Section 2.5: Continuity
Instructor: Ms. Hoa Nguyen
([email protected])
Limits and Continuity
To deﬁne continuity, two things must be satisﬁed:
•
f
(
a
) is deﬁned (i.e.,
a
is in the domain of
f
).
•
lim
x
→
a
f
(
x
) exists.
Deﬁnition
: A function
f
is
continuous at a number
a
if lim
x
→
a
f
(
x
) =
f
(
a
).
Hence,
f
is
discontinuous at
a
if:
•
f
(
a
) is NOT deﬁned.
•
lim
x
→
a
f
(
x
) does not exist.
•
lim
x
→
a
f
(
x
)
6
=
f
(
a
).
Example 2 a, b, c of Section 2.5
(textbook):
Remark
:
•
f
is continuous at
a
⇒
limit of
f
(
x
), as
x
approaches
a
, exists.
•
Limit of
f
(
x
), as
x
approaches
a
, exists
;
f
is continuous at
a
.
Continuity from one side
A function
f
is
continuous from the right at a number
a
if lim
x
→
a
+
f
(
x
) =
f
(
a
).
A function
f
is
continuous from the left at a number
a
if lim
x
→
a

f
(
x
) =
f
(
a
).
Example 10 of Section 2.3 and Examples 2 d and 3 of Section 2.5
(textbook):
Continuity on an interval
A function
f
is
continuous on an interval
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 Fall '08
 Noohi
 Calculus, Continuity, Derivative, Limits, Continuous function, Inverse function, Ms. Hoa Nguyen

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