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I need the soluation for question 3-1 in the first attachment and question 1 with a b in the second attatment.

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Since differentiation is also a limit operation, this is another “interchange of limit operations” result. D2f is uniformly continuous on A x J. Exercises 3.1 (a) If f is integrable, prove that f? is integrable. Hint: Given € > 0, let h and k be step functions such that hf $k and S (k - h) < /M, where M is the maximum value of \k(x) + h(x). Then prove that h2 and kare step functions with h? Sf? Sk? (we may assume that I shfk since f is integrable if and only if \f| is--why?), and that ſ (k2 – ha)< €. Then apply Theorem 3.3. (b) If f and g are integrable, prove that fg is integrable. Hint: Express fg in terms of (f+g)2 and (f-g)2; then apply (a). 3.2 Let f: R → R be a bounded function which vanishes outside the interval Q. Show that fis integrable with Sf=I if and only if lim R(f, PK, Ix)=1 k-00 for every sequence of partitions {Pulis and associated selections {Svi such that limk>. (mesh of Pk)=0. 3.3 Let A and B be contented sets, and f: Ax BA a uniformly continuous function. If : A R is defined by 11 YEA Textbook, Page 233-234, 3.1, 3.7, plus, 1. Consider two intervals A = (0,2) > [1,4 and B = [1,3] x [0,3) in R. Let f = a4A and g = b4B, where 4A and B are the characteristic functions of A and B, respectively, and a, b are constants. (a) Find a step function expression for f +g. (a) Evaluate the integralſ (8 + 9). 3.7. For each positive integer n, the Bessel function Jn't may be defined by In W) cosat at S 1. tj 1/2 103.5 . (2n-1) 1) T - prove that Jn (x) Satisfies Bessel's differential equation -)Jn In + Į Jon't llom . 14 1 +2 1ylch M® MIQUELRI y=f2+ R2 to o=Y+ f2-42- 0=lik, ne-ythan 0=(142) Pyt fz- (6) Cuish M® MIQUELRIUS fytzf-2f 1 falififing zf- fiffinf . in hefs schim 1 оч. and subject to the condition ) with bounder undery condition се --- C- \ Gampel considera JORDI LABANDA M® MIQUELRIUS ЧС 7 + во - о * (1) лъ ix- 'n stolz SWE ral bellof у м. A>> Д що шӘ\ kaf neco, hef oo - ч- \ 4, 5, 14, , , 16. - - - " - бу анъа - КА" - ч QH Х У У. си- си? х 4 Син АСУ ТУС >4- pro (5ч гәче че чре ч f in this case and f is integrable iss given esa ] step function SiRn R, bounded have bounded се рәч M® MIQUELRIUS So far dy Fox B CRM is contended Sopp ig dyr = is a function of J for each fued I Say=fung ва идэ х әess = CuS R R utuid if мә~49 hp xp 2p critics as : 3 SXəp° 9 ХЭР r to'c SS S - до ѕ No 2lfoxif asss yo [he] (62] [211] - nou an f. • Руә\ « Bans CSS. up chin soss (с 11- hat aftpt-ching is if He
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Explanation & Answer

attached is the solution to question 3-1 and question 1 with a b as required

Question 1:
(a) Let A1   A  B  , A2  A \ B , A3  A  B and A4  B \ A . For i 1, 2,3, 4 , let  i be the value
c

of f  g on Ai .
Since A2 , A3 are contained in A but A1 and A4 are disjoint with A , we can deduce that a A  0 on A1

and A4 and a A  1on A2 and A3 . Similarly, since A3 , A4  B while A1 and A2 are disjoint with A , it
implies that bB  0 on A3 and A4 , and b B  1 on A1 and A2 .
We will calculate each of the values of  i ’s as follow:

1   f  g  A  f
1

A1

2   f  g  A  f

A2

3   f  g  A  f

A3

4   f  g  A  f

A4

2

3

4

g
g
g
g

A1

 a A

A1

 a B

A1

 00  0

A2

 a A

A2

 a B

A2

A3

 a A

A3

 a B

A3

A4

 a A

A4

 a B

A4

 a0 a
 ab
 0b  b

Hence, a step function expression for f  g can be shown below:
4

f  g    i  Ai
i 1

 1  A1   2  A2   3  A3   4  A4
 0   A1  a   A2   a  b    A3  b   A4
(b) Before calculating the integral  f  g , we have to calculate the areas Ai ’s, for i 2,3, 4 . Notice
that A3  1, 2  1,3 , and the area of rectangular  a, b  c, d  is  b  a  d  c  . Hence, the
calculation is shown below:

A3   2  1   3 �...


Anonymous
This is great! Exactly what I wanted.

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