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Precalculus

12.3 Continuity

Precalculus12.3 Continuity
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  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

Learning Objectives

In this section, you will:
  • Determine whether a function is continuous at a number.
  • Determine the numbers for which a function is discontinuous.
  • Determine whether a function is continuous.

Arizona is known for its dry heat. On a particular day, the temperature might rise as high as 118 F 118 F and drop down only to a brisk 95 F. 95 F. Figure 1 shows the function T ,T, where the output of T( x ) T( x ) is the temperature in Fahrenheit degrees and the input x xis the time of day, using a 24-hour clock on a particular summer day.

Graph of function that maps the time since midnight to the temperature. The x-axis, labelled x, represents the hours since midnight from 0 to 24. The y-axis, labelled T(x), represents the temperature from 0 to 120. The function is continuous that peaks at (16, 118).
Figure 1 Temperature as a function of time forms a continuous function.

When we analyze this graph, we notice a specific characteristic. There are no breaks in the graph. We could trace the graph without picking up our pencil. This single observation tells us a great deal about the function. In this section, we will investigate functions with and without breaks.

Determining Whether a Function Is Continuous at a Number

Let’s consider a specific example of temperature in terms of date and location, such as June 27, 2013, in Phoenix, AZ. The graph in Figure 1 indicates that, at 2 a.m., the temperature was 96 F 96 F . By 2 p.m. the temperature had risen to 116 F, 116 F, and by 4 p.m. it was 118 F. 118 F. Sometime between 2 a.m. and 4 p.m., the temperature outside must have been exactly 110.5 F. 110.5 F. In fact, any temperature between 96 F 96 F and 118 F 118 F occurred at some point that day. This means all real numbers in the output between 96 F 96 F and 118 F 118 F are generated at some point by the function according to the intermediate value theorem,

Look again at Figure 1. There are no breaks in the function’s graph for this 24-hour period. At no point did the temperature cease to exist, nor was there a point at which the temperature jumped instantaneously by several degrees. A function that has no holes or breaks in its graph is known as a continuous function. Temperature as a function of time is an example of a continuous function.

If temperature represents a continuous function, what kind of function would not be continuous? Consider an example of dollars expressed as a function of hours of parking. Let’s create the function D ,D, where D( x ) D( x ) is the output representing cost in dollars for parking x xnumber of hours. See Figure 2.

Suppose a parking garage charges $4.00 per hour or fraction of an hour, with a $25 per day maximum charge. Park for two hours and five minutes and the charge is $12. Park an additional hour and the charge is $16. We can never be charged $13, $14, or $15. There are real numbers between 12 and 16 that the function never outputs. There are breaks in the function’s graph for this 24-hour period, points at which the price of parking jumps instantaneously by several dollars.

Graph of function that maps the time since midnight to the temperature. The x-axis represents the hours parked from 0 to 24. The y-axis represents dollars amounting from 0 to 28. The function is a step-function.
Figure 2 Parking-garage charges form a discontinuous function.

A function that remains level for an interval and then jumps instantaneously to a higher value is called a stepwise function. This function is an example.

A function that has any hole or break in its graph is known as a discontinuous function. A stepwise function, such as parking-garage charges as a function of hours parked, is an example of a discontinuous function.

So how can we decide if a function is continuous at a particular number? We can check three different conditions. Let’s use the function y=f( x ) y=f( x ) represented in Figure 3 as an example.

Graph of an increasing function with a discontinuity at (a, f(a)).
Figure 3

Condition 1 According to Condition 1, the function f( a ) f( a ) defined at x=a x=a must exist. In other words, there is a y-coordinate at x=a x=a as in Figure 4.

Graph of an increasing function with a discontinuity at (a, 2). The point (a, f(a)) is directly below the hole.
Figure 4

Condition 2 According to Condition 2, at x=a x=a the limit, written lim xa f(x) , lim xa f(x) ,must exist. This means that at x=a x=a the left-hand limit must equal the right-hand limit. Notice as the graph of f fin Figure 3 approaches x=a x=a from the left and right, the same y-coordinate is approached. Therefore, Condition 2 is satisfied. However, there could still be a hole in the graph at x=a x=a .

Condition 3 According to Condition 3, the corresponding y ycoordinate at x=a x=a fills in the hole in the graph of f. f. This is written lim xa f(x)=f(a). lim xa f(x)=f(a).

