MAST20026 Assignment 5 Semester 2 2022
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MAST20026 Assignment 5
Semester 2 2022
This assignment consists of 2 pages.
Due Date: Friday 21 Oct
Assignments are due in Gradescope on the due date listed above. To ease submission, please prepare assignment solutions for each question on its own page. Do not include your name anywhere on your assignment.
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PART A
Read the definition of gauge integral from Wikipedia (under the title Henstock - Kurzweil integral). Read the open letter https://math.vanderbilt.edu/schectex/ccc/gauge/letter/. Write a short response (< 250 words) response that considers the following questions.
● Do you think the definition of gauge integral have similar conceptual difficulty as that of Riemann integral?
● Would you be happier if you were taught gauge integral in Calculus or Real Analysis?
5 marks
PART B
(2) (5 marks) Prove the Mean Value Theorem using the six steps outlined at the end of 6.1.1. In your work you may assume derivatives of linear functions can be computed as you expect and that linear functions are continuous on their domain.
(3) Consider the following theorem and its proof.
Theorem. Let f : [a; b] → R be bounded. Then f is integrable on [a; b] if and only if for every 
> 0 there exists a partition Pe such that
U (f; Pe ) _ L(f; Pe ) < 
1 Proof. Assume f is integrable on [a; b]. Let 
> 0. Let 
\ = 
.
2 There exists a e (L(f; P) I P e 
} such that a > L(f) _ 
\ .
3 Similarly, there exists b e (U (f; P) I P e 
} such that b < U (f) + 
\ .
4 Since a e (L(f; P) I P e 
} there exists P e 
such that L(f; P) = a.
5 Similarly, there exists Q e 
such that U (f; Q) = b.
6 Therefore L(f; P) + 
\ > L(f) and U (f) > U (f; Q) _ 
\ .
7 Since f is integrable on [a; b] we have L(f) = U (f).
8 And so L(f; P) + 
\ > U (f; Q) _ 
\ .
9 Rearranging, we have U (f; Q) _ L(f; P) < 
\ + 
\ = 
.
10 Consider the partition P u Q. Notice P u Q is a refinement of both P and Q.
11 We have L(f; P) < L(f; P u Q) and U (f; Q) > U (f; P u Q).
12 Therefore U (f; P u Q) _ L(f; P u Q) < 
.
13 Therefore for every 
> 0 there exists a partition Pe such that U (f; Pe ) _ L(f; Pe ) < 
.
14 To prove the converse, we proceed by contradiction.
15 Assume for every . L 0 there exists a partition Q such that l (f< Q) _ P(f< Q) > ., but f is not integrable
16 on [a< b].
17 Since f is not integrable on [a< b] we have P(f) > l (f).
18 Let r = l (f) _ P(f). Notice r L 0. Let . = r .
19 By hypothesis, there exists a partition Qe such that l (f< Qe ) _ P(f< Qe ) > ..
20 Notice l (f< Qe ) _ P(f< Qe ) = (l (f< Qe ) _ l (f)) + (P(f) _ P(f< Qe )) + (l (f) _ P(f)).
21 By definition l (f< Qe ) _ l (f) > 0 and P(f) _ P(f< Qe ) > 0.
22 Therefore l (f< Qe ) _ P(f< Qe ) > 0 + 0 + r = ..
23 This is a contradiction.
24 Therefore if for every . L 0 there exists a partition Qe such that l (f< Qe ) _ P(f< Qe ) > ., then f is
25 integrable on [a< b].
口
(a) (2 mark ) How do you know the statement on line 2 is true. Cite a result or definition from the subject material to justify your response.
(b) (2 mark ) How do you know the second sentence on line 10 is true. Cite a result or definition from the subject material to justify your response.
(c) ( 1 mark ) How do you know the inequalities on line 11 are true. Cite a result or definition from the subject material to justify your response.
(d) ( 1 mark ) Which definition(s) are being referenced on line 21?
(e) ( 1 mark ) What is the contradiction on line 23? In particular, which two lines contradict one another.
(4) Let f : [a< b] → R be a bounded function such that for every z< y e [a< b], if z < y then f (z) < f (y).
(a) (3 marks) Let n e N＋ and let Qn = (z0 < , , , < zn } be the partition of [a< b] with zk _ zk - 1 = 
for
k e (1< 2< , , , n}.
Prove
l (f< Qn ) _ P(f< Qn ) =
(b) (3 marks) Prove f is integrable on [a< b] using the theorem from Question (3).
PART C (Optional – nothing to submit, no marks, you will not be assessed on the content of PART C)
● The LATEX source file for this assignment can be found in the Assignment 5 folder in Files on Canvas. This is a great starting point if you want to prepare your solution for this assignment using LATEX.
● In this folder you will also find a named Assignment5PartC.zip. Open, unzip, and compile these files in your LATEX editor and read about graphics and figures. In this zip file you will find a .txt file with more information on how to do this.
2022-11-03