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15. These functions are not generally either positive or continuous, but
they will be able to play the roˆ le of inscribed and circumscribed
polygons for us in an analytical setting. The use of properties (i) and (ii)
in qns 12
—
15 is to establish the reasonableness of the definition of the
integral of a step function when it comes in qn 15.
12 If
f
(
x
)
A
on [
a
,
b
], show that there is a unique definition of
f
which is compatiblewith (i) abov.
B
A
a
c
c + h
b
13 If
f
(
x
)
A
on [
a
,
c
] and
f
(
x
)
B
on (
c
,
b
], with
A
B
usethe
equation
f
f
f
f
to show that
(
c
a
)
A
hA
(
b
c
h
)
B
f
(
c
a
)
A
hB
(
b
c
h
)
B
so
(
c
a
)
A
(
b
c
)
B
h
(
B
A
)
f
(
c
a
)
A
(
b
c
)
B
and, if this is to hold for all
h
, however small,
f
must be defined
as (
c
a
)
A
(
b
c
)
B
.
Is the same definition necessary if
A
B
?
14 Show that thevalueof theintgral found in qn 13 must also bethe
valueof theintegral for thefunction,
f
, defined by
