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8/31/2009
1
Continuity
Section 2.6
Definition of Continuity
A function is
continuous at c
if the following
three conditions are met:
1.
is defined
2.
exists
3.
)
(
c
f
)
(
lim
x
f
c
x
→
)
(
)
(
lim
c
f
x
f
c
x
=
→
Examples:
o
Use the definition of continuity to determine
if the following function is continuous at 1.
o
Use the definition of continuity to determine
if the following function is continuous at 1.
≤
+
<
+
=
1
,
1
1
,
1
)
(
3
x
x
x
x
x
f

≤
=
1
,
1
1
,
)
(
x
x
x
x
x
f
Types of Discontinuities
o
Removable (Holes in the graph)
o
Nonremovable
n
Infinite
n
Finite Jump
Continuity on Intervals
o
A function is continuous on an
open
interval
(a,b)
if
it is continuous at each point in the interval.
o
A function is continuous on a
closed
interval
[a,b]
if
it is continuous on the open interval (a,b)
and
and
.
(i.e., nothing "bad" happens at the endpoints)
o
We say a function is
continuous
if it is continuous
on the entire real line.
)
(
)
(
lim
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 Spring '11
 PamelaSatterfield
 Continuity

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