Suppose for each n ∈ ℕ f_{n} is a continuous function defined on some interval
[a,b]
. Also
suppose that for each fixed x ∈
[a,b]
, lim_{n→∞}f_{n}
(x )
= f
(x)
. This is called pointwise
convergence. Does it follow that f is continuous on
[a,b]
? The answer is NO. Consider
the following
f (x) ≡ xn for x ∈ [0,1]
n
Then lim_{n→∞}f_{n}
(x )
exists for each x ∈
[0,1]
and equals
{
f (x) ≡ 1 if x = 1
0 if x ⁄= 1
You should verify this is the case. This limit function is not continuous. Indeed, it has a
jump at x = 1. Here are graphs of the first few of these functions.
PICT
If you want the convergence to carry continuity with it you need something more than
point-wise convergence. You need uniform convergence. The concept seems to have been
understood by Cauchy but was not at all clear. Weierstrass is the first to formalize this
concept.
Definition 4.9.1Let
{fn}
be a sequence of functions defined on D. Then f_{n}issaid to converge uniformly to f on D if
The following picture illustrates the above definition.
PICT
The dotted lines define sort of a tube centered about the graph of f and the graph of
the function f_{n} fits in this tube for all n sufficiently large. The tube can be made as
narrow as desired.
It is convenient to observe the following properties of
∥⋅∥
_{∞}, written
∥⋅∥
for
short.
Lemma 4.9.2The norm
∥⋅∥
_{∞}satisfies the following properties.
∥f∥ ≥ 0 and equals 0 if and only if f = 0 (4.1)
(4.1)
For α a number,
∥αf∥ = |α|∥f∥ (4.2)
(4.2)
∥f + g∥ ≤ ∥f∥ + ∥g∥ (4.3)
(4.3)
Proof:The first claim 4.1 is obvious. As to 4.2, it follows fairly easily.
and so this shows that f is continuous. To show the claim about uniform continuity, use
the same string of inequalities above where δ is chosen so that for any pair x,y with
|x − y|
< δ,
|fn (y)− fn (x)|
<
ε
3
. Then the above shows that if
|x− y|
< δ,
then
|f (x)− f (y)|
< ε which satisfies the definition of uniformly continuous.
■
This implies the following interesting corollary about a uniformly Cauchy sequence of
continuous functions.
Definition 4.9.4Let
{fn}
be a sequence of continuous functions defined on
[a,b]
. It is said to beuniformly Cauchy if for every ε > 0 there exists n_{ε}such that ifm,k > n_{ε}
∥fm − fk∥ < ε
Corollary 4.9.5Suppose
{fn}
is a uniformly Cauchy sequence of functionsdefined on D. Then there exists a unique continuous function f such that
lim_{n→∞}
∥fn − f∥
= 0. If each f_{n}is uniformly continuous, then so is f.
Proof: The hypothesis implies that
{fn(x)}
is a Cauchy sequence in ℝ for
each x. Therefore, by completeness of ℝ, this sequence converges for each x.
Let f