Mathematical Analysis (Math 1010)

with Richard Kent.

Office: KH 313

Office hours: Wednesday and Thursdays 11am to noon.

TA: Zhongyang Li

Recitation: TBA



Solutions to Final.


Announcement. No class on Tuesday, February 3rd.


Text: Mathematical Analysis by Tom Apostol. Handout.


Final exam: 9:00 AM on May 6, 2009 in ???.


Notice: We will observe most of Reading Period,
using some days during that time to make up for
classes I cancel due to travel.



Homework (negative homeworks not to turn in):


HW 1 (due Feb 5): 1.6, 1.11, 1.15, 1.19, 1.21, and

A) Prove carefully that a composition of injective functions is injective.

B) Prove that the set of rational numbers is countable using the following outline:
        i) Show that it suffices to show that Q⁺,
            the set of nonnegative rationals, is countable.
        ii) Then show that i) follows from the fact that N × N,
            the product of the set of natural numbers with itself, is countable.
        iii) Show that N × N is countable.



HW 2 (due Feb 19): 2.5, 2.6, 2.11, 2.12, 2.14, 2.15, 2.21,
3.2, 3.5, 3.6, 3.7, 3.15, and 3.16.



Solution to 2.11.



HW −1 (never due):

a. If I and J are two intervals in the line, show that their intersection contains
an interval provided the intersection is nonempty.

b. Given an open interval I in the real line R,
find an explicit bijection between I and R.

c. Prove that (0,1) × (0,1) has the same cardinality as (0,1).

d. Prove that Rⁿ and R have the same cardinality.



HW 3 (due Mar 5): 3.17, 3.19, 3.21, 3.22, 3.23, 3.27, 3.29, 3.31, 3.33, 3.39, and

1) Prove that there is an uncountable closed subset of the line
that does not contain an interval.



HW −2 (never due): 3.20, 3.24, 3.25, 3.34, 3.37, 3.40, 3.41, 3.42, and

a) Show that if a < b are irrational numbers, then Q ∩ [a,b] is
not a compact subset of Q (with the subspace metric).



HW 4 (due April 7):

0) Enjoy your break.

1) Let X be a metric space and let f and g be continuