CalculuS - Amazon S3€¦ · Continuous Functions A function is continuous on an interval if and...
Transcript of CalculuS - Amazon S3€¦ · Continuous Functions A function is continuous on an interval if and...
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CalculuS Continuity Lesson 1
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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
If a function f is not continuous at a point c ,
we say that f is discontinuous at c and c is a
point of discontinuity of f.
Note that c need not be in the domain of f.
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CONTINUITY Functions such as f(x) = 2x–1 are considered
continuous which implies that as it graphs there are no skips.
Are these continuous?
1.
2. 3.
4. 5. 6.
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2. 2.
6. 5.
Which rule is violated by each of these?
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Continuous Functions
A function is continuous on an interval if and
only if it is continuous at every point of the
interval. A continuous function is one that is
continuous at every point of its domain. A
continuous function need not be continuous on
every interval.
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Do #4 and 5 p.58 Packet
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Continuity is also interfered with in
problems when in function form.
1,4
1,3.3
0,3
0,3.2
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Assignment
Finish Packet
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Intermediate Value Theorem
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Slide 2- 21
Properties of Continuous Functions
If the functions and are continuous at , then the
following combinations are continuous at .
1. Sums :
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
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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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Slide 2- 23
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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Slide 2- 24
Intermediate Value Theorem for Continuous Functions
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.
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Assignment
Finish Packet
IVT 1.4 WS
Review for Quiz