hw6 - false(a An equicontinuous pointwise bounded subset of...

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Homework 6 - Math 321, Spring 2015 Due on Friday February 27 1. Given a nonconstant non-decreasing function α : [ a, b ] R , let R α [ a, b ] denote the collection of all bounded functions on [ a, b ] which are Riemann-Stieltjes integrable with respect to α . Is R α [ a, b ] a vector space, a lattice, an algebra? 2. This problem focuses on computing the Riemann-Stieltjes integral for specific choices of integrators. (a) Let x 0 = a < x 1 < x 2 < · · · < x n = b be a fixed collection of points in [ a, b ], and let α be an increasing step function on [ a, b ] that is constant on each of the open intervals ( x i - 1 , x i ) and has jumps of size α i = α ( x i +) - α ( x i - ) at each of the points x i . For i = 0 and n , we make the obvious adjustments α 0 = α ( a +) - α ( a ) , α n = α ( b ) - α ( b - ) . If f B [ a, b ] is continuous at each of the points x i , show that f ∈ R α [ a, b ] and Z b a f dα = n X i =0 f ( x i ) α i . (b) If f is continuous on [1 , n ], compute R n 1 f ( x ) d [ x ], where [ x ] is the greatest integer in x . What is the value of R t 1 f ( x ) d [ x ] if t is not an integer? 3. Determine, with adequate justification, whether each of the following statements is true or
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Unformatted text preview: false. (a) An equicontinuous, pointwise bounded subset of C [ a,b ] is compact. (b) The function χ Q is Riemann integrable on [0 , 1]. (c) The function χ Δ is Riemann integrable on [0 , 1], where Δ denotes the Cantor middle-third set. (We have already run into this set in Homework 2, Problem 5). (d) T α {R α [ a,b ] : α increasing } = C [ a,b ]. (e) If f is a monotone function and α is both continuous and non-decreasing, then f ∈ R α [ a,b ]. (f) There exists a non-decreasing function α : [ a,b ] → R and a function f ∈ R α [ a,b ] such that f and α share a common-sided discontinuity. (g) If f ∈ R α [ a,b ] with m ≤ f ≤ M and if ϕ is continuous on [ m,M ], then ϕ ◦ f ∈ R α [ a,b ]. 1...
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