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Math 365C: Real Analysis I
The University of Texas at Austin , Spring 2021
Official times : MWF 1-2PM CST Media : Canvas , Zoom , Piazza (see “Course delivery and resources")
Instructor : Joe Kileel Teaching assistant : Hunter Vallejos Consultant : João Pereira Office hours : schedule on Canvas
This course is an introduction to analysis. Analysis, together with algebra and topology, form the central core of modern pure mathematics. Analysis is also in- dispensable to the applied and computational mathematician. Beginning with the notion of limit from calculus and continuing with ideas about convergence and the concept of function that arose with the description of heat flow using Fourier series, analysis is primarily concerned with infinite processes , the study of spaces where these processes act and the application of differential and integral methods. Specifically in this course, we will have three principle focuses. We will rigorously develop the operations and concepts of single-variable calculus (limits, continuity, derivative, Taylor series, Riemann integral). We will study abstract metric spaces (capturing the general notion of distance). Time permitting, we will sketch the be- ginnings of functional analysis (where functions are regarded as points in a space). At risk of sounding melodramatic, learning real analysis is a rite of passage for un- dergraduate math majors (both those with pure and applied interests). At the same time, the present course is important to theoretically minded students in adjacent disciplines, such as physics, computer science, statistics, electrical engineering, fi- nance and economics. You might also find analysis good for your health: promoting rigorous and abstract thinking, and empowering you to turn pictures into proofs.
Textbook
The course textbook is Walter Rudin’s Principles of Mathematical Analysis , 3rd ed. We will cover the first five chapters and the seventh chapter. For Riemann integra- tion, we will follow Chapter 7 of Stephen Abbott’s Understanding Analysis , 2nd ed. As for optional supplementary reading, Real Mathematical Analysis , 2nd ed. by Charles Pugh is nice for intuition and for having many pictures. Introduction to Met- ric and Topological Spaces , 2nd ed. by Wilson Sutherland goes beyond our scope, but certain parts might be helpful for understanding metric spaces.
Course delivery and resources
Our course will be delivered using the following resources and structure:
- Lectures will be asynchronous : they will be prerecorded and posted on Panopto integrated in Canvas (typically not later than the corresponding official lecture time MWF 1-2PM CST). Handwritten notes produced during the lectures will be saved as PDFs and also uploaded to Canvas.
- Exam review problem-solving sessions will be synchronous , occurring live on Zoom , and will include students working in breakout rooms and some pre- senting to the class. Also, the first informal organizational lecture on Wednes- day, January 20 will occur live on Zoom.
- Office hours in this course will be plentiful: most weeks we will have six office hours, typically at least one hour Tuesday–Friday. The instructor, teaching assistant and course consultant will each generally offer two hours per week over Zoom (see Canvas for a schedule). Please take advantage of office hours.
- Discussion boards will be set up on Piazza integrated with Canvas. Course- related questions should be posted on Piazza rather than emailed to course staff (with the exception of questions on personal matters). Course staff will answer questions on Piazza and moderate discussion there. Students are en- couraged to try their hand in answering some of their classmates’ questions on Piazza , as this is a great way to learn yourself. Note that making mistakes in Piazza is absolutely acceptable; we personally think it is also a good way to learn. For students falling very near boundaries between letter grades, Pi- azza participation might be taken into discretionary consideration; however, for this to help boost a boundary case, it must be the case that the student has posted to Piazza non-anonymously (visible to other students) exclusively.
Legal notes: course materials should not be distributed outside the course. The or- ganizational meeting and some problem-solving sessions may be recorded. Class recordings are reserved only for students in this class and are protected under FERPA. The recordings should not be shared outside the class in any form. Violation of this restriction by a student could lead to Student Misconduct proceedings.
Grading
Numerical scores will be computed per this breakdown:
- Homework: 20% (lowest two dropped)
- Midterm I: 20%
- Midterm II: 20%
- Final: 40%
Letter grades with plus and minuses will then be determined according to a curve. The end distribution of letters will be similar to previous iterations of Math 365C.
