INTEGRATION LECTURE NOTES
by E B Davies
1. Introduction
If
f
is a realvalued function on a bounded interval [
a, b
] the integral
b
a
f
(
x
) d
x
is
intended to be a measure of the area under the graph of the function.
If the function
takes both positive and negative values then one defines the integral to be the difference
of the areas of
A
and
B
where
A
=
{
(
x, y
) :
a
≤
x
≤
b
and 0
< y < f
(
x
)
}
B
=
{
(
x, y
) :
a
≤
x
≤
b
and 0
> y > f
(
x
)
}
.
The above idea of integral transfers the problem of giving a precise definition of integral
to that of giving a precise definition of area, which turns out to be no easier. We shall
not attempt to define integration from first principles in these notes, but concentrate on
the properties that a successful integration procedure should have, and then develop the
theory from there.
The first issue is that one cannot hope to integrate every conceivable function, ob
taining a welldefined real number as its integral. In some cases the integral is infinite, in
others both sets
A
and
B
have infinite areas, and there is no sensible meaning to
∞  ∞
,
while in yet others the function may be so irregular that it is not clear how to start to
define its integral.
In these notes we only consider the integrals of piecewise continuous functions on a
bounded interval [
a, b
].
Piecewise continuous functions
f
: [
a, b
]
→
R
are functions for
which there exist numbers
a
=
a
0
< a
1
<
· · ·
< a
n
=
b
such that
f
is continuous on each interval (
a
k

1
, a
k
) and the limits
lim
x
→
a
k

0
f
(
x
)
,
lim
x
→
a
k
+0
f
(
x
)
exist for all relevant
k
. In other words, this means that
f
has a jump discontinuity of size
f
(
a
k
+0)

f
(
a
k

0) at each point
a
k
, but is otherwise continuous. The space
PC
of such
functions contains the space
S
of step functions and the space
C
of continuous functions
on [
a, b
] (recall that a step function
f
is a piecewise continuous functions in the sense of
our definition, which takes constant values on each interval (
a
k

1
, a
k
)).
It is evident from the definition that if
f
and
g
are piecewise continuous functions on
[
a, b
] and
α, β
∈
R
then the combination (
αf
+
βg
)(
x
) =
αf
(
x
) +
βg
(
x
) is also piecewise
continuous. The set of jump points of
αf
+
βg
is just the union of the jump points of
f
and of
g
, or possibly a smaller set if the new function has a jump of size zero between
two consecutive intervals.
We summarize this by saying that
PC
is a vector space of
functions. It is not finitedimensional because it contains the polynomials of all orders.
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 Spring '09
 Calculus, Derivative, Continuous function

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