Pointwise and Uniform Convergence of Sequences
of Functions (7.1)
For a sequence of numbers, convergence means only one thing (at
least in the context of advanced calculus). However, there are many
different ways of defining convergence of a sequence of functions. We
will see two. The first is pointwise convergence.
Definition: A sequence fn of functions defined on a set E converges
pointwise to a function f defined on E if
lim fn (x) = f (x)
n→∞
holds for all x ∈ E. Written in logical notation, fn converges pointwise
to f on E if ∀ x ∈ E and ∀ ǫ > 0, ∃ N ∈ N so that ∀ n ≥ N we
have |fn (x) − f (x)| < ǫ. Note that since N appears after ǫ and x are
quantified, the value of N may depend on both ǫ and x.
Example 13. You have already seen this idea in a somewhat different form. For example, it is a standard fact from basic calculus that
for any x with |x| < 1, we have limn→∞ xn = 0. Interpret this now as
a statement about a sequence of functions. This is illustrated in Figure
1, where y = xn is plotted for a few values of n. As we look at any fixed
x value between −1 and 1, (in the figure, x is 0.6), the values of the
functions approach 0. Thus we say that {xn } approaches 0 pointwise
on the set (−1, 1). The pointwise limit is not 0 if we look at the larger
set (−1, 1]: on that set, {xn } converges pointwise to
0 −1 < x < 1
f (x) =
.
1
1=x
Example 13 shows that a pointwise limit of continuous functions
need not be continuous. In fact, there is hardly any elementary calculus
property which is conserved by pointwise convergence. In other words,
the pointwise limit of a sequence of differentiable functions need not
be differentiable, the pointwise limit of integrable functions need not
be integrable, and so on. Here are a couple of examples.
16
1
0.5
0
-1
-0.5
0
0.5
1
x
-0.5
-1
Figure 1. Pointwise convergence
Example 14. Let {rn }, n = 1, · · · be an enumeration of Q ∩ [0, 1].
Define fn by
1 x = r1 , · · · , rn
fn (x) =
.
0
otherwise
Each fn is Riemann integrable on [0, 1] since it is non-zero except for
a finite number of points. However, the pointwise limit of {fn } is the
Dirichlet function, which we have seen is not Riemann integrable.
Example 15. For n ≥ 2, define a piecewise linear function fn on
[0, 1] by

n2 x
0 ≤ x ≤ 1/n

2
2n − n x 1/n ≤ x ≤ 2/n .
fn (x) =

0
2/n ≤ x ≤ 1
(See figure 2.) These are often called tent functions. First to show that
fn (x) goes to zero pointwise on [0, 1].
There are two cases: if x ∈ (0, 1], we have fn (x) = 0 for all n > 2/x,
and if x = 0, ´then fn (0) = 0 for all n.
1
However, 0 fn (x) dx = 1 for all n, so that in this case,
ˆ 1
ˆ 1
ˆ 1
1 = lim
0 dx = 0.
lim fn (x) dx =
fn (x) dx 6=
n→∞
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2012,
T. Vogel
0
0 n→∞
17
0
n
1/n
2/n
x
Figure 2. A tent function
At this point you might suspect that finding the pointwise limit of
a sequence of functions is not all that useful, since the pointwise limit
might not inherit any useful properties of the sequence. The reason for
studying pointwise limits is that once you find a pointwise limit, you
can often show that the sequence converges in a stronger, more useful
sense called uniform convergence.
Definition: A sequence {fn } defined on a set E converges uniformly to a function f if for every ǫ > 0 there exists an N so that n ≥ N
implies that |fn (x) − f (x)| < ǫ holds for all x ∈ E. In logical notation,
∀ǫ > 0, ∃N ∈ N so that ∀n ≥ N , ∀x ∈ E, there holds |fn (x)−f (x)| < ǫ.
The difference between this definition and the definition of pointwise
convergence is that for uniform convergence, N depends on ǫ, but not
on x. In other words, if n ≥ N , then |fn (x) − f (x)| < ǫ holds uniformly
across E (see Figure 3).
Note: If fn converges to f uniformly on E, then fn converges
to f pointwise as well. Therefore, if you are asked to prove that a
sequence fn converges uniformly on a set E, it makes sense to first find
the pointwise limit, and then attempt to prove that the convergence is
uniform. The method used in example 16 to show uniform convergence
is common: first find the pointwise limit f (x) of {fn }, then bound
|fn (x) − f (x)| on E by a sequence of constants which approaches 0.
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2012,
T. Vogel
18
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
x
Figure 3. Uniform convergence
Example 16. Prove that the sequence {fn } =
uniformly on the interval [1, 2].
n
nx2 +1
nx+1
o
converges
Proof: We must first find a candidate for the uniform limit by
taking the pointwise limit.
Since
nx2 + 1
x2 + 1/n
(2)
lim
= lim
=x
n→∞ nx + 1
n→∞ x + 1/n
for all x ∈ [1, 2], we have that fn converges pointwise to x on [1, 2]. To
show uniform convergence, we must look at |fn (x) − x|. We have that
2
1 − x 1 + |x|
nx + 1
3
(3)
nx + 1 − x = nx + 1 ≤ nx + 1 ≤ 1 + n
3
goes to zero as n tends to infinity, fn will
for all x ∈ [1, 2]. Since 1+n
converge uniformly to x on [1, 2]. More precisely, if ǫ > 0 is arbitrary,
3
there is an N so that n ≥ N implies that 1+n
< ǫ. By (3), n ≥
N implies that |fn (x) − x| < ǫ for all x ∈ [1, 2], proving uniform
convergence.
Some properties are conserved under uniform convergence. One is
continuity.
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2012,
T. Vogel
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