Graphics Reference
In-Depth Information
corresponding claim for
s
? Deduce that the lower sum from the
lower step function
s
the upper sum from the upper step function
S
. The idea of uniting the two dissections is due to Cauchy, 1823.
You have proved that every lower sum
every upper sum.
21 Show that for any bounded function
f
:[
a
,
b
]
R, the lower in
t
egral
must be less than or equal to the upper integral, that is
f
f
,
by showing that if this were false, we could use the basic property
of supremum and infimum (see qns 4.64 and 4.72) to contradict the
result of qn 20.
Putting together the definitions of upper and lower integral with
thersults of qns 20 and 21, weobtain
s
f
f
S
,
for all lower step functions
s
and all upper step functions
S
.
M
b
a
b a
c
b a
a
b
b
a
b a
m
22 Verify that any number lying between the upper integral and the
lower integral would satisfy the conditions we stated before qn 12
for theintegral
f
.
The Riemann integral
Question 22 implies that no unique integral satisfying the
conditions (i) and (ii), as stated before qn 12, can exist unless
f
f
,
and then
f
must equal the common value. When this is the case the
bounded function
f
is defined to be
integrable
(or more precisely
Riemann integrable
)on[
a
,
b
], and thecommon valueof
f
and
f
is
called the
Riemann integral
of thefunction.
