MATH1010:
Chapter 2 cont…
1
LIMITS AND DERIVATIVES cont…
Continuity (Section 2.5 of Stewart, pg. 124)
Question:
How does
⎩
⎨
⎧
=
≠
+
=
3
3
2
1
)
(
x
x
x
x
f
compare with
1
)
(
+
=
x
x
g
?
Definition:
A function
f
is
continuous
at a number
a
if
)
(
)
(
lim
a
f
x
f
a
x
=
→
So, what actually has to hold?
Graphical Example:
What happens if we are given a formula for the function?…how do we determine where
the function is continuous?
Example:
1
4
5
)
(
2
+
+
+
=
x
x
x
x
f
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Chapter 2 cont…
2
Example:
⎪
⎩
⎪
⎨
⎧
=
≠
−
=
3
3
6
3
1
)
(
x
x
x
x
f
Application:
Suppose that a certain country has the following income tax rates, where
x
is income in thousands of dollars.
⎪
⎩
⎪
⎨
⎧
>
≤
≤
<
=
20
20
10
10
15
.
0
1
.
0
0
)
(
x
x
x
x
f
Definition:
A function
f
is
continuous from the right
at a number
a
if
)
(
)
(
lim
a
f
x
f
a
x
=
+
→
and
f
is
continuous from the left
at
a
if
)
(
)
(
lim
a
f
x
f
a
x
=
−
→
Graphical Example:
Example:
⎩
⎨
⎧
>
≤
+
−
+
=
0
0
7
2
1
4
)
(
x
x
x
x
x
f
So far, we’ve only talked about continuity at a point, but what do we mean when we say a
function is continuous on e.g. [0, 6]?
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 Spring '08
 Kim
 Calculus, Continuity, Derivative, Limits, Continuous function, income tax rates, graphical example

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