 2

limx→2

g�x� � g�2� � 3.

limx→2�

g�x� � limx→2�

�x2 � 1� � 3

limx→2�

g�x� � limx→2�

�5 � x� � 3

x � 2.x � 2.��1, 3�,

�2, 3�,��1, 2�x2 � 15 � x

g�x� � �5 � x,x2 � 1,

�1 ≤ x ≤ 2 2 < x ≤ 3

T R Y I T 4

Discuss the continuity of f �x� � �x � 2,14 � x2,

�1 ≤ x < 3 3 ≤ x ≤ 5

.

8

7

6

5

4

3

2

1

1 2 3 4−1

x

5 − x, −1 ≤ x ≤ 2

x2 − 1, 2 < x ≤ 3

y

g (x) =

F I G U R E 1 . 6 5

1

−3

−2

2

1

−3 −2 −1 2 3−1

x

yf (x) = x[[ ]]

F I G U R E 1 . 6 6 Greatest IntegerFunction

Use a graphing utility tocalculate the following.

(a) (b) (c) 0�3.53.5

T E C H N O L O G Y

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R. R. Donnelley & Sons Company isone of the world’s largest commer-cial printers. It prints and binds amajor share of the national publica-tions in the United States, includingTime, Newsweek, and TV Guide.In 2004, the printing and binding of books accounted for 15% ofDonnelley’s business. The other part of its business came from theprinting and binding of catalogs,magazines, directories, and othertypes of publications. In the photo,an employee is aligning a roll ofpaper on a printing press in one of Donnelley’s plants.

E X A M P L E 5 Modeling a Cost Function

A bookbinding company produces 10,000 books in an eight-hour shift. The fixedcost per shift amounts to $5000, and the unit cost per book is $3. Using the greatest integer function, you can write the cost of producing x books as

Sketch the graph of this cost function.

SOLUTION Note that during the first eight-hour shift

which implies

During the second eight-hour shift

which implies

The graph of C is shown in Figure 1.67. Note the graph’s discontinuities.

� 10,000 � 3x.

C � 5000�1 � x � 110,000�� 3x

10,001 ≤ x ≤ 20,000 x � 110,000� � 1,

C � 5000�1 � x � 110,000�� 3x � 5000 � 3x.

1 ≤ x ≤ 10,000 x � 110,000� � 0,

C � 5000�1 � x � 110,000�� 3x.

F I G U R E 1 . 6 7

Cos

t (in

dol

lars

)

Number of books

Cost of Producing Books

10,000 20,000 30,000

110,000

100,000

90,000

80,000

70,000

60,000

50,000

40,000

30,000

20,000

10,000

x

C

First s

hift

Secon

d shif

t

Third s

hift

C = 5000 + 3x1 + ( (x − 110,000[ [[ [

T R Y I T 5

Use a graphing utility to graphthe cost function in Example 5.

R.R. Donnelley & Sons

360040_0106.qxd 1/5/05 1:04 PM 6


Step Functions and Compound Functions

To graph a step function or compound function with a graphing utility, you must be familiar with the utility’s programming language.

For instance, different graphing utilities have different “integer truncation”functions. One is IPart and it yields the truncated integer part of x. Forexample, IPart and IPart The other function isInt which is the greatest integer function. The graphs of these two functions are shown below. When graphing a step function, you should set your graphing utility to dot mode.

On some graphing utilities, you can graph a piecewise-defined functionsuch as

.

The graph of this function is shown below.

Consult the user’s guide for your graphing utility for specific keystrokesyou can use to graph these functions.

f�x� � �x2 � 4,�x � 2,

x ≤ 2 2 < x

�x�,�3.4� � 3.��1.2� � �1

�x�,

T E C H N O L O G Y

Graph of f�x� � IPart�x�

Graph of f�x� � Int�x�

−2

3−3

2

−2

3−3

2

9

−6

−9

6

360040_0106.qxd 1/5/05 1:04 PM Page 67

Extended Application: Compound Interest

Banks and other financial institutions differ on how interest is paid to an account.If the interest is added to the account so that future interest is paid on previouslyearned interest, then the interest is said to be compounded. Suppose, for example, that you deposited $10,000 in an account that pays 6% interest, com-pounded quarterly. Because the 6% is the annual interest rate, the quarterly rateis or 1.5%. The balances during the first five quarters are shownbelow.

Quarter Balance

1st $10,000.00

2nd

3rd

4th

5th 10,456.78 � �0.015��10,456.78� � $10,613.63

10,302.25 � �0.015��10,302.25� � $10,456.78

10,150.00 � �0.015��10,150.00� � $10,302.25

10,000.00 � �0.015��10,000.00� � $10,150.00

14�0.06� � 0.015


You can use a spreadsheetor the table feature of a

graphing utility to create a table.Try doing this for the data shownat the right. (Consult the user’smanual of a spreadsheet softwareprogram for specific instructionson how to create a table.)

