CM221A
ANALYSIS I
NOTES ON WEEK 6
There will be no lectures on the week starting 8 November.
Use the
opportunity to revise the material covered in lectures.
Many of you
need to go through solutions to the class test and exercise sheets to see
what you got wrong and why.
DO NOT LEAVE THIS UNTIL THE SPRING VACATION!
CONTINUOUS FUNCTION II
A function
f
defined on an interval
I
is continuous if lim
x
→
c
f
(
x
) =
f
(
c
) for all
c
∈
I
. If
c
is an end point of
I
and
c
is included in
I
then we consider the right or
the left limit at
c
. If
c
does not belong to
I
, we do not impose any condition on
the right or left limit of
f
(
x
) at
c
. Clearly, if
f
is continuous on
I
then it is also
continuous on any smaller interval
I
1
⊂
I
.
Definition.
We say that a function is bounded if its range (the set of its values)
is a bounded subset of
R
.
If a function
f
is bounded then its range has finite g.l.b.
m
and l.u.b.
M
and
m
6
f
(
x
)
6
M
for all
x
from the domain of definition. The numbers
m
and
M
are called the greatest lower bound and, respectively,the least upper bound of the
function
f
. Note that the function
f
may not attain the maximum value
M
and/or
the minimum value
m
, even if it is continuous.
Example.
The function
f
(
x
) = (1

x
) sin(1
/x
) is continuous and bounded on
(0
,
1] and it has the least upper bound 1, but it does not take the value 1 anywhere,
and there is no obvious way of defining it at
x
= 0.
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 Spring '09
 Topology, Intermediate Value Theorem, Continuous function, Order theory, Metric space

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