5. Continuity
P. K. Lamm
09/10/11 (15:22)
p. 1 / 8
Lecture Notes:
5. Continuity
These classnotes are intended to be supplementary to the textbook and are necessarily limited
by the time allotted for classes. For full and precise statements of definitions and theorems,
as well as material covering other topics and examples, please consult the textbook.
1. Continuity of Functions
Consider the following two limit problems:
lim
x
→
3
9

x
2
3

x
= lim
x
→
3
(3

x
)(3 +
x
)
3

x
= lim
x
→
3
3 +
x
= 6
,
and
lim
x
→
3
x
2
= 3
2
= 9
.
In the first example where
f
(
x
) =
9

x
2
3

x
, we have
lim
x
→
3
f
(
x
)
6
=
f
(3)
because there is no value on the curve at
x
= 3; thus for the first example there is a hole in the
curve above the point
x
= 3.
In the second example where
f
(
x
) =
x
2
, we have
lim
x
→
3
f
(
x
) =
f
(3)
,
so that there is no gap or jump in the curve at
x
= 3.
We say that the function
f
(
x
) =
x
2
is
continuous
at
x
= 3, while the curve
y
=
f
(
x
) =
9

x
2
3

x
is
discontinuous
or
not continuous
at
x
= 3.
1
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5. Continuity
P. K. Lamm
09/10/11 (15:22)
p. 2 / 8
Definition:
Suppose
f
is defined on an interval surrounding the point
x
0
.
We say
f
is
continuous at
x
0
provided
(1)
f
(
x
0
) exists;
(2)
lim
x
→
x
0
f
(
x
) exists; and
(3)
f
(
x
0
) = lim
x
→
x
0
f
(
x
) .
Definition:
We say
f
is
discontinuous at
x
0
if
f
is not continuous at
x
0
.
Definition:
We say
f
is
continuous on
(
a, b
) if
f
is continuous at every point
x
0
∈
(
a, b
).
Definition:
We say
f
is
continuous on
[
a, b
] if
f
is continuous on (
a, b
) and
lim
x
→
a
+
f
(
x
) =
f
(
a
)
,
and
lim
x
→
b

f
(
x
) =
f
(
b
)
.
Definition:
We say
f
is
continuous
if it is continuous on its domain.
If a function
f
is continuous on [
a, b
] then we can use a pencil to draw its graph from (
a, f
(
a
)) to
(
b, f
(
b
)) without ever having to cause the pencil to leave the page. A continuous function has no
holes, gaps, or jumps.
Some common continuous functions:
1. Polynomials are continuous on (
∞
,
∞
).
2. The trig functions sin
x
and cos
x
are continuous on (
∞
,
∞
).
3. The absolute value function
f
(
x
) =

x

is continuous on (
∞
,
∞
).
4. For positive integers
m
and
n
, the function
f
(
x
) =
x
m/n
is continuous on (
∞
,
∞
) if
n
is
odd, and continuous on [0
,
∞
) if
n
is even.
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 Fall '10
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 Calculus, Continuity, Continuous function, P. K. Lamm

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