T E C H N O L O G Y

E X A M P L E 6 Graphing Compound Interest

Sketch the graph of the balance in the account described above.

SOLUTION Let A represent the balance in the account and let t represent thetime, in years. You can use the greatest integer function to represent the balance,as shown.

From the graph shown in Figure 1.68, notice that the function has a discontinuityat each quarter.

A � 10,000�1 � 0.015�4t

T R Y I T 6

Write an equation that gives the balance of the account in Example 6 if theannual interest rate is 8%.

Compound Interest

If P dollars is deposited in an account, compounded n times per year, with an annual rate of r (in decimal form), then the balance A after t years is given by

Sketch the graph of each function. Which function is continuous? Describe the differences in policy between a bank that uses the first formula and a bank that usesthe second formula.

a. b. A � P �1 �rn�

ntA � P �1 �

rn�

nt

A � P �1 �rn�

nt.

T A K E A N O T H E R L O O K

Time (in years)

Bal

ance

(in

dol

lars

)

Quarterly Compounding

t

A

10,000

10,100

10,200

10,300

10,400

10,500

10,600

10,700

1 134 4

12

54

F I G U R E 1 . 6 8

360040_0106.qxd 1/5/05 1:04 PM Page 68


In Exercises 1–10, determine whether the function is continuouson the entire real line. Explain your reasoning.

1. 2.

3. 4.

5. 6.

7. 8.

9.

10.

In Exercises 11–34, describe the interval(s) on which the functionis continuous.

11. 12.

13. 14.

15.

16.

17.

18.

19.

20.

21.

22. f �x� �x � 1

x2 � x � 2

f �x� �x � 5

x2 � 9x � 20

f �x� �1

x2 � 1

f �x� �x

x2 � 1

f �x� �x � 3x2 � 9

f �x� �x

x2 � 1

f �x� � 3 � 2x � x2

f �x� � x2 � 2x � 1

1210

14

86

2 6

2

−2−6x

y

1 2 3−1−2−3

1

2

3

−3

x

y

f �x� �x3 � 8x � 2

f �x� �x2 � 1x � 1

3

2

1

−1

−2

−3

−3 3x

y

321−2−3

−3

−2

−1

x

y

f �x� �1

x2 � 4f �x� �

x2 � 1x

g�x� �x2 � 9x � 20

x2 � 16

g�x� �x2 � 4x � 4

x2 � 4

f �x� �x � 4

x2 � 6x � 5 f �x� �

2x � 1x2 � 8x � 15

f �x� �3x

x2 � 1f �x� �

14 � x2

f �x� �1

9 � x2f �x� �1

x2 � 4

f �x� � �x2 � 1�3f �x� � 5x3 � x2 � 2

E X E R C I S E S 1 . 6

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section.

In Exercises 1–4, simplify the expression.

1. 2.

3. 4.

In Exercises 5–8, solve for x.

5. 6.

7. 8.

In Exercises 9 and 10, find the limit.

9. 10. limx→�2

�3x3 � 8x � 7�limx→3

�2x2 � 3x � 4�

x3 � 5x2 � 24x � 03x2 � 8x � 4 � 0

x2 � 4x � 5 � 0x2 � 7x � 0

x3 � 16xx3 � 2x2 � 8x

2x2 � 2x � 124x2 � 24x � 36

x2 � 5x � 6x2 � 9x � 18

x2 � 6x � 8x2 � 6x � 16

P R E R E Q U I S I T ER E V I E W 1 . 6

360040_0106.qxd 1/5/05 1:04 PM Page 69

23. 24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

In Exercises 35–38, discuss the continuity of the function on the closed interval. If there are any discontinuities, determinewhether they are removable.

Function Interval

35.

36.

37.

38.

In Exercises 39–44, sketch the graph of the function and describethe interval(s) on which the function is continuous.

39. 40.

41.

42.

43.

44.

In Exercises 45 and 46, find the constant a (Exercise 45) and theconstants a and b (Exercise 46) such that the function is continuouson the entire real line.

45.

46.

In Exercises 47–52, use a graphing utility to graph the function.Use the graph to determine any x-values at which the function isnot continuous.

47.

48.

49.

50.

51.

52.

In Exercises 53–56, describe the interval(s) on which the functionis continuous.

53. 54.

