CONTINUITY
We have seen that the limit of a function as
x
approaches
a
can sometimes
be found by calculating the value of the function at
x
=
a
.
Functions with this
property are called
continuous at
a
. Mathematical definition is as follows.
Continuity
A function
f
is continuous at
a
if
lim
x
→
a
f
(
x
) =
f
(
a
)
.
In fact, this definition requires three conditions of continuity. Those are
1. Existence of
f
(
a
),
2. Existence of the limit lim
x
→
a
f
(
x
), and
3. lim
x
→
a
f
(
x
) =
f
(
a
)
.
If any one of the above is not fulfilled, we say that
f
is
discontinuous at
a
.
f
is
continuous on an interval
if it is continuous at every point on the interval. We can
also define continuity from the left(right) by using the left(right)hand limit.
Example 1.
Find the points at which each of the following functions are dis
continuous.
(a)
f
(
x
) =
x
3

1
x

1
(b)
f
(
x
) =
H
(
x
)
(c)
f
(
x
) =
(
x
2
+
x
+ 1
if
x
6
= 1
2
if
x
= 1
.
Solution
In (a),
f
(1) does not exist, thus it is discontinuous at 1.
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 Fall '09
 BenedictGross
 Continuity, Intermediate Value Theorem, Continuous function

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