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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Ask a homework question - tutors are online