Satisfying all three conditions means that the function is continuous. All three conditions are satisfied for the function represented in Figure 5 so the function is continuous as x=a. x=a.

Graph of an increasing function with filled-in discontinuity at (a, f(a)).
Figure 5 All three conditions are satisfied. The function is continuous at x=a x=a .

Figure 6 through Figure 9 provide several examples of graphs of functions that are not continuous at x=a x=a and the condition or conditions that fail.

Graph of an increasing function with a discontinuity at (a, f(a)).
Figure 6 Condition 2 is satisfied. Conditions 1 and 3 both fail.
Graph of an increasing function with a discontinuity at (a, 2). The point (a, f(a)) is directly below the hole.
Figure 7 Conditions 1 and 2 are both satisfied. Condition 3 fails.
Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)), which is closed, and another increasing segment from (a, f(a)-1), which is open, to positive infinity.
Figure 8 Condition 1 is satisfied. Conditions 2 and 3 fail.
Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)) and another increasing segment from (a, f(a) - 1) to positive infinity. This graph does not include the point (a, f(a)).
Figure 9 Conditions 1, 2, and 3 all fail.

Definition of Continuity

A function f( x ) f( x ) is continuous at x=a x=a provided all three of the following conditions hold true:

  • Condition 1: f(a) f(a) exists.
  • Condition 2: lim xa f(x) lim xa f(x) exists at x=a x=a .
  • Condition 3: lim xa f(x)=f(a) lim xa f(x)=f(a) .

If a function f( x ) f( x ) is not continuous at x=a , x=a ,the function is discontinuous at x=a x=a .

Identifying a Jump Discontinuity

Discontinuity can occur in different ways. We saw in the previous section that a function could have a left-hand limit and a right-hand limit even if they are not equal. If the left- and right-hand limits exist but are different, the graph “jumps” at x=a x=a . The function is said to have a jump discontinuity.

As an example, look at the graph of the function y=f( x ) y=f( x ) in Figure 10. Notice as x xapproaches a ahow the output approaches different values from the left and from the right.

Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)), which is closed, and another increasing segment from (a, f(a)-1), which is open, to positive infinity.
Figure 10 Graph of a function with a jump discontinuity.

Jump Discontinuity

A function f( x ) f( x ) has a jump discontinuity at x=a x=a if the left- and right-hand limits both exist but are not equal: lim x a f(x) lim x a + f(x) lim x a f(x) lim x a + f(x) .

Identifying Removable Discontinuity

Some functions have a discontinuity, but it is possible to redefine the function at that point to make it continuous. This type of function is said to have a removable discontinuity. Let’s look at the function y=f( x ) y=f( x ) represented by the graph in Figure 11. The function has a limit. However, there is a hole at x=a x=a . The hole can be filled by extending the domain to include the input x=a x=a and defining the corresponding output of the function at that value as the limit of the function at x=a x=a .

Graph of an increasing function with a removable discontinuity at (a, f(a)).
Figure 11 Graph of function f fwith a removable discontinuity at x=a x=a .

Removable Discontinuity

A function f( x ) f( x ) has a removable discontinuity at x=a x=a if the limit, lim xa f(x) , lim xa f(x) , exists, but either

  1. f( a ) f( a ) does not exist or
  2. f( a ), f( a ), the value of the function at x=a x=a does not equal the limit, f(a) lim xa f(x). f(a) lim xa f(x).

Example 1

Identifying Discontinuities

Identify all discontinuities for the following functions as either a jump or a removable discontinuity.

  1. f(x)= x 2 2x15 x5 f(x)= x 2 2x15 x5
  2. g(x)={ x+1, x<2 x, x2 g(x)={ x+1, x<2 x, x2

Try It #1

Identify all discontinuities for the following functions as either a jump or a removable discontinuity.

  1. f(x)= x 2 6x x6 f(x)= x 2 6x x6
  2. g(x)={ x , 0x<4 2x, x4 g(x)={ x , 0x<4 2x, x4

Recognizing Continuous and Discontinuous Real-Number Functions

Many of the functions we have encountered in earlier chapters are continuous everywhere. They never have a hole in them, and they never jump from one value to the next. For all of these functions, the limit of f( x ) f( x ) as x xapproaches a ais the same as the value of f( x ) f( x ) when x=a. x=a. So lim xa f(x)=f(a). lim xa f(x)=f(a). There are some functions that are continuous everywhere and some that are only continuous where they are defined on their domain because they are not defined for all real numbers.