Course announcements
Important course announcements will typically be double-posted, on Canvas An- nouncements and on Piazza. In any case, students are responsible for making them- selves aware of important course announcements.
Accommodations
The University of Texas provides, upon request, appropriate academic accommoda- tions for qualified students with disabilities. For more information, contact Services for Students with Disabilities at 512-471-6259 or [email protected]
Academic integrity
The University of Texas holds you to the following Standards of Conduct: https://deanofstudents.utexas.edu/conduct/standardsofconduct.php. Violations of these Standards shall be treated seriously, and punished appropriately.
Blanket caveat
This syllabus is subject to change. Students are responsible for making themselves aware of syllabus changes announced in Canvas Announcements and Piazza.
Lecture # Date Topics Text
1 (informal, live) Wed, Jan 20 (^) course, what’s a real number anyway?everyone says hi, organization of the ° 2 Fri, Jan 22 R^ as ordered field with least-upper-boundproperty, archimedean property Chapter 1 3 Mon, Jan 25 Euclidean space, Dedekind cuts (sketch) Chapter 1 4 Wed, Jan 27 countable sets Chapter 2 5 Fri, Jan 29 metric space: basic definitions & examples Chapter 2 6 Mon, Feb 1 metric space: basic definitions & examples Chapter 2 7 (drop date) Wed, Feb 3 compact sets Chapter 2 8 Fri, Feb 5 compact sets Chapter 2 9 Mon, Feb 8 connected sets Chapter 2 10 Wed, Feb 10 convergent sequences in metric spaces Chapter 3 11 Fri, Feb 12 Cauchy sequences in metric spaces,complete metric spaces Chapter 3 12 Mon, Feb 15 STORM ° 13 Wed, Feb 17 STORM ° 14 Fri, Feb 19 STORM ° 15 Mon, Feb 22 STORM ° 16 Wed, Feb 24 sequential compactness, completeness Chapter 3 17 Fri, Feb 26 numerical examples, lim sup & lim inf Chapter 3 18 (midterm review, live) Mon, Mar 1 problems session on Zoom ° 19 (midterm review, live) Wed, Mar 3 problems session on Zoom °
20 (no lecture, exam) Fri, Mar 5 MIDTERM I (covering lectures 2-11, 16-17) ° 21 Mon, Mar 8 BREAK ° 22 Wed, Mar 10 root & ratio tests Chapter 3 23 Fri, Mar 12 power series, absolute convergence Chapter 3 ° Mon, Mar 15 –Fri, Mar 19 SPRING BREAK ° 24 Mon, Mar 22 addition, multiplication, rearrangment of series Chapter 3 25 Wed, Mar 24 continuity of functions on metric spaces Chapter 4 26 Fri, Mar 26 continuity & compactness Chapter 4 27 Mon, Mar 29 uniform continuity, continuity & connectedness Chapter 4 28 Wed, Mar 31 examples, one-sided limits, limits at 1 Chapter 4 29 Fri, Apr 2 derivative of real-valued function (basics) Chapter 5 30 Mon, Apr 5 local extrema, Rolle’s theorem, mean value theorem Chapter 5 31 Wed, Apr 7 continuity of derivatives, L’Hhigher derivatives^ ospital’s rule,ˆ Chapter 5 32 Fri, Apr 9 Taylor’s theorem, remainder term, examples Chapter 5 33 (midterm review, live) Mon, Apr 12 problems session on Zoom ° 34 (midterm review, live) Wed, Apr 14 problems session on Zoom °
35 (no lecture, exam) Fri, Apr 16 MIDTERM II (covering lectures 17-28) ° 36 Mon, Apr 19 (^) construction of Riemann-Stieltjes integralupper & lower Riemann integrals, Abbott 37 Wed, Apr 21 refinements, continuous functions are integrable Abbott 38 Fri, Apr 23 monotonic functions, bounded functions withfinitely many discontinuities Abbott 39 Mon, Apr 26 (^) (e.g., linearity, supremum bound), change of variablefirst properties of Riemann-Stieltjes integral Abbott 40 Wed, Apr 28 fundamental theorem of calculus,integration by parts Abbott 41 Fri, Apr 30 pointwise limit of functions, swappingorder of limits (counterexamples) Chapter 7 42 Mon, May 3 uniform convergence, examples Chapter 7 43 Wed, May 5 uniform convergence & continuity,uniform convergence & integration Chapter 7 44 (final review, live) Fri, May 7 problems session on Zoom °
° date TBA (^) with emphasis on lectures 32-43) FINAL^ (covering all lectures, °
Real Analysis: Homework 2
Due: 23:59 CST, February 7, 2021
- Given a set T , let P(T ) denote the set of all subsets of T. One calls P(T ) the power set of T. For example, P({ 1 , 2 }) = {;, { 1 }, { 2 }, { 1 , 2 }}. Consider the following subset of the power set of the natural numbers: I := {S 2 P(N) : S is infinite}. Exhibit an explicit bijection f : I! R, or prove a bijection doesn’t exist. Hint: first see if you can biject I with the interval (0, 1] ✓ R using binary strings. If so, can you biject (0, 1] with (0, 1), and then (0, 1) with R?
