8.38
Example (Ruler function.)
We now present an example of an integrable function
that
is not monotonic on any interval of positive length. Define
![$R:[0,1]\to\mbox{{\bf R}}$](img2169.gif)
by
This formula defines
![$R(x)$](img556.gif)
uniquely: If
![$\displaystyle { {q\over {2^n}}={p\over {2^m}}
} $](img2171.gif)
where
![$p$](img246.gif)
and
![$q$](img1923.gif)
are odd, then
![$m=n$](img2172.gif)
. (If
![$m>n$](img2173.gif)
, we get
![$2^{m-n}q=p$](img2174.gif)
,
which says
that an even number is odd.) The set
![$S_0^1R$](img2175.gif)
under the graph of
![$R$](img47.gif)
is shown
in
the figure.
This set resembles the markings giving fractions of an inch on a
ruler, which motivates the name
ruler function for
![$R$](img47.gif)
.
It is easy to see that
![$R$](img47.gif)
is not monotonic on any interval of length
![$>0$](img2177.gif)
.
For each
![$p\in\mbox{{\bf R}}$](img2178.gif)
let
![$\delta_p\colon\mbox{{\bf R}}\to\mbox{{\bf R}}$](img2179.gif)
be defined by
We have seen that
![$\delta_p$](img2181.gif)
is integrable on any interval
![$[a,b]$](img1071.gif)
and
![$\displaystyle {\int_a^b \delta_p=0}$](img2182.gif)
. Now define a sequence of functions
![$F_j$](img2183.gif)
by
Each function
![$F_j$](img2183.gif)
is integrable with integral
![$0$](img48.gif)
and
I will now show that
is integrable.
Let
be a partition-sample sequence for
. I'll show
that
.
Let
be a generic element in
.
Observe that if
then
Hence by the Archimedian property, we can
choose
![$M \in \mbox{${\mbox{{\bf Z}}}^{+}$}$](img2190.gif)
so that
![$\displaystyle { {1\over
{2^M}}<\epsilon}$](img2192.gif)
. Then
Now since
![$\displaystyle { 0\leq R(x)-F_M(x)\leq {1\over {2^{M+1}}}<{1\over 2}\epsilon}$](img2196.gif)
for
all
![$x\in [0,1]$](img2197.gif)
, we have
Since
![$F_M$](img2199.gif)
is integrable and
![$\int F_M=0$](img2200.gif)
, we have
![$\{\sum (F_M,P_n,S_n)\}\to
0$](img2201.gif)
so there is an
![$N\in\mbox{${\mbox{{\bf Z}}}^{+}$}$](img2202.gif)
such that
![$\displaystyle {\vert\sum (F_M,P_n,S_n)\vert<{\epsilon\over
2}}$](img2203.gif)
for all
![$n\in\mbox{{\bf Z}}_{\geq N}$](img2204.gif)
. By equation (
8.40) we have
Hence
![$\{\sum(R,P_n,S_n)\}\to 0$](img2189.gif)
, and hence
![$R$](img47.gif)
is integrable and
![$\displaystyle {\int_0^1 R=0}$](img2206.gif)
.
8.41
Exercise.
A
Let
![$R$](img47.gif)
be the ruler function. We just gave a complicated proof that
![$R$](img47.gif)
is
integrable and
![$\displaystyle {\int_0^1 R=0}$](img2206.gif)
. Explain why if you
assume ![$R$](img47.gif)
is
integrable, then it is easy to show that
![$\displaystyle {\int_0^1 R=0}$](img2206.gif)
.
Also show that
if you assume that the non-integrable function
in equation (8.37)
is integrable then it
is easy
to show that
.