? As in the case of integrals of functions of one variable, this is an important
question. It turns out the Riemannn integrable functions are characterized by being continuous except on a
very small set. This has to do with Jordan content.
Definition 20.4.1A bounded set E, has Jordan content 0 or content 0if for every ε > 0 there exists agrid G such that
∑
v(Q) < ε.
Q∩E ⁄=∅
This symbol says to sum the volumes of all boxes from G which have nonempty intersection withE.
Next it is necessary to define the oscillation of a function.
Definition 20.4.2Let f be a function defined on ℝ^{n}and let
ωf,r(x) ≡ sup {|f (z)− f (y)| : z,y ∈ B (x,r)}.
This is called the oscillation of f on B
(x,r)
. Note that this function of r is decreasing in r. Define theoscillation of f as
ωf (x) ≡ rli→m0+ ωf,r(x).
Note that as r decreases, the function ω_{f,r}
(x)
decreases. It is also bounded below by 0,
so the limit must exist and equals inf
{ωf,r(x) : r > 0}
. (Why?) Then the following simple
lemma whose proof follows directly from the definition of continuity gives the reason for this
definition.
Lemma 20.4.3A function f is continuous at x if and only if ω_{f}
(x)
= 0.
This concept of oscillation gives a way to define how discontinuous a function is at a point. The
discussion will depend on the following fundamental lemma which gives the existence of something called
the Lebesgue number.
Definition 20.4.4Letℭbe a set whose elements are sets of ℝ^{n}and let K ⊆ ℝ^{n}. The set ℭ iscalled a cover of K if every point of K is contained in some set of ℭ. If the elements of ℭ are opensets, it is called an open cover.
Lemma 20.4.5 Let K be sequentially compact and let ℭ be an open cover of K. Then there existsr > 0 such that whenever x ∈ K,B(x,r) is contained in some set of ℭ.
Proof:Suppose this is not so. Then letting r_{n} = 1∕n, there exists x_{n}∈ K such that B
(xn,rn)
is
not contained in any set of ℭ. Since K is sequentially compact, there is a subsequence, x_{nk}
which converges to a point x ∈ K. But there exists δ > 0 such that B
(x,δ)
⊆ U for some
U ∈ℭ. Let k be so large that 1∕k < δ∕2 and
|xnk − x|
< δ∕2 also. Then if z ∈ B
(xnk,rnk)
, it
follows
δ δ
|z − x| ≤ |z − xnk|+ |xnk − x | < +-= δ
2 2
and so B
(xnk,rnk)
⊆ U contrary to supposition. Therefore, the desired number exists after all.
■
Theorem 20.4.6Let f be a bounded function which equals zero off a bounded set and let W denote theset of points where f fails to be continuous. Then f ∈ℛ
(ℝn)
if W has content zero. That is, for all ε > 0
there exists a grid G such that
∑
v(Q) < ε (20.10)
Q∈GW
(20.10)
where
GW ≡ {Q ∈ G : Q ∩ W ⁄= ∅} .
Proof: Let W have content zero. Also let
|f (x )|
< C∕2 for all x ∈ ℝ^{n}, let ε > 0 be given, and let G be
a grid which satisfies (20.10). Since f equals zero off some bounded set, there exists R such that f equals
zero off of B
( R)
0,2
. Thus W ⊆ B
( R)
0,2
. Also note that if G is a grid for which (20.10) holds, then this
inequality continues to hold if G is replaced with a refined grid. Therefore, you may assume the diameter of
every box in G which intersects B
(0,R )
is less than
R-
3
and so all boxes of G which intersect the set where f
is nonzero are contained in B
(0,R)
. Since W is bounded, G_{W} contains only finitely many boxes.
Letting
∏n
Q ≡ [ai,bi]
i=1
be one of these boxes, enlarge the box slightly as indicated in the following picture.
PICT
The enlarged box is an open set of the form,
∏n
^Q ≡ (ai − ηi,bi + ηi)
i=1
where η_{i} is chosen small enough that if
∏n ( ^)
( bi + ηi − (ai − ηi)) ≡ v Q ,
i=1
and
^
GW
denotes those
^
Q
for Q ∈G which have nonempty intersection with W, then
∑ ( ^)
v ^Q < ε (20.11)
^Q∈^GW
(20.11)
where
^
^Q
is the box,
∏n
((ai − 2ηi), bi + 2ηi).
i=1
For each x ∈ ℝ^{n}, let r_{x}< min
(η1∕2,⋅⋅⋅,ηn∕2)
be such that
ωf,rx (x) < ε +ωf (x). (20.12)
(20.12)
Now let ℭ denote all intersections of the form
^
Q
∩ B
(x,rx)
such that x ∈B
(0,R)
so that ℭ is an open
cover of the compact set B
(0,R)
. Let δ be a Lebesgue number for this open cover of B
(0,R )
and let ℱ be a refinement of G such that every box in ℱ has diameter less than δ. Now let
ℱ_{1} consist of those boxes of ℱ which have nonempty intersection with B
(0,R ∕2)
. Thus all
boxes of ℱ_{1} are contained in B
(0,R )
and each one is contained in some set of ℭ. Let ℭ_{W} be
those open sets of ℭ,
^
Q
∩ B
(x,rx)
, for which x ∈ W. Thus each of these sets is contained in
some
^^Q
where Q ∈G_{
W}. Let ℱ_{W} be those sets of ℱ_{1} which are subsets of some set of ℭ_{W}.
Thus
∑
v(Q) < ε. (20.13)
Q ∈ℱW
(20.13)
because each Q in ℱ_{W} is contained in a set
^^
Q
described above and the sum of the volumes of these is less
than ε by (20.11). Then
U (f)− ℒ (f) = ∑ (M (f)− m (f ))v (Q )
ℱ ℱ Q∈ℱ Q Q
W
∑
+ (MQ (f)− mQ (f ))v (Q ).
Q ∈ℱ1∖ℱW
If Q ∈ℱ_{1}∖ℱ_{W}, then Q must be a subset of some set of ℭ ∖ℭ_{W} since it is not in any set of ℭ_{W}. Say
Q ⊆
Q^
1
∩ B
(x,r )
x
where x
∈∕
W. Therefore, from (20.12) and the observation that x
Consider the inside sum with the aid of the following picture.
PICT
In this picture, the little rectangles represent the boxes Q_{j}×
[ai,ai+1]
for fixed j. The part of P having
x contained in Q_{j} is between the two surfaces, x_{n+1} = g
(x)
and x_{n+1} = f
(x)
and there is a zero placed in
those boxes for which
MQj× [ai,ai+1](XP)− mQj ×[ai,ai+1](XP ) = 0.
You see, X_{P} has either the value of 1 or the value of 0 depending on whether
(x,y)
is contained in P. For
the boxes shown with 0 in them, either all of the box is contained in P or none of the box is contained in
P. Either way,
MQj ×[ai,ai+1](XP )− mQj×[ai,ai+1](XP ) = 0
on these boxes. However, on the boxes intersected by the surfaces, the value of
MQ × [a,a ](XP )− mQ ×[a ,a ](XP )
j i i+1 j i i+1
is 1 because there are points in this box which are not in P as well as points which are in P. Because of the
construction of G^{′} which included all values of
The first of the sums in (20.15) contains all possible terms for which
MQj× [ai,ai+1](XP )− mQj ×[ai,ai+1](XP )
might be 1 due to the graph of the bottom surface gX_{E} while the second sum contains all possible terms
for which the expression might be 1 due to the graph of the top surface fX_{E}.