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f
f
, wesay that thefunction
f
is
not integrable
on
[
a
,
b
]. A function which is not bounded cannot have an integral in this
sense.
When
23 Use the basic property of an infimum (see qn 4.72) and the
analogous property of a supremum, to
sh
ow that if the function
f
is
integrable on [
a
,
b
] (that is
f
) then it is possible to
find an upper sum and a lower sum for
f
which arearbitrarily close
to each other; that is, given
f
f
0, it is possibleto find an uppr
step function
S
and a lower step function
s
such that
S
s
.
24 (Converse of qn 23.) If for a function
f
:[
a
,
b
]
R it is possibleto
find upper sums and lower sums whose differences arearbitrarily
small, use the chain of inequalities before qn 22 to prove that the
upper and lower integrals are equal, so that the function is
integrable.
Question 24 provides the standard test for integrability. We proved
in qns 12
—
15 that a step function was necessarily integrable. In qns 7
and 8 we have an argument which shows that a bounded monotonic
function is integrable even if it is not continuous.
25 Provethat thefunction
f
: [0, 1]
[0, 1], given by
f
(1/
n
)
1
f
(
x
)
0
when
n
1, 2, 3, . . .,
otherwise,
is integrable.
Summary
-
definition of the Riemann integral
Step functions
Definition
qn 15
R
A function
s
:[
a
,
b
]
is called a
step function
if, for somesubdivision
a
x
x
x
...
x
b
,
s
(
x
)
A
when
x
x
x
,
i
1, 2, . . .,
n
;
s
(
x
when
i
0, 1, . . .,
n
.
The integral of this step function is
)
B
s
x
A
(
x
).
