copyright © 2016, 2012, and 2010 pearson education, inc. 1 what you ’ ll learn about definition...

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 1

What you’ll learn about

Definition of continuity at a point Types of discontinuities Sums, differences, products, quotients, and

compositions of continuous functions Common continuous functions Continuity and the Intermediate Value Theorem

…and whyContinuous functions are used to describe how a body moves through space and how the speed of a chemical reaction changes with time.

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Continuity at a Point

Any function whose graph can be sketched in

one continuous motion without lifting the pencil is an example of a continuous function.

y f x

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Example Continuity at a Point

Find the points at which the given function is continuous and the points at which it is discontinuous.

o

Points at which is continuousfAt 0 x

At 6 x

At 0 < < 6 but not 2 3 c cPoints at which is discontinuousfAt 2xAt 0, 2 3, 6 c c c

0

lim 0

x

f x f

6

lim 6

x

f x f

lim

x cf x f c

2

lim does not existxf x

these points are not in the domain of f

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You Try

For which intervals is the function continuous? Where is the function discontinuous?

Slide 2.3- 4

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Continuity at a Point

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

bf x f a f x f b

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Continuity at a Point

If a function   is   , we say that is    and   is a point of discontinuity of   . 

Note that need not be in the domain of   .

f fc f

c f

not continuous at a pointdiscontinuous at

cc

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Continuity at a Point

The typical discontinuity types are:

a) Removable (2.21b and 2.21c)

b) Jump (2.21d)

c) Infinite(2.21e)

d) Oscillating (2.21f)

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Continuity at a Point

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Example Continuity at a Point

There is an infinite discontinuity at 1.x

2

3Find and identify the points of discontinuity of 1

yx

[5,5] by [5,10]

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Continuous Functions

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 functionion need not be

continuous on every interval.

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Continuous Functions

2

22

yx

[5,5] by [5,10]

The given function is a continuous function because it is continuous at every point of its domain. It does have a point of discontinuity at 2 because it is not defined there.x

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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 cx c

f gf gf gk f kfg

ovided 0g c

Properties of Continuous FunctionsProperties of Continuous Functions

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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

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Intermediate Value Theorem for Continuous Functions

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

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Intermediate Value Theorem for Continuous Functions

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The Intermediate Value Theorem for Continuous Functions is the reason why the graph of a function continuous on an interval cannot have any breaks. The graph will be connected, a single, unbroken curve. It will not have jumps or separate branches.

Intermediate Value Theorem for Continuous FunctionsIntermediate Value Theorem for

Continuous Functions