- I proved in lectures that the set R is uncountable. Prove nonetheless that the metric space R (with its usual metric) is second-countable. A metric space (M, d) is said to be second-countable if there exists a countable collection U = {U (^) i }^1 i=1 of open sets U (^) i ✓ M such that any open set in M may be written as a union of a subcollection of U. Hint: Q is countable.
- For x = (x 1 ,... , x (^) k ), y = (y 1 ,... , y (^) k ) 2 R k^ , define
d 1 (x, y) :=
X^ k
i=
|x (^) i y (^) i |.
(i) Prove that d 1 is a metric on R k^. (ii) People sometimes call d 1 the Manhattan metric. By means of a sketch in R 2 , explain why this is reasonable. Hint: If you don’t know, the streets in Manhattan are arranged in a grid. (iii) Write d 2 for the metric on R k^ induced by the Euclidean inner product, d 2 (x, y) := kx yk 2 =
qP k i=1 (x^ i^ ^ y^ i^ )^ (^2). Prove that for all x, y 2 R k^ ,
p^1 k d^1 (x,^ y)^ ^ d^2 (x,^ y)^ ^ d^1 (x,^ y). Hint: for one of these inequalities, use Cauchy-Schwarz. (iv) Deduce that (R k^ , d 1 ) and (R k^ , d 2 ) have the same open sets. We can express this by saying that d 1 and d 2 are topologically equivalent. (v) In R 2 , sketch the open ball centered at the origin of unit radius with respect to d 1 , and likewise with respect to d 2.
- Let M be a metric space with metric d. For x 2 M and A ✓ M , define
d(x, A) := inf a 2 A d(x, a).
(Recall from HW1 that ‘inf’ denotes infimum, or greatest lower bound.) (i) Show that if y is another point of M ,
d(y, A) d(y, x) + d(x, A).
(ii) Fill in the blanks: d(x, A) = 0 if and only if x is a point in A or a of A if and only if x is a point in the of A.
- Fix a prime number p. Given n 1 , n 2 2 Z, define d (^) p (n 1 , n 2 ) to be p r^ where p r^ (r 2 Z (^) 0 ) is the largest power of p that divides n 1 n 2 if n 1 6 = n 2 , and define d (^) p (n 1 , n 2 ) to be 0 if n 1 = n 2. For example, d 2 (5, 17) = 2 ^2 because 2^2 divides 5 17 = 12 but 2^3 does not. Prove that d (^) p satisfies the ultrametric inequality:
d (^) p (n 1 , n 3 ) max (d (^) p (n 1 , n 2 ), d (^) p (n 2 , n 3 )) 8 n 1 , n 2 , n 3 2 Z.
Deduce that d (^) p defines a metric on Z. This is called the p-adic metric.
for all (x 1 , y 1 ), (x 2 , y 2 ) 2 M ⇥ N.
(i) Prove that d (^) M ⇥N is a metric on M ⇥ N. We call this a product metric.
