MATH1131 – Mathematics 1A
Calculus Chapter 03
Properties of Continuous Functions
Dr. Thanh Tran
School of Mathematics and Statistics
The University of New South Wales
Sydney, Australia
1
1
Combining Continuous Functions
2
The Intermediate Value Theorem
3
The MaximumMinimum Theorem
2
Continuity at a point (revision)
Definition 1
Suppose that
f
is defined on some
open
interval containing the point
a
.
We say that
f
is
continuous
at
a
if
lim
x
→
a
f
(
x
)
exists and
lim
x
→
a
f
(
x
) =
f
(
a
)
.
Combining Continuous Functions
3
Theorem 2
If the functions f and g are continuous at a, then
f
±
g is continuous at a
fg is continuous at a
f
/
g is continuous at a, provided g
(
a
)
negationslash
=
0
Proof: We look only at continuity of
fg
. As
f
and
g
are continuous at
a
we
have
lim
x
→
a
f
(
x
) =
f
(
a
)
and
lim
x
→
a
g
(
x
) =
g
(
a
)
.
Next
lim
x
→
a
(
fg
)(
x
) =
lim
x
→
a
(
f
(
x
)
g
(
x
)
)
(defn of
fg
)
=
parenleftBig
lim
x
→
a
f
(
x
)
parenrightBigparenleftBig
lim
x
→
a
g
(
x
)
parenrightBig
(props of limits)
=
f
(
a
)
×
g
(
a
)
(
f
,
g
are cts at
a
)
=
(
fg
)
(
a
)
(defn of
fg
)
Since the limit of
fg
as
x
→
a
is the value of
fg
at
x
=
a
,
fg
is continuous at
a
.
Combining Continuous Functions
4
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Theorem 3
Let g be continuous at a and let f be continuous at g
(
a
)
, then f
◦
g is
continuous at a.
Proof: Use rule for composition of limits — for you to do!
Combining Continuous Functions
5
Example 4
Let
f
:
R
→
R
be given by
f
(
x
) =
braceleftBigg
cos
(
ax
)
for
x
≤
π
bx
for
x
> π
.
For what values of
a
and
b
will
f
be continuous?
From the definition of
f
we see that
f
is continuous when
x
negationslash
=
π
. In fact,
since
f
(
x
) =
cos
(
ax
)
when
x
< π
, and the cosine function is
continuous,
f
is continuous.
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 Spring '11
 Calculus, Statistics, Topology, Intermediate Value Theorem, Continuous function, Properties of Continuous Functions

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