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non-empty and bounded above it has a supremum, which is called
the
lower integral
for
f
, and is denoted by
f
.
17 Every bounded function on a closed interval has a lower integral.
Find the lower integral for the function in qn 11.
18 Consider a function
f
:[
a
,
b
]
bea step
function such that
S
(
x
)
f
(
x
) for all values of
x
. Such a step
function is called an
upper step function
for
f
.
Wecall
[
m
,
M
]. Let
S
:[
a
,
b
]
R
S
,an
upper sum
for
f
.
Why must
M
(
b
a
) bean uppr sum for
f
?
Why must every upper sum be greater than or equal to
m
(
b
a
)?
Since, for the given bounded function
f
, the set of all upper sums is
non-empty and bounded below it has an i
n
finimum, which is called
the
upper integral
for
f
, and is denoted by
f
.
19 Find the upper integral for the function in qn 11.
It is in proving the existence of upper and lower integrals for
bounded functions that completeness is needed in building a theory of
integration.
x
0
x
1
x
2
x
3
S
s
y
0
z
0
y
1
z
1
y
2
z
3
y
3
z
5
z
2
z
4
20 (
Darboux
, 1875) For a given bounded function
f
:[
a
,
b
]
R,let
S
be
an upper step function, which is constant on the intervals (
x
,
x
)
where
a
x
b
.
Let
s
be a lower step function which is constant on the intervals
(
y
x
x
...
x
,
y
) where
a
y
y
y
...
y
b
.
Let
z
,
z
,
z
,...,
z
x
,...,
x
y
,...,
y
where
a
z
b
.
Why areboth
S
and
s
constant on each of the intervals (
z
z
z
...,
z
,
z
)?
Provethat
s
(
x
)
S
(
x
) on such an interval.
Does
S
have the same value whether it is calculated on the
intervals (
x
,
x
) or on theintrvals (
z
,
z
)? What is the