(ii) Let W be an open set in M ⇥ N. Prove that W =
S
↵ U^ ↵^ ⇥^ V^ ↵^ for some open sets^ U^ ↵^ in M and some open sets V (^) ↵ in N. We say M ⇥ N has the product topology.
(iii) Prove that if (M, d (^) M ) and (N, d (^) N ) are both compact, then so is (M ⇥ N, d (^) M ⇥N ). Hint: Consider an open cover of M ⇥ N. You need to show there exists a finite subcover. By part (ii), argue that you may assume the open cover is of the form {U (^) ↵ ⇥ V (^) ↵ } (^) ↵ where U (^) ↵ and V (^) ↵ are open sets in M and N , respectively. Now for each x 2 M , consider {V (^) ↵ : x 2 U (^) ↵ }. Argue that this is an open cover of N. Now invoke compactness of N. Now figure out how to use compactness of M. Alternative Hint: If you prefer not to work with the open cover definition of compactness here, you can instead do this question using the characterization that a set K is compact if and only if every infinite subset of K has a limit point in K.
(iv) Prove that if (M, d (^) M ) and (N, d (^) N ) are both connected, then so is (M ⇥ N, d (^) M ⇥N ). Hint: Suppose M ⇥ N = W 1 t W 2 (disjoint union) for open sets W 1 , W 2 ✓ M ⇥ N with W 1 6 = ;. You need to show W 2 = ;. Use part (ii) and reason somewhat similarly to part (iii), first hinted approach.
Real Analysis: Homework 4
Due: 23:59 CST, March 2, 2021
- Directly using the ✏ N definition of the limit of a sequence, determine
lim n!
p n 2 + 3n n.
- Find the limit inferior and superior of the sequence (s (^) n ) in R given by:
s 1 = 0; s (^2) m := s (^2) m 1 2
; s (^2) m+1 :=
where the above recursion holds for all m 2 N.
- Let (x (^) n ) be a sequence in a compact metric space M. Suppose (x (^) n ) does not converge. Prove that it has two convergent subsequences with di↵erent limits.
- Suppose (a (^) n ) (^1) n=1 is a Cauchy sequence in a metric space M such that there exists a subsequence (a (^) n (^) k ) (^1) k=1 converging to p 2 M. Prove that the whole sequence (a (^) n ) (^1) n=1 converges to p.
- Let M denote the set of all bounded real sequences x = (x (^) n ). For x, y 2 M , define d(x, y) := sup (^) n |x (^) n y (^) n |.
(i) Prove that (M, d) is a metric space. We call d the supremum metric.
(ii) Prove that (M, d) is complete.
(iii) Exhibit a bounded sequence in M with no convergent subsequence.
(g)
P
n!
n n^
(h)
P 1
n=
log(n) log(n)^
(i)
P
log( n+1 n )
(j)
P 1
1+z n^ where^ z^2 R^ is fixed (your answer may depend on^ z)
- For each of the following power series in z 2 R, determine (with justification) the radius of convergence R and whether the series converges absolutely, converges non-absolutely or diverges at the boundary points z = ±R (in cases when R is finite). Here each ⌃ is over n from 1 to 1. Hint: If you try to apply the lim sup expression for the radius of convergence, remember that the limit superior of a sequence simply equals the limit when the limit exists.
(a)
P
n 3 z n
(b)
P 2 n n! z^
n
(c)
P (^2) n n 2 z^
n
(d)
P (^) n 3 3 n^ z^
n
- Consider the series ⌃a (^) n given by
1 2 ^
1 2 +^
1 4 ^
1 4 +^
1 4 ^
1 4 +^
1 8 ^
1 8 +^
1 8 ^
1 8 +^
1 8 ^
1 8 +^
1 8 ^
1 8 +^
1 16 ^
1 16 +^... (a) Explain why ⌃a (^) n converges non-absolutely.
(b) Discuss (with justification) an explicit rearrangement ⌃a (^0) n of ⌃a (^) n such that
lim sup n!
s (^0) n = 100 and lim inf n! s (^0) n = 1 ,
where (s (^0) n ) is the sequence of partial sums of the rearrangement
P
a (^0) n.
