1.4. Uniform and Pointwise Convergence 7
y0 = a y1 y2 ··· yn−1 ym = b is another parti-
tion for the same step function with value dj on (yj−1, yi),
then
n
i=1
ci(xi xi−1) =
m
j=1
di(yj yj−1).
In other words, the value of the integral of a step function
depends only on the function, not on the choice of partition.
Hint: the union of the sets of points defining the two par-
titions defines a third partition and the integral using this
partition is equal to the integral using each of the partitions.
(3) Prove that the integral of step functions as given in Defini-
tion 1.3.2 satisfies properties I–V of §1.2.
1.4. Uniform and Pointwise Convergence
Throughout the text we will be interested in the following question:
If a sequence of functions {fn} “converges” to a limit function f does
the sequence of numbers {
b
a
fn(x) dx} converge to a limit equal to
the integral of the limit function? Put another way, we are interested
in when lim and commute, i.e., when is
lim
n→∞
b
a
fn(x) dx =
b
a
lim
n→∞
fn(x) dx?
The answer, as we will see, depends on what we mean by the sequence
of functions “converging,” i.e., what does lim fn mean. It turns out
there are many interesting (and very different) choices for what we
might mean. Among the types of convergence that we will consider
the strongest is called uniform convergence. We recall its definition.
Definition 1.4.1. (Uniform convergence). A sequence of func-
tions {fm} is said to converge uniformly on [a, b] to a function f if
for every ε 0 there is an M (independent of x) such that for all
x [a, b],
|f(x) fm(x)| ε whenever m M.
We contrast this with the following much weaker notion of con-
vergence.
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