INTRO CLASSICAL ANALYSIS
INTRO CLASSICAL ANALYSIS MATH 615
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This 3 page Class Notes was uploaded by Evert Christiansen on Wednesday October 21, 2015. The Class Notes belongs to MATH 615 at Texas A&M University taught by Natarajan Sivakumar in Fall. Since its upload, it has received 20 views. For similar materials see /class/225999/math-615-texas-a-m-university in Mathematics (M) at Texas A&M University.
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Date Created: 10/21/15
Siuakumar 1 O CO 4 0 d3 1 00 M515 080 Example Sheet 11 Suppose that f and g belong to BV1 1 Recall that HfHoo 3 sup m 3 a S 90 S b and Moo 3 SUPHWEH 3 a S 90 S b Show that fg belongs to BV1 b and Vfglmbl S HfHonglmbl HgHoonlmbl Suppose f E BV1 b Show that Vf1 b 0 if and only if f is constant throughout 11 ii Prove that the function Hme 3 WW Vfla7bl7 f E BVlaybly is a norm on BV1 b True or false lf fn is a sequence of functions such that fquot E BV1 b for every n and fn converges uniformly to f on 11 then f E BV1 b Suppose fn is a sequence of functions satisfying the following conditions a fquot E BV1 b for every n b there exists a constant A such that an1b A for every n and c fn converges pointwise to f on 11 Prove that f E BV1b and Vf1b A ii Prove that BV1b MM is a Banach space Let 1 and b be real numbers with 1 lt I Let c be a xed number in the open interval 1 1 De ne 7 0 ifa xltcg Hquot1 ifcgmgb Show that H is Riemann integrable on 11 Suppose that f is Riemann integrable on 11 and that x 2 0 for every 1 g x g I Let 12 a alt b Showthatffsz Suppose that f 11 a R satis es each of the following conditions x 2 0 for every 1 g x g 1 ii 1 is Riemann integrable on 11 and iii 1 is right continuous at every point in 1 b and left continuous at b Prove if ff 0 then x 0 for every x 6 11 Prove that the function HIH1Ow Megan is a norm on CO ii Show that C0 1 is not complete CD Suppose that a and b are real numbers with a lt I Let a g 01 lt 02 lt lt cm be xed and de ne fm1 1fmck k1n 0 otherwise 12 Show that f f O 1 ii Let T1 T2 be an enumeration of the rationals in 0119 and de ne for every n E N the function 1 1fm r1rn 0 otherwise 1W 7 b a Observe that f f 0 for every n b Find the pointwise limit of f and show that it is not Riemann integrable on 0 1 Thus the pointwise limit of a sequence of Riemann integrable functions need not be Riemann integrable 10 De ne f 01 7 R as follows ifz gcdmn1 f90 0 if z is irrational i Show that f is discontinuous at every rational number in 01 and continuous at every irrational number in 0 1 ii Show that f 0 0 De nition Suppose P is a partition of an interval 0119 The norm of P is de ned as follows maxzk 7 7671 1g k g n 11 Let f be a bounded function de ned on the compact interval 0 b b i Suppose f is Riemann integrable on 0119 and let I f Prove that for every 6 gt 0 there is a 6 66 gt 0 such that SP f 7 I lt e for everay partition P with lt 6 and any Riemann partition sum SP 1 corresponding to P ii Suppose that there exists a number I such that for every 6 gt 0 there is a 6 66 gt 0 such that SPf 7 I lt e for every partition P with lt 6 and any Riemann partition sum SP 1 corresponding to P Prove that f is Riemann integrable on 0 b and I is the Riemann integral of f 12 Suppose f is Riemann integrable on ab Prove that 1 127aquot kb7a 7 1 Jim 7 gflt T 13 Suppose that g is a nondecreasing function on 11 i Prove lf f1 and f2 are integrable with respect to 97 and f1z f2 for every a g x b7 then f1 d9 S f2 d9 ii Provealf m g g M for every a g x g b and f is integrable with respect to g on b M then mltgltbgt 7 M g fdg Mltgltbgt ego iii Prove the First Mean Value Igheorem for Riemann Stieltjes Integrals If f is continuous 12 on 11 then there is a point c in 11 such that fdg fcgb 7 9a Use ii and the Intermediate Value Theorem 14 Suppose that f E BVO7 27139 and that n is a xed positive integer Prove that S Vf 0 27139 71 02W 35 cosnm dz
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