- For a sequence (b (^) n ) (^1) n=1 ✓ R, the infinite product
Y^1
n=
b (^) n = b 1 b 2 b 3...
is defined to be the limit of partial products,
Y^1
n=
b (^) n := lim n! p (^) N where p (^) N :=
Y^ N
n=
b (^) n = b 1 b 2... b (^) N ,
when this limit exists (else the infinite product is divergent).
(a) Consider infinite products in which each factor is at least 1, i.e.,
Y^1
n=
(1 + a (^) n ) where a (^) n 2 R (^) 0.
Prove this infinite product converges if and only if the series ⌃ (^1) n=1 a (^) n converges. Hint: for one direction, you may assume without proof that log 3 (1 + x) x for all x 0.
(b) The following formula for ⇡ was discovered in the 1600s:
Y^1
n=
4 n 2 4 n 2 1
Using part (a), prove that the infinite product indeed converges (you need not verify the limit is ⇡). Using a computer or calculator, compute the partial products 2
Q N
n=
4 n 2 4 n 2 1 to three digits after the decimal for N = 1,... , 10.
- [Extra Credit] In this bonus exercise, we consider attaching a limit definition to a double series. Let (a (^) m,n : m, n 2 N) be a doubly indexed infinite array of real numbers. - For each m, we call
P 1
n=1 a^ m,n^ := lim^ N^!
P N
n=1 a^ m,n^ the^ m-th row series.
P 1
m=1 a^ m,n^ := limM^!
P M
m=1 a^ m,n^ the^ n-th column series.
P 1
m=
P 1
n=1 a^ m,n^ := lim^ M^!
P M
m=
lim (^) N!
P N
n=1 a^ m,n
the row-first iter- ated series.
P 1
n=
P 1
m=1 a^ m,n^ := lim^ N^!
P N
n=
lim (^) M!
P M
m=1 a^ m,n
the column-first iterated series.
P 1
m,n=1 a^ m,n^ := limP^!
P P
m,n=1 a^ m,n^ the^ double series. We say that an iterated series converges if and only if each inner limit converges and the series of such (the outer limit) also converges.
(a) Can you find (a (^) m,n ) for which each row series diverges to + 1 , each column series diverges to 1, yet the double series converges?
(b) Prove that if the iterated series
P 1
m=
P 1
n=1 |a^ m,n^ |^ converges, then X^1
m=
X^1
n=
a (^) m,n ,
X^1
n=
X^1
m=
a (^) m,n ,
X^1
m,n=
a (^) m,n
all converge to the same real number.
is a closed subset of R n^. One also calls this set the solution set to the polynomial system defined by f 1 ,... , f (^) k (or in other language, the real algebraic variety cut out by f 1 ,... , f (^) k ).
- Let f : (X, d (^) X )! (Y, d (^) Y ) be a function between metric spaces. Define the graph of f to be
(^) f := {(x, f (x)) : x 2 X} ✓ X ⇥ Y.
Equip X ⇥ Y with the product metric from HW3 Q5,
d (^) X⇥Y ((x 1 , y 1 ), (x 2 , y 2 )) :=
p d (^) X (x 1 , x 2 ) 2 + d (^) Y (y 1 , y 2 ) 2 for all (x 1 , y 1 ), (x 2 , y 2 ) 2 X ⇥ Y.
Prove that if f is continuous then its graph f is a closed subset of (X ⇥ Y, d (^) X⇥Y ).
Remark : The converse also holds, so that f is continuous if and only if its graph is closed.
- Record a video in which you discuss the following capstone result:
The continuous image of a compact set is compact. As a corollary, we have the extreme value theorem: any real-valued continuous function with compact domain is bounded and attains its bounds.
In your own words, you should explain the meaning of this result, outline the proof in reason- able but not necessarily total detail, and provide an example as well as a non-example. The video should include mathematical formulas as my lectures do, which you can write out by hand or in slides. Please keep the video under six minutes. See Canvas Announcements or Piazza for the instructions on submitting your video, which must be done separately to the submission of this homework.
