calculus - amazon s3€¦ · continuous functions a function is continuous on an interval if and...
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CalculuS Continuity Lesson 1
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.
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.
2. 2.
6. 5.
Which rule is violated by each of these?
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.
Do #4 and 5 p.58 Packet
Continuity is also interfered with in
problems when in function form.
1,4
1,3.3
0,3
0,3.2
1
2.1
x
xxy
xx
xxy
x
xy
Assignment
Finish Packet
Intermediate Value Theorem
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
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
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
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.
Assignment
Finish Packet
IVT 1.4 WS
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