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(iii) Now supposethat thedivision of theintrval [
a
,
b
] has been
into
n
equal lengths, so that
x
x
(
b
a
)/
n
for all
values of
i
1, 2, . . .,
n
. Check that in this case the difference
between the upper sum and the lower sum
b
a
n
(
M
m
).
If it were possible to prove that this may be made less than
any given
0, we would have succeeded in proving that
f
was integrable.
Now usethefact that a function which is continuous on a
closed interval is uniformly continuous on that interval to
show that, for any given
0, there exists a
0 such that
x
y
f
(
x
)
f
(
y
)
/(
b
a
).
Show how to choose
n
so that the difference of upper and
lower sums previously calculated is less than
.
This establishes that a continuous function on a closed interval is
integrable.
40 Does the converse of the result at the end of qn 39 hold? If a
function is integrable must it be continuous?
41
f
0 and
f
(
x
)
0 for all
x
[
a
,
b
]. If
f
is continuous, provethat
f
(
x
)
0 for all
x
[
a
,
b
]. What if
f
were not necessarily continuous?
42 If
f
and
g
areintegrableon [
a
,
b
] and
(
k
·
f
l
·
g
)
0, under
what conditions could you deduce that
k
·
f
(
x
)
l
·
g
(
x
) for all
x
[
a
,
b
]?
43
(i) If
f
is a continuous function on [
a
,
b
], provethat
lies between the upper sum and the lower sum calculated in
qn 39.
(ii) Deduce that its limit as
n
b
a
n
i
n
(
b
a
)
f
a
f
.
(iii) Check that the limit obtained coincides with the measurement
of area in qns 1 and 2.
is
The notation for integrals, proposed by Leibniz, which is in
standard usetoday, namly
f
(
x
)
dx
, is deliberately suggestive of the
limit of thesum
f
(
x
)
x
, where
x
denotes the difference
x
x
,
and the limit is taken as the greatest of the
x
0.
