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 .