- Recall from HW2 Q4 how we define the distance between the subset of a metric space and a point in the metric space. If (M, d) is a metric space, A ✓ M nonempty and x 2 M , then
d(x, A) := inf a 2 A
d(x, a).
(a) Show that if A is compact, then this infimum is attained, i.e., there exists a ⇤^2 A such that d(x, a ⇤^ ) = d(x, A).
(b) Give an example where A is not compact and the infimum is not attained.
- [Extra Credit] Consider the great circle C on the surface of Earth passing through your favorite point in Austin, Texas and the North Pole. Prove that at any instant in time there must exist two antipodal (or diametrically opposite) points p, p^0 2 C at which the surface temperature exactly matches. Please assume the surface of Earth is a perfect sphere so that C is a perfect circle, and that temperature is a continuous real-valued function of position.
Real Analysis: Homework 7
Due: 23:59 CDT, April 13, 2021
- In this exercise, we gain experience in applying the fact the continuous image of a connected set is connected and in applying its corollary, the intermediate value theorem.
(a) Consider a polynomial function f : R! R of odd degree. This means f (x) = c (^) d x d^ + c (^) d 1 x d ^1 +... + c 0 for x 2 R and some fixed coe cients c (^) d ,... , c 0 2 R with c (^) d 6 = 0 and d an odd positive integer. Prove that f must have a real root, i.e., there exists r 2 R such that f (r) = 0. Hint: What can you say about f (x) when x is a very large positive number and when x is a very large negative number? Please prove it. Now assuming HW6 Q2, apply the intermediate value theorem to conclude.
(b) Recall HW2 Q1. Back then, one part of our solution was to construct a bijection from (0, 1] to (0, 1). Prove that there does not exist a continuous bijection g : (0, 1]! (0, 1). Hint: Assume for a contradiction that there does exist a continuous bijection g : (0, 1]! (0, 1). What can you say about g ((0, 1))?
- (a) Directly using the " definition of the limit of a function, prove the Squeeze Theorem:
Let I ✓ R be an interval having a 2 R as a limit point. Let f, u, ` : I \ {a}! R. Assume
- `(x) f (x) u(x) for all x 2 I \ {a}, and
- lim (^) x!a `(x) = lim (^) x!a u(x) = L for some L 2 R. Then lim (^) x!a f (x) exists and equals L as well.
(b) Consider the function F : R! R defined by
F (x) =
x + x 2 sin( (^) x^1 ) x 6 = 0, 0 x = 0.
Show that F is di↵erentiable everywhere and determine F 0 (x) for each x 2 R. Hint: To show F is di↵erentiable at 0, use the Squeeze Theorem (part (a)).
(c) For F as in part (b), show that while F 0 (0) > 0 there does not exist > 0 such that F is monotonically increasing on ( , ). Hint: Since F 0 exists, show from the limit definition of a derivative that if F were monotonically increasing on ( , ) then we would have F 0 (x) 0 for all x 2 ( , ). Now consider the expression for F 0 (x) from part (b).
Real Analysis: Homework 8
Due: 23:59 CDT, April 30, 2021
The first two exercises are based on Taylor’s theorem (Lecture 24). The second is for extra credit. The last three exercises are about Riemann integrability.
- Let f (x) =
p x. Express f (1 + h) as a quadratic in h plus a remainder term involving h 3. By taking h = 0 .02, find an approximate value for
p 2 and prove it is accurate to seven digits. Hint: Notice
p 0 .98 = 0. 7
p
[This entire exercise is for extra credit.] Suppose f : [a, b]! R is twice di↵erentiable, f (a) < 0, f (b) > 0, there exists > 0 such that f 0 (x) for all x 2 [a, b] and there exists M 0 such that 0 f 00 (x) M for all x 2 [a, b]. So f is strictly increasing (by f 0 > 0) and convex (by f 00 0). Since f is continuous (being di↵erentiable), strictly increasing and f (a) < 0 and f (b) > 0, please note there exists a unique ⇠ 2 (a, b) such that f (⇠) = 0. In this bonus exercise, we are going to develop Newton’s method, which is an important numerical method for computing ⇠. One sometimes describes Newton’s method as a root-finding method.
