MATH 321 - HOMEWORK #4
Due Friday, Feb 26.
P ROBLEM 1. Let γ : [a, b] → R2 be a continuous rectifiable curve. Prove that
L(x) = Λxa γ
is continuous on [a, b]. (Here Λxa γ is the length of γ on the interval [a, x].)
P ROBLEM 2. Find sequences of functions {fn } and {gn }, such that fn → f and gn → g
uniformly, but fn · gn does not converge uniformly.
P ROBLEM 3. Let {fn } be a sequence of continuous functions on [a, b]. Prove that fn → f
uniformly if and only if for any sequence of points {xn } ⊂ [a, b] that converges to x,
n→∞
fn (xn ) −→ f(x).
(Hint: for the if direction use the compactness of [a, b] and argue by contradiction.)
P ROBLEM 4. We defined the Lp norm on the space of continuous functions on E:
Z
1
kfkp =
|f|p dx p , p ≥ 1,
E
kfk∞ = sup |f(x)|.
x∈E
A sequence of continuous functions {fn } on E converges to f in the Lp norm if
n→∞
kfn − fkp −→ 0.
Give examples of sequences of continuous functions that converge in one norm but not
the other. When the domain E is not bounded, find fn and f, such that each function is
supported on some finite interval [a, b] (but the interval [a, b] may depend on the function).
(a) E = [a, b], fn converges in L1 but not in L∞ .
(b) E = R, fn converges in L∞ but not in L1 .
(c) E = R, fn converges in L2 but not in L1 .
(d) E = [a, b], fn converges pointwise to a continuous f, but not in L1 .
(e) E = [a, b], fn converges in L1 but not pointwise at any point x ∈ E. (Hint: if the domain
E is the circle S1 , think of the graph as a wave that travels around the circle. The wave
keeps the same height, but gets narrower as n → ∞.)
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