8 1. The Regulated and Riemann Integrals

Definition 1.4.2. (Pointwise convergence). A sequence of func-

tions {fm} is said to converge pointwise on [a, b] to a function f if

for each ε 0 and each x ∈ [a, b] there is an Mx (depending on x)

such that

|f(x) − fm(x)| ε whenever m ≥ Mx.

On first encountering these two types of convergence of functions

it is diﬃcult to appreciate how different they are and how different

their consequences can be. The point of Examples 1.4.3 and 1.5.4

below and Exercise 1.5.6 parts (6) and (7) is to illustrate some of

the ways these concepts differ and to emphasize the importance of

the distinction. It should be immediately clear that a sequence of

functions which converges uniformly to f also converges pointwise to

f. The following example shows that the converse of this statement

is not true.

Example 1.4.3. For m ∈ N define the functions fm : [0, 1] → R by

fm(x) =

0, if x = 1;

xm,

otherwise.

Then for any fixed x0 ∈ [0, 1] it is clear that lim

n→∞

f

n(x0)

= 0. That

is, the sequence {fn} converges pointwise to the constant function 0.

On the other hand, it does not converge uniformly to 0. For

example, if ε = 1/3 we can never have |fm(x)−0| ε for all x ∈ [0, 1]

since if xm =

1/21/m,

then f(xm) = 1/2.

1.5. Regulated Integral

We now want to define the integral of a more general class of func-

tions than just step functions. Since we know how to integrate step

functions it is natural to try to take a sequence of better and better

step function approximations to a more general function f and define

the integral of f to be the limit of the integrals of the approximat-

ing step functions. For this to work we need to know that the limit

of the integrals exists and that it does not depend on the choice of

approximating step functions. It turns out that all of this works if

the more general function f can be uniformly approximated by step

functions, i.e., if there is a sequence of step functions which converges