−2

4

2

−4 −2 2x

(−3, 0)

y

−2

−1

2

1

21x

y

f �x� � x�x � 3f �x� �x

x2 � 1

f �x� � 2x � 1

f �x� � x � 2 x

f �x� � �3x � 1,x � 1,

x ≤ 1 x > 1

f �x� � �2x � 4,x2 � 2x,

x ≤ 3 x > 3

k �x� �x � 4

x2 � 5x � 4

h�x� �1

x2 � x � 2

f �x� � �2,ax � b,�2,

x ≤ �1 �1 < x < 3 x ≥ 3

f �x� � �x3,ax2,

x ≤ 2 x > 2

f �x� � �x2 � 4,2x � 4,

x ≤ 0 x > 0

f �x� � �x2 � 1,x � 1,

x < 0 x ≥ 0

f �x� �x � 3

4x2 � 12x

f �x� �x3 � x

x

f �x� �2x2 � x

xf �x� �

x2 � 16x � 4

�0, 4�f �x� �x

x2 � 4x � 3

�1, 4�f �x� �1

x � 2

��2, 2�f �x� �5

x2 � 1

��1, 5�f �x� � x2 � 4x � 5

g�x� � x2 � 5f �x� �1

x � 1,h�x� � f �g�x��,

g�x� � x � 1, x > 1f �x� �1�x

,h�x� � f �g�x��,

f �x� � x � x

f �x� � x � 1

f �x� ��4 � x�4 � x

f �x� ��x � 1�x � 1

f �x� � �x2 � 4,3x � 1,

x ≤ 0 x > 0

f �x� � �12x � 1, 3 � x,

x ≤ 2x > 2

f �x� � �3 � x,x2 � 1,

x ≤ 2 x > 2

f �x� � ��2x � 3,x2,

x < 1 x ≥ 1

1 2−1−2

1

2

−2

x

y

x

y

−2−3 1 2 3

−3

1

2

3

f �x� �x2

� x f �x� � 2x � 1


360040_0106.qxd 1/5/05 1:04 PM Page 70

55. 56.

Writing In Exercises 57 and 58, use a graphing utility to graphthe function on the interval Does the graph of the func-tion appear to be continuous on this interval? Is the function infact continuous on Write a short paragraph about the importance of examining a function analytically as well asgraphically.

57.

58.

59. Compound Interest A deposit of $7500 is made in anaccount that pays 6% compounded quarterly. The amountA in the account after t years is

(a) Sketch the graph of A. Is the graph continuous? Explainyour reasoning.

(b) What is the balance after 7 years?

60. Environmental Cost The cost C (in millions of dollars)of removing x percent of the pollutants emitted from thesmokestack of a factory can be modeled by

(a) What is the implied domain of C? Explain your reasoning.

(b) Use a graphing utility to graph the cost function. Is thefunction continuous on its domain? Explain your reasoning.

(c) Find the cost of removing 75% of the pollutants fromthe smokestack.

61. Consumer Awareness A shipping company’s chargefor sending an overnight package from New York to Atlantais $9.80 for the first pound and $2.50 for each additionalpound or fraction thereof. Use the greatest integer functionto create a model for the charge C for overnight delivery ofa package weighing x pounds. Use a graphing utility tograph the function, and discuss its continuity.

62. Consumer Awareness A cab company charges $3 forthe first mile and $0.25 for each additional mile or fractionthereof. Use the greatest integer function to create a modelfor the cost C of a cab ride n miles long. Use a graphingutility to graph the function, and discuss its continuity.

63. Consumer Awareness A dial-direct long distance callbetween two cities costs $1.04 for the first 2 minutes and$0.36 for each additional minute or fraction thereof.

(a) Use the greatest integer function to write the cost C ofa call in terms of the time t (in minutes). Sketch thegraph of the cost function and discuss its continuity.

(b) Find the cost of a nine-minute call.

64. Salary Contract A union contract guarantees a 9%yearly increase for 5 years. For a current salary of $28,500,the salary for the next 5 years is given by

where represents the present year.

(a) Use the greatest integer function of a graphing utility tograph the salary function, and discuss its continuity.

(b) Find the salary during the fifth year (when ).

65. Inventory Management The number of units in inven-tory in a small company is

where the real number t is the time in months.

(a) Use the greatest integer function of a graphing utility tograph this function, and discuss its continuity.

(b) How often must the company replenish its inventory?

66. Owning a Franchise You have purchased a franchise.You have determined a linear model for your revenue as afunction of time. Is the model a continuous function?Would your actual revenue be a continuous function oftime? Explain your reasoning.

67. Biology The gestation period of rabbits is only 26 to 30days. Therefore, the population of a form (rabbits’ home)can increase dramatically in a short period of time. Thetable gives the population of a form, where t is the time inmonths and N is the rabbit population.

Graph the population as a function of time. Find any pointsof discontinuity in the function. Explain your reasoning.

0 ≤ t ≤ 12N � 25�2 t � 22 � � t�,

t � 5

t � 0

S � 28,500�1.09�t

C �2x

100 � x.

t ≥ 0.A � 7500�1.015�4t,

f �x� �x3 � 8x � 2

f �x� �x2 � x

x

��4, 4�?

��4, 4�.

4

3

2

1

321x

y

−2

2

1

3−3 −1−2 21x

y

f �x� �x � 1�x

f�x� �12

2x


t 0 1 2 3 4 5 6

N 2 8 10 14 10 15 12

360040_0106.qxd 1/5/05 1:04 PM Page 71

SECTION 1.6 1ccw/calculus/Chapter_01/Page61-71.pdf · Before looking at this definition, consider...

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