ContinuitySection 2.3

Review Quiz

Review Quiz (Continued)

2 2

In Exercises 4 – 6, find the remaining functions in the list of functions:

, , , .

2 1 14. , 1

5

5. , sin , domain of [0, )

f g f g g f

xf x g x

x x

f x x g f x x g

2

3

16. 1, , 0

7. Use factoring to solve 2 9 5 0

8. Use graphing to solve 2 1 0

g x x g f x xx

x x

x x

Review Quiz (Answers)

2

31

1 1

1

2

2 2

2

2

2 1

no limi

3 2 11. Find lim

4

2. Let int . Find each limit.

a lim b lim

c lim d 1

4 5, 23. Let

4 , 2

Find each limit.

a lim b lim

l

t

1 2

i

1

c m

x

x x

x

x x

x

x x

x

f x x

f x f x

f x f

x x xf x

x x

f x f x

dno limit 22 f x f

Review Quiz (Answers)

2 2

2

In Exercises 4 – 6, find the remaining functions in the list of functions:

, , , .

2 1 14. , 1

5

5. , sin , domain of [0, )

2 3 4, 0 , 5

6 1 2 1

sin , 0 sin , 0

f g f g g f

xf

x xf g x x

x g xx x

f x

g f x xx x

g x x x f g x x x

x g f x x g

Review Quiz (Answers)

2

2

3

16. 1, , 0

7. Use factoring to solve 2 9 5 0

8. Use graphing to solve 2 1 0

11, 0 , 1

1

1, 5

2

0.453

xf x x f g x

g x x g f x xx

x x

x x

xxx

x

x

What you’ll learn . . .• Continuity at a point

• Continuous Function

• Algebraic Combinations

• Composites

• Intermediate Value Theorem for Continuous Functions

Continuity at a Point

Recall, any function y=f(x) whose graph can be sketched in one continuous motion without lifting the pencil is an example of a continuous function. You have seen this in both Algebra II and Pre-Calculus.

Interior Point: A function is continuous at an interior point of its

domain if lim

Endpoint: A function is continuous at a left

endpoint or is continuous

x c

y f x c

f x f c

y f x

a

at a right endpoint of its domain if

lim or lim respectively.x a x b

b

f x f a f x f b

Definition: Continuity

A function is continuous at x = a when (i) f(a) is defined(ii) lim𝑥→𝑎 𝑓 𝑥 exists(iii) lim

𝑥→𝑎𝑓 𝑥 = f(a)

Otherwise, f is said to be discontinuous at x = a.

Types of Discontinuity

• REMOVABLE

• JUMP

• INFINITE

• OSCILLATING

Example: Continuity at a Point

2

3Find and identify the points of discontinuity of

1y

x

There is an infinite discontinuity at 1.x

Continuous Function

A function is if and only if

it is continuous at every point of the interval. A

is one that is continuous at every

point of its domain.  A continuous funct

continuous on an interval

continuous function

ion need not be

continuous on every interval.

Example: Finding where a Rational Function is Continuous

Determine where 𝑓 𝑥 =𝑥2+2𝑥−3

𝑥−1is continuous

Example: Removing a DiscontinuityMake the function from the last slide continuous everywhere by redefining it at a single point.

Example: Nonremovable Discontinuity

Find all discontinuities of 𝑓 𝑥 =1

𝑥2and 𝑔 𝑥 = cos

1

𝑥.

Consider . . . All polynomials are continuous everywhere. Additionally, sinx, cos x, 𝑡𝑎𝑛−1(𝑥) and 𝑒𝑥are continuous everywhere. 𝑛 𝑥 is continuous for all x, when n is odd and for x > 0, when x is even. We also have ln 𝑥 is continuous for x > 0 and 𝑠𝑖𝑛−1(𝑥) and 𝑐𝑜𝑠−1(𝑥) for -1<x<1.

Example: “Fixing” a FunctionDetermine values of a and b that make the given function continuous.

𝑓 𝑥 =

2𝑠𝑖𝑛𝑥

𝑥𝑖𝑓 𝑥 < 0

𝑎 𝑖𝑓 𝑥 = 0𝑏𝑐𝑜𝑠 𝑥 𝑖𝑓 𝑥 > 0

Properties of Continuous Functions

If the functions and are continuous at , then the

following combinations are continuous at .

1. Su ms:

2. Differences:

3. Products:

4. Constant multiples: , for any number

5. Quotients: , pr

f g x c

x c

f g

f g

f g

k f k

f

g

ovided 0g c

Example: Continuity for a Rational Function

Determine where f is continuous, for 𝑓 𝑥 =𝑥4−3𝑥2+2

𝑥2−3𝑥−4.

Think about why you don’t see anything peculiar about the graph at x = -1.

Composite of Continuous Functions

If is continuous at and is continuous at , then the

composite is continuous at .

f c g f c

g f c

Intermediate Value Theorem for Continuous Functions***** STAR STAR STAR STAR STAR *****

0 0

A function that is continuous on a closed interval [ , ]

takes on every value between and . In other words,

if is between and , then for some in [ , ].

y f x a b

f a f b

y f a f b y f c c a b

Note: While this result seems reasonable, the proof is a lot more complicated than you might think, and it waits for a more advanced Calculus course.

Example: Intermediate Value Theorem

Show that 𝑓 𝑥 = 𝑥4 + 𝑥 − 3 has a root between -2 and 0 and another one between 0 and 2.