(a) By trying various values of f , suppose we find x 1 2 (a, b) such that f (⇠) > 0. Then notice x 1 2 (⇠, b) since f is strictly increasing and f (⇠) = 0 by definition of ⇠. Let us now define a sequence of real numbers (x (^) n ) (^1) n=1 by the recursion
x (^) n+1 := x (^) n f (x (^) n ) f 0 (x (^) n )
(n 2 N).
Please interpret this formula geometrically in terms of a tangent to the graph of f. Remark : Generating such a sequence is what we mean by running Newton’s method. We call each iteration (passing from x (^) n to x (^) n+1 ) a Newton step. (b) Using the Mean Value Theorem, prove ⇠ x (^) n+1 < x (^) n (n 2 N). Also show
lim n! x (^) n = ⇠.
(c) Using Taylor’s theorem, show that there exists t (^) n 2 (⇠, x (^) n ) (n 2 N) such that
x (^) n+1 ⇠ =
f 00 (t (^) n ) 2 f 0 (x (^) n )
(x (^) n ⇠) 2.
(d) Set A := M/(2 ). Using part (c) and the assumed bound on f 00 , deduce
0 x (^) n+1 ⇠
A
[A(x 1 ⇠)] 2
n (n 2 N).
Remark : Notice the exponent is 2n^ , which grows rapidly with n. So, if the bracketed quantity A(x 1 ⇠) is strictly less than 1 (this depends on the initialization x 1 ), then the bound on the RHS above decays rapidly to 0 indeed. The number of correct digits of x (^) n (as compared to ⇠) roughly doubles every time we perform a constant number of Newton steps. In numerical analysis, one refers to this behavior as quadratic convergence. (e) Consider f : R! R given by f (x) = x 1 /^3. Using computer software or a calculator, perform Newton’s method (the procedure in part (a)). What happens? Please reconcile.
- (a) Let f : [0, 1]! R be the indicator function of Q \ [0, 1]. This means, for x 2 [0, 1],
f (x) =
1 if x is rational; 0 if x is irrational.
Prove that f is not Riemann integrable. Hint: What is U(f )? What is L(f )? (b) Recall HW6 Q1(c). Consider the function defined there restricted to [0, 1]. That is, let g : [0, 1]! R be given by
g(x) =
1 /q if x 2 Q and x = p/q in lowest terms with p 2 Z, q 2 Z (^) > 0 ; 0 if x / 2 Q.
Prove that g is Riemann integrable and
R 1
0 g^ = 0.^ Hint:^ Try to verify the^ "^ ^ P characterization of integrability.
- Let f : [a, b]! R be increasing throughout [a, b] (i.e., f (x) f (y) whenever a x y b). Prove that f is integrable on [a, b]. Hint: Verify the " P characterization of integrability by considering a partition of equispaced breakpoints.
- Recall the “polished” definition I presented in lecture of the Riemann integral: a bounded
function f : [a, b]! R is integrable with
R (^) b a f^ =^ A^ if its upper integral and lower integral both equal A, that is, U(f ) = A = L(f ). Actually, Riemann originally defined his integral di↵erently, in a way corresponding to the Riemann sums you may have seen in a calculus class:
Riemann’s Original Definition: A bounded function f : [a, b]! R is integrable with R (^) b a f^ =^ A^ if for all^ "^ >^0 there exists^ ^ >^0 such that for any partition^ P^ =^ {x^0 ,... , x^ n^ }^ of [a, b] with a = x 0 < x 1 <... < x (^) n 1 < x (^) n = b and any sample points c (^) k 2 [x (^) k 1 , x (^) k ] (each k) such that x (^) k x (^) k 1 < (each k), it holds that
X^ n
k=
(x (^) k x (^) k 1 )f (c (^) k ) A
<^ ".
Prove if f satisfies Riemann’s original definition, then f satisfies the definition from lectures. Remark : Actually, the converse holds too (proof omitted). So Riemann’s original definition is equivalent to the one from lectures. In your opinion, is the construction I gave easier?
