20 1. The Regulated and Riemann Integrals
Hence,
b
a
u0(x) dx
b
a
v0(x) dx ε/2 + ε/2 = ε,
and u0 and v0 are the desired functions.
There are several facts about the relation with the regulated in-
tegral that must be established. Every regulated function is Riemann
integrable, but there are Riemann integrable functions which have no
regulated integral. Whenever a function has both types of integral
the values agree. We start by giving an example of a function which
is Riemann integrable, but not regulated.
Example 1.7.4. Define the function f : [0, 1] R by
f(x) =
1, if x =
1
n
for n N;
0, otherwise.
Then f(x) is Riemann integrable and
1
0
f(x) dx = 0, but it is not
regulated.
Proof. We define a step function um(x) by
um(x) =
1, if 0 x
1
m
;
f(x), otherwise.
A partition for this step function is given by
x0 = 0 x1 =
1
m
x2 =
1
m 1
··· xm−1 =
1
2
xm = 1.
Note that um(x) f(x). Also,
1
0
um(x) dx =
1
m
. This is because it
is constant and equal to 1 on the interval [0,
1
m
] and except for a finite
number of points it is constant and equal to 0 on the interval [
1
m
, 1].
Hence,
inf
u∈U(f )
1
0
u(x) dx inf
m∈N
1
0
um(x) dx = inf
m∈N
1
m
= 0.
Also, the constant function 0 is f(x) and its integral is 0, so
0 sup
v∈L(f )
1
0
v(x) dx .
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