Examples of Continuous Functions

The following functions are continuous everywhere:

Polynomial functions Ex: f(x)= x 4 9 x 2 f(x)= x 4 9 x 2
Exponential functions Ex: f(x)= 4 x+2 5 f(x)= 4 x+2 5
Sine functions Ex: f(x)=sin( 2x )4 f(x)=sin( 2x )4
Cosine functions Ex: f(x)=cos( x+ π 3 ) f(x)=cos( x+ π 3 )
Table 1

The following functions are continuous everywhere they are defined on their domain:

Logarithmic functions Ex: f(x)=2ln( x ) f(x)=2ln( x ) , x>0 x>0
Tangent functions Ex: f(x)=tan( x )+2, f(x)=tan( x )+2, x π 2 +kπ, x π 2 +kπ, k kis an integer
Rational functions Ex: f(x)= x 2 25 x7 , f(x)= x 2 25 x7 , x7 x7
Table 2

How To

Given a function f( x ), f( x ), determine if the function is continuous at x=a. x=a.

  1. Check Condition 1: f(a) f(a) exists.
  2. Check Condition 2: lim xa f(x) lim xa f(x) exists at x=a. x=a.
  3. Check Condition 3: lim xa f(x)=f(a). lim xa f(x)=f(a).
  4. If all three conditions are satisfied, the function is continuous at x=a. x=a. If any one of the conditions is not satisfied, the function is not continuous at x=a. x=a.

Example 2

Determining Whether a Piecewise Function is Continuous at a Given Number

Determine whether the function f(x)={ 4x, x3 8+x, x>3 f(x)={ 4x, x3 8+x, x>3 is continuous at

  1. x=3 x=3
  2. x= 8 3 x= 8 3
Try It #2

Determine whether the function f(x)={ 1 x , x2 9x11.5, x>2 f(x)={ 1 x , x2 9x11.5, x>2 is continuous at x=2. x=2.

Example 3

Determining Whether a Rational Function is Continuous at a Given Number

Determine whether the function f(x)= x 2 25 x5 f(x)= x 2 25 x5 is continuous at x=5. x=5.

Analysis

See Figure 12. Notice that for Condition 2 we have

lim x5 x 2 25 x5 = lim x3 (x5) (x+5) x5                     = lim x5 (x+5)                     =5+5=10                     Condition 2 is satisfied. lim x5 x 2 25 x5 = lim x3 (x5) (x+5) x5                     = lim x5 (x+5)                     =5+5=10                     Condition 2 is satisfied.

At x=5, x=5, there exists a removable discontinuity. See Figure 12.

Graph of an increasing function with a removable discontinuity at (5, 10).
Figure 12
Try It #3

Determine whether the function f(x)= 9 x 2 x 2 3x f(x)= 9 x 2 x 2 3x is continuous at x=3. x=3. If not, state the type of discontinuity.

Determining the Input Values for Which a Function Is Discontinuous

Now that we can identify continuous functions, jump discontinuities, and removable discontinuities, we will look at more complex functions to find discontinuities. Here, we will analyze a piecewise function to determine if any real numbers exist where the function is not continuous. A piecewise function may have discontinuities at the boundary points of the function as well as within the functions that make it up.

To determine the real numbers for which a piecewise function composed of polynomial functions is not continuous, recall that polynomial functions themselves are continuous on the set of real numbers. Any discontinuity would be at the boundary points. So we need to explore the three conditions of continuity at the boundary points of the piecewise function.

How To

Given a piecewise function, determine whether it is continuous at the boundary points.

  1. For each boundary point a aof the piecewise function, determine the left- and right-hand limits as x xapproaches a, a, as well as the function value at a. a.
  2. Check each condition for each value to determine if all three conditions are satisfied.
  3. Determine whether each value satisfies condition 1: f(a) f(a) exists.
  4. Determine whether each value satisfies condition 2: lim xa f(x) lim xa f(x) exists.
  5. Determine whether each value satisfies condition 3: lim xa f(x)=f(a). lim xa f(x)=f(a).
  6. If all three conditions are satisfied, the function is continuous at x=a. x=a. If any one of the conditions fails, the function is not continuous at x=a. x=a.

Example 4

Determining the Input Values for Which a Piecewise Function Is Discontinuous

Determine whether the function f fis discontinuous for any real numbers.

f(x)={ x+1, x<2 3, 2x<4 x 2 11, x4 f(x)={ x+1, x<2 3, 2x<4 x 2 11, x4

Analysis

See Figure 13. At x=4, x=4, there exists a jump discontinuity. Notice that the function is continuous at x=2. x=2.

