1.5. Regulated Integral 11

Definition 1.5.3. (Regulated integral). If f is a regulated func-

tion on [a, b], we define the regulated integral by

b

a

f(x) dx = lim

n→∞

b

a

fn(x) dx

where {fn} is any sequence of step functions converging uniformly to

f.

One might well ask if we can take the same approach and define

an integral for functions which are the limits of pointwise conver-

gent sequences of functions. Unfortunately, this does not work as the

following example shows.

Example 1.5.4. For each n ∈ N define a step function on [0, 1] by

fn(x) =

2n2,

if x ∈ [

1

2n

,

1

n

];

0, otherwise.

Notice that if x0 ∈ (0, 1] and 1/n x0, then fn(x0) = 0; so clearly

lim fn(x0) = 0 for every x0. Also, fn(0) = 0 for all n. In other words,

the sequence of functions {fn} converges pointwise to the constant

function f = 0.

However,

1

0

fn(x) dx = n so the sequence of integrals diverges

while the integral of the limit function f = 0 has the value 0.

This should be contrasted with part (6) of Exercise 1.5.6 which

shows that if a sequence of functions {fn} converges uniformly to a

function f, then under very general hypotheses,

lim

n→∞

b

a

fn(x)dx =

b

a

f(x)dx.

For the definition of regulated integral to be interesting it is im-

portant that there are lots of regulated functions which we might

want to integrate. This is indeed the case since the regulated func-

tions include all continuous functions on a closed interval [a, b].

Theorem 1.5.5. (Continuous functions are regulated). Every

continuous function f : [a, b] → R is a regulated function.