Graphics Reference
In-Depth Information
31 If thefunction
f
is defined by
f
(
x
)
0
when
f
(
x
)
0,
when
f
(
x
)
0,
f
(
x
)
provethat
s
is a lower step function for
f
and that
S
is an
upper step function for
f
. Then by considering
s
and
S
on the
intervals on which both are constant, as in qn 20, show that the
difference between the upper and lower sums which they give is less
than or equal to
. Deduce that, if
f
is integrable, then so is
f
.
32 If thefunction
f
is defined by
0
f
(
x
)
when
f
(
x
)
0,
when
f
(
x
)
0,
f
(
x
)
show that
f
(
x
) for all values of
x
.
Deduce from qns 27 and 31 that, if
f
is integrable, then so is
f
.
(
x
)
(
f
)
33 If thefunction
f
is defined by
f
(
x
)
f
(
x
)
when
f
(
x
)
0,
when
f
(
x
)
0,
f
(
x
)
check that
f
f
f
, and deduce that, if
f
is integrable, then
so is
f
.
34 Givean exampleto show that
f
may be integrable even when
f
is
not.
35 Usetheinequality
f
f
f
f
,
which follows from thetriangleinequality of qn 2.61 to provethat
f
f
, when
f
is integrable on [
a
,
b
].
36 (This is a hard question on which nothing further in this topic
depends, but it opens up the question as to what functions may be
Riemann integrable in a most dramatic way.)
(
homae
, 1875) In qn 6.72 we examined the continuity of the ruler
function
f
defined on [0, 1] by
T
f
(
x
)
0
when
x
is irrational or zero
f
(
p
/
q
)
1/
q
when
p
and
q
are non-zero integers with no
common factor.
Wefound that this function was continuous at each irrational point
