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Transcript of 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.
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 2
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
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 3
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
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 4
You Try
For which intervals is the function continuous? Where is the function discontinuous?
Slide 2.3- 4
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 5
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
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 6
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
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 7
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)
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 8
Continuity at a Point
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 9
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]
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 10
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.
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 11
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
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 12
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
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 13
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
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 14
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
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 15
Intermediate Value Theorem for Continuous Functions
Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 16
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