Unformatted text preview: Homework 2 – Math 118B, Winter 2010
Due on Thursday, January 28, 2010
1. Suppose α increases on [a, b], a ≤ x0 ≤ b, α is continuous at x0 , f (x0 ) =
1, and f (x) = 0 if x = x0 . Prove that f ∈ R(α) and that f dα = 0.
b
a 2. Suppose that f ≥ 0, f is continuous on [a, b], and
that f (x) = 0 for all x ∈ [a, b]. f (x)dx = 0. Prove 3. Suppose f is a bounded real function on [a, b] and f 2 ∈ R on [a, b].
Doest it follow that f ∈ R?
4. Let P be the Cantor set. Let f be a bounded real function on [0, 1]
which is continuous at every point outside P . Prove that f ∈ R on
[0, 1].
5. Suppose f is a real function on (0, 1] and f ∈ R on [c, 1] for every
c > 0. Deﬁne
1 1 f (x) dx = lim c→0 0 f (x) dx,
c if this limit exists and is ﬁnite.
(a) If f ∈ R on [0, 1], show that this deﬁnition of the integral agrees
with the old one.
(b) Construct a function f such that the above limit exists, although
it fails to exist with f  in place of f .
6. Let p, q be positive real numbers such that
11
+ = 1.
pq
Prove that if f and g are real functions in R(α), then
q f  dα
a 1/q b p f g dα ≤
a 1/p b b g  dα . a This is called H¨lder’s inequality, and it reduces to the CauchySchwarz
o
inequality when p = q = 2. 1 ...
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 Fall '09
 Garcia
 Math, Calculus, Continuous function, Cantor, Suppose, real function

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