CONTINUITY

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BASIC IDEA
Intuitively, a function is continuous at x = c if you can
draw it without lifting your pen from the paper. In the
diagram below, the function on the left is continuous
throughout, but the function on the right is not. It is
"discontinuous" at x = c.
CONTINUITY
SOURCE: www.portledge.org/homework/jsillat/documents/1.4lecture_000.ppt
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FORMAL DEFINITION
EXAMPLE OF DISCONTINUITY
Formally, a function is continuous at x = c on the
open interval (a, b) if the following three conditions
are met:
f(c) is defined
lim f ( x) exists
x→c
lim f ( x ) = f (c )
x →c
3
• F is continuous at x=c if there are no holes,
jumps or gaps at c
4
Removable and
Nonremovable discontinuity
Graphically:
Removable
discontinuity
Nonremovable
discontinuity
Removable
discontinuity
f (x) is continuous at c.
lim f ( x) = f (c )
x →c
5
Discuss the continuity of the following graph.
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Continuity at a Point
• A function can be discontinuous at a point
– The function jumps to a different value at a
point
– The function goes to infinity at one or both
sides of the point, known as a pole
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8
Discuss the continuity of each of
the following functions:
f ( x) =
The domain of f is all nonzero real numbers. You can
conclude that f is continuous at every x-value in its
domain. In other words, there is no way to define f(0) so
as to make the function continuous at x = 0.
1
x
f ( x) =
1
x
Nonremovable Discontinuity
9
Discuss the continuity of each of
the following functions:
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• The domain of f is all real numbers except x = 1.
Therefore, you can conclude that f is continuous at
every x-value in its domain.
• At x = 1, the function has a removable discontinuity.
• If f(1) is defined as 2, the “newly defined” function is
continuous for all real numbers.
x2 − 1
f ( x) =
x −1
x2 − 1
f ( x) =
x −1
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Removable Discontinuity
12
Discuss the continuity of the
function:
• The domain of f(x) is all real numbers.
• Therefore, you can conclude that the function is continuous on its entire
domain.
• When a function is continuous from (-∞,∞),
f ( x ) = sin x
• We say that the function is everywhere continuous.
f ( x) = sin x
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Everywhere Continuous
14
Theorem Properties of
Continuity
Continuity Theorem
• A function will be continuous at any
number x = c for which f(c) is defined,
when …
ƒ
ƒ
ƒ
ƒ
ƒ
f(x) is a polynomial
f(x) is a power function
f(x) is a rational function
f(x) is a trigonometric function
f(x) is an inverse trigonometric function
15
16
Continuity of a Composite Function
Removing a discontinuity:
f ( x) =
lim
x →1
x3 − 1
x2 −1
has a discontinuity at x = 1 .
Write an extended function that is continuous
at x = 1 .
( x − 1) ( x 2 + x + 1) 1 + 1 + 1
x3 − 1
=
lim
=
x →1
( x + 1)( x − 1)
2
x2 −1
 x3 − 1
 x 2 − 1 , x ≠ 1
f ( x) = 
 3 , x =1
 2
=
3
2
Note: There is another
discontinuity at x = −1 that can
not be removed.
17
18
Properties of Continuous Functions
Continuity on an Interval
• If f and g are functions, continuous at x = c
Then …
ƒ s ⋅ f ( x ) is continuous (where s is a constant)
• The function f is said to be continuous on
an open interval (a, b) if
ƒ
ƒ
f(x) + g(x) is continuous
f ( x) ⋅ g ( x) is continuous
ƒ
f ( x)
is continuous
g ( x)
ƒ
f(g(x)) is continuous
– It is continuous at each number/point of the
interval
• It is said to be continuous on a closed
interval [a, b] if
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– It is continuous at each number/point of the
interval and
– It is continuous from the right at a and
continuous from the left at b
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Intermediate Value Theorem
Continuity on an Interval
If a function is continuous between a and b, then it takes
• On what intervals are the following
functions continuous?
on every value between f ( a ) and f ( b ) .
g ( x) = sin x
f (b)
f ( x) =
Because the function is
continuous, it must take on
every y value between f ( a )
and f ( b ) .
f (a)
x2 −1
x2 − 4
b
a
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Intermediate Value Theorem
The Intermiate Value Theorem implies that if f
is continuous on [a,b] and k is any number
between f(a) and f(b), then there exists a
number c in [a,b] such that f(c) = k.
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→
Locating Roots with Intermediate Value Theorem
• Given f (a) and f (b) have opposite sign
– One negative, the other positive
• Then there must be a root between a and b
a
b
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