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8.9
Definition (Operations on functions.)
Let
and
be functions where
are sets.
Let
. We define functions
,
,
,
and
as
follows:
Remark: These operations of addition, subtraction, multiplication
and division for functions satisfy the associative, commutative and
distributive laws that you expect them to. The proofs are straightforward and
will
be omitted.
8.10
Definition (Partition-sample sequence.)
Let
be an interval. By a
partition-sample sequence for
I
will mean a pair of sequences
where
is a sequence
of
partitions of
such that
, and for each
in
,
is a sample for
.
8.11
Theorem (Sum theorem for integrable functions.)
Let be integrable functions on an interval . Then and
are integrable on and
and
Proof: Suppose and are integrable on . Let
be
a partition-sample sequence for . If
and
, then
Since and are integrable, we have
By the sum theorem for sequences,
Hence is integrable and
.
The proof of the second statement is left as an exercise.
8.12
Notation (
)
If
is integrable on an interval
we will sometimes write
instead of
. The ``
" in this expression is
a dummy variable, but the ``
" is a part of the notation and may not be
replaced by another symbol. This notation will be used mainly in cases where
no particular name is available for
. Thus
means
where
is the function on
defined by
for all
. The ``
" here stands for difference,
and
is a ghost of the differences
that appear in the
approximations
for the integral. The
notation is due to Leibniz.
8.13
Example.
Let
This function is integrable over every closed bounded subinterval of
,
since it is a sum of five functions that are known to be
integrable. By several applications of the sum theorem for
integrals we get
8.15
Theorem (Inequality theorem for integrals.) Let and
be integrable functions on the interval such that
Then
8.16
Exercise.
Prove the inequality theorem for integrals.
A
Proof: We have
Hence by the inequality theorem for integrals
Hence
It follows that
8.18
Theorem.
Let be real numbers with , and let be a function
from to
. Suppose is integrable on and is
integrable on
. Then is integrable on and
.
Proof:
Since is integrable on and on , it follows that
is bounded on and on , and hence is bounded
on .
Let
be a partition-sample sequence for .
For each in
we define a partition of and a
partition
of , and a sample for ,
and a
sample
for
as follows:
Then there is an index such that
.
Let
We have
where
Let be a bound for on . Then
Also,
Now
Since
it follows from the squeezing rule that
and hence
.
From equation (8.21)
we have
|
(8.22) |
Since
and
, we
see
that
is a partition-sample sequence on
, and
is a partition-sample
sequence on
. Since was given to be integrable on and on , we
know
that
and
Hence it follows from (8.22) that
i.e., is integrable on and
8.24
Definition (Spike function.)
Let
be an interval. A function
is called
a
spike function, if there exist numbers
and
, with
such that
Proof: Case 1: Suppose Observe that is increasing on the interval and decreasing
on the interval , so is integrable on each of these intervals.
The set of points under the graph of is the union of a horizontal
segment and a vertical segment, and thus is a zero-area set. Hence
By the previous theorem, is integrable on , and
.
Case 2: Suppose . Then by case 1 we see that is integrable with
integral equal to zero,
so by the sum theorem for integrals too.
8.28
Exercise.
Prove corollary
8.26, i.e., explain why it follows from theorem
8.25.
8.29
Definition (Piecewise monotonic function.)
A function
from an interval
to
is
piecewise monotonic
if there are points
in
with
such that
is monotonic on each of the intervals
.
8.30
Example.
The function whose graph is sketched below is piecewise monotonic.
8.31
Theorem.
Every piecewise monotonic function is integrable.
Proof: This follows from corollary 8.23.
8.32
Exercise.
A
Let
Sketch the graph of
. Carefully explain why
is integrable, and find
.
8.33
Example.
Let
Then
Hence
is integrable on
, and
Next: 8.3 A Non-integrable Function
Up: 8. Integrable Functions
Previous: 8.1 Definition of the
  Index
Ray Mayer
2007-09-07