MATH1010:
Chapter 2 cont…
1
LIMITS AND DERIVATIVES cont…
Continuity (Section 2.5 of Stewart, pg. 119)
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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View Full DocumentMATH1010:
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:
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 Fall '08
 Mihai
 Calculus, Continuity, Derivative, Intermediate Value Theorem, Limits, Continuous function, Inverse function, Complex number

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