Welcome to Math 4320 Reall Analysis II. Take a look at the course syllabus, and if you're ready you can start on Homework #1.. Homework will (usually) be posted here on Monday or Tuesday, and due in class the following Monday (unless stated otherwise). We will usually go over the homework problems on the day they are due, but I will not collect or grade your homework solutions. Sometimes there will be a short quiz after we discuss the homework which will be graded.

Section 10.1 #2,7

Section 10.2 #1,2

Section 10.3 #1,2,7,11

Section 11.1 #4,5,6,9

Section 11.2 #1,6,7,9,10

Section 12.1 #3,4,5

Section 12.2 # 1,2,8,9,10

Also, according to the syllabus, Exam #1 is Wednesday, February 21. The exam will cover the material we covered in chapters 10,11,12, including the "bonus material" on the topological definition of compactness. Monday, 2/19 will be a review day.

1) What is the boundary of the set Q = rational numbers (subset of R^1)?

2) Prove that x is on the boundary of A if and only if for every epsilon, the open ball B_epsilon(x) intersects A and A^c (A compliment).

3) Use the topological definition of compactness (via open covers) to prove that any closed subset of a compact set is compact. (Don't use the closed and bounded charactarization of compact subsets in Euclidian space.)

Section 13.1 #1,3,4,7,11

Section 13.2 #6

Correction: we will go over the homework problems from chapter 13 on wednesday, 3/7.

Section 13.3 #2,3,4,6,9

Section 14.1 #5,7,8,11,15

Section 14.2 #1,3

I may add some (review) problems here in the next few days. Enjoy Spring break!

And here they are! This is an old exam with some interesting problems on it. We haven't covered all the material on the test, but you should be able to tell what is relevent and what isn't! We can go over these problems before the exam in a few weeks.

Section 14.2 #1,3

Section 14.3 #1,7,11

We will go over HW#9 and the practice test next monday. Bring in any questions you might have as well.

Section 15.2 #4,5,8

Suppose F: R^2 -> R^2 is F_1(x,y) = sqrt(x) + sqrt(y), and F_2(x,y) = cuberoot(x) - cuberoot(y). Use the Inverse Function Theorem to approximate F^{-1}(1.9, .05), using the fact that F(1,1) = (2,0). (You will need a calculator!)

Section 17.1 #12,13

Section 17.4 #14

Prove that the Jordan Content of the graph of y = sin(1/x), x>0, is zero.

As promised, here's a fun problem on Borel sets. Let f(x) be a real valued function (arbitrary). Show that the set {x : f is continuous at x} is a Borel set. Hint: Let A_n = {x : for some h, u,v in (x-h,x+h) implies |f(u)-f(v)| less than 1/n}. Show A_n is open. Then what?

Here's another one. Let f_n be a sequence of continuous functions (R to R). Show the set of convergence points is a Borel set.

I hope you enjoyed the course, and learned some analysis!