Graph of a piecewise function that has disconuity at (4, 3).
Figure 13 Graph is continuous at x=2 x=2 but shows a jump discontinuity at x=4. x=4.
Try It #4

Determine where the function f(x)={ πx 4 ,  x<2 π x ,    2x6 2πx,  x>6 f(x)={ πx 4 ,  x<2 π x ,    2x6 2πx,  x>6 is discontinuous.

Determining Whether a Function Is Continuous

To determine whether a piecewise function is continuous or discontinuous, in addition to checking the boundary points, we must also check whether each of the functions that make up the piecewise function is continuous.

How To

Given a piecewise function, determine whether it is continuous.

  1. Determine whether each component function of the piecewise function is continuous. If there are discontinuities, do they occur within the domain where that component function is applied?
  2. For each boundary point x=a x=a of the piecewise function, determine if each of the three conditions hold.

Example 5

Determining Whether a Piecewise Function Is Continuous

Determine whether the function below is continuous. If it is not, state the location and type of each discontinuity.

f(x)={ sin(x), x<0 x 3 , x>0 f(x)={ sin(x), x<0 x 3 , x>0

Analysis

See Figure 14. There exists a removable discontinuity at x=0; x=0; lim x0 f(x)=0, lim x0 f(x)=0, thus the limit exists and is finite, but f( a ) f( a ) does not exist.

Graph of a piecewise function where from negative infinity to 0 f(x) = sin(x) and from 0 to positive infinity f(x) = x^3.
Figure 14 Function has removable discontinuity at 0.

Media

Access these online resources for additional instruction and practice with continuity.

12.3 Section Exercises

Verbal

1.

State in your own words what it means for a function f fto be continuous at x=c. x=c.

2.

State in your own words what it means for a function to be continuous on the interval ( a,b ). ( a,b ).

Algebraic

For the following exercises, determine why the function f fis discontinuous at a given point a aon the graph. State which condition fails.

3.

f(x)=ln | x+3 |,a=3 f(x)=ln | x+3 |,a=3

4.

f(x)=ln | 5x2 |,a= 2 5 f(x)=ln | 5x2 |,a= 2 5

5.

f(x)= x 2 16 x+4 ,a=4 f(x)= x 2 16 x+4 ,a=4

6.

f(x)= x 2 16x x ,a=0 f(x)= x 2 16x x ,a=0

7.

f( x )={ x,   x3 2x,x=3  a=3 f( x )={ x,   x3 2x,x=3  a=3

8.

f( x )={ 5,  x0 3,  x=0   a=0 f( x )={ 5,  x0 3,  x=0   a=0

9.

f( x )={ 1 2x , x2 3, x=2   a=2 f( x )={ 1 2x , x2 3, x=2   a=2

10.

f( x )={ 1 x+6 , x=6 x 2 , x6   a=6 f( x )={ 1 x+6 , x=6 x 2 , x6   a=6

11.

f( x )={ 3+x, x<1 x, x=1 x 2 , x>1     a=1 f( x )={ 3+x, x<1 x, x=1 x 2 , x>1     a=1

12.

f( x )={ 3x, x<1 x, x=1 2 x 2 , x>1     a=1 f( x )={ 3x, x<1 x, x=1 2 x 2 , x>1     a=1

13.

f( x )={ 3+2x, x<1 x, x=1 x 2 , x>1     a=1 f( x )={ 3+2x, x<1 x, x=1 x 2 , x>1     a=1

14.

f( x )={ x 2 , x<2 2x+1, x=2 x 3 , x>2     a=2 f( x )={ x 2 , x<2 2x+1, x=2 x 3 , x>2     a=2

15.

f( x )={ x 2 9 x+3 , x<3 x9, x=3 1 x , x>3     a=3 f( x )={ x 2 9 x+3 , x<3 x9, x=3 1 x , x>3     a=3

16.

f( x )={ x 2 9 x+3 , x<3 x9, x=3 6, x>3     a=3 f( x )={ x 2 9 x+3 , x<3 x9, x=3 6, x>3     a=3

17.

f( x )= x 2 4 x2 ,  a=2 f( x )= x 2 4 x2 ,  a=2

18.

