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# hw2 - Homework 2 – Math 118B Winter 2010 Due on Thursday...

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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 Cauchy-Schwarz o inequality when p = q = 2. 1 ...
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