517 HW6
1. Give an example of a sequence of equicontinuous functions {fn } that converges pointwise
but not uniformly.
2. Let fn : [a, b] → R be monotone for each n. Prove that if {fn } converges pointwise to a
continuous function, then it converges uniformly.
3. Let X be compact, fn : X → R, and {fn } equicontinuous. Prove that if {fn } converges
pointwise then it converges uniformly.
4. Let φ : [0, 1] × R → R be continuous. Suppose there is 0 < M < 1 such that |φ(r, s)| ≤ M
and |φ(r, s) − φ(r, t)| ≤ M |s − t| for all r ∈ [0, 1], s, t ∈ R. Prove there is a solution to
y 0 = φ(x, y),
y(0) = c
by taking the following route:
Let Z be the set of continuous functions [0, 1] → R with the sup metric (see HW5, Problem 1),
define Ψ : Z → Z by
Z x
(Ψf )(x) = c +
φ(t, f (t)) dt,
x ∈ [0, 1],
0
and let
E = {f ∈ Z : |f (x) − c| ≤ M and |f (x) − f (y)| ≤ M |x − y| for all x, y ∈ [0, 1]}.
Use the Arzela-Ascoli theorem to show that that E is compact. Then show1 that Ψ|E is a
contraction mapping (see HW 4, Problem 1).
Note that the assumption M < 1 can be removed: how?
1
You can use without justification standard facts about integration.
1