f( x )= 25 x 2 x 2 10x+25 ,  a=5 f( x )= 25 x 2 x 2 10x+25 ,  a=5

19.

f( x )= x 3 9x x 2 +11x+24 ,  a=3 f( x )= x 3 9x x 2 +11x+24 ,  a=3

20.

f( x )= x 3 27 x 2 3x ,  a=3 f( x )= x 3 27 x 2 3x ,  a=3

21.

f(x)= x |x| ,  a=0 f(x)= x |x| ,  a=0

22.

f( x )= 2| x+2 | x+2 ,  a=2 f( x )= 2| x+2 | x+2 ,  a=2

For the following exercises, determine whether or not the given function f fis continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

23.

f( x )= x 3 2x15 f( x )= x 3 2x15

24.

f( x )= x 2 2x15 x5 f( x )= x 2 2x15 x5

25.

f( x )=2 3 x+4 f( x )=2 3 x+4

26.

f( x )=−sin( 3x ) f( x )=−sin( 3x )

27.

f( x )= | x2 | x 2 2x f( x )= | x2 | x 2 2x

28.

f( x )=tan( x )+2 f( x )=tan( x )+2

29.

f( x )=2x+ 5 x f( x )=2x+ 5 x

30.

f( x )= log 2 ( x ) f( x )= log 2 ( x )

31.

f(x)=ln  x 2 f(x)=ln  x 2

32.

f( x )= e 2x f( x )= e 2x

33.

f(x)= x4 f(x)= x4

34.

f( x )=sec( x )3 f( x )=sec( x )3 .

35.

f( x )= x 2 +sin( x ) f( x )= x 2 +sin( x )

36.

Determine the values of b band c csuch that the following function is continuous on the entire real number line.

f(x)= { x+1, 1<x<3 x 2 +bx+c, | x2 |1 f(x)= { x+1, 1<x<3 x 2 +bx+c, | x2 |1

Graphical

For the following exercises, refer to Figure 15. Each square represents one square unit. For each value of a ,a, determine which of the three conditions of continuity are satisfied at x=a x=a and which are not.

Graph of a piecewise function where at x = -3 the line is disconnected, at x = 2 there is a removable discontinuity, and at x = 4 there is a removable discontinuity and f(4) exists.
Figure 15
37.

x=3 x=3

38.

x=2 x=2

39.

x=4 x=4

For the following exercises, use a graphing utility to graph the function f(x)=sin( 12π x ) f(x)=sin( 12π x ) as in Figure 16. Set the x-axis a short distance before and after 0 to illustrate the point of discontinuity.

Graph of the sinusodial function with a viewing window of [-10, 10] by [-1, 1].
Figure 16
40.

Which conditions for continuity fail at the point of discontinuity?

41.

Evaluate f(0). f(0).

42.

Solve for x xif f(x)=0. f(x)=0.

43.

What is the domain of f( x )? f( x )?

For the following exercises, consider the function shown in Figure 17.

Graph of a piecewise function where at x = -1 the line is disconnected and at x = 1 there is a removable discontinuity.
Figure 17
44.

At what x-coordinates is the function discontinuous?

45.

What condition of continuity is violated at these points?

46.

Consider the function shown in Figure 18. At what x-coordinates is the function discontinuous? What condition(s) of continuity were violated?

Graph of a piecewise function where at x = -1 the line is disconnected and where at x = 1 and x = 2 there are a removable discontinuities.
Figure 18
47.

Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at x=7 x=7 and x=1. x=1.

48.

The function f(x)= x 3 1 x1 f(x)= x 3 1 x1 is graphed in Figure 19. It appears to be continuous on the interval [ 3,3 ], [ 3,3 ], but there is an x-value on that interval at which the function is discontinuous. Determine the value of x xat which the function is discontinuous, and explain the pitfall of utilizing technology when considering continuity of a function by examining its graph.

Graph of the function f(x) = (x^3 - 1)/(x-1).
Figure 19
49.

Find the limit lim x1 f(x) lim x1 f(x) and determine if the following function is continuous at x=1: x=1:

fx={ x 2 +4 x1 2 x=1 fx={ x 2 +4 x1 2 x=1
50.

The graph of f(x)= sin(2x) x f(x)= sin(2x) x is shown in Figure 20. Is the function f( x ) f( x ) continuous at x=0? x=0? Why or why not?

Graph of the function f(x) = sin(2x)/x with a viewing window of [-4.5, 4.5] by [-1, 2.5]
Figure 20
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