4.9 Uniform Convergence of Continuous Functions
Suppose for each n ∈ ℕ fn is a continuous function defined on some interval
suppose that for each fixed x ∈
. This is called pointwise
Does it follow that f
is continuous on
? The answer is
exists for each
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.
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
Definition 4.9.1 Let
be a sequence of functions defined on D. Then fn is
said to converge uniformly to f on D if
∞ is called a norm.
The following picture illustrates the above definition.
The dotted lines define sort of a tube centered about the graph of f and the graph of
the function fn 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
Lemma 4.9.2 The norm
∞ satisfies the following properties.
For α a number,
Proof: The first claim 4.1 is obvious. As to 4.2, it follows fairly easily.
The last follows from
Now with this preparation, here is the main result.
Theorem 4.9.3 Let fn be continuous on D and
. Then f is also continuous. If each fn is uniformly
continuous, then f is uniformly continuous.
Proof: Let ε > 0 be given and let x ∈ D. Let n be such that
continuity of fn
there exists δ >
0 such that if
Then for such
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
. Then the above shows that if
which satisfies the definition of uniformly continuous.
This implies the following interesting corollary about a uniformly Cauchy sequence of
Definition 4.9.4 Let
be a sequence of continuous functions defined on
. It is said to be uniformly Cauchy if for every ε >
0 there exists nε such that if
m,k > nε
Corollary 4.9.5 Suppose
is a uniformly Cauchy sequence of functions
defined on D. Then there exists a unique continuous function f such that
. If each fn is uniformly continuous, then so is f.
Proof: The hypothesis implies that
is a Cauchy sequence in
. Therefore, by completeness of ℝ
, this sequence converges for each x
. Then by continuity of
provided n is sufficiently large. Since x is arbitrary, this shows that
if n is large enough. this says limn→∞
Now the continuity of f
. How many such functions f
are there? There can be only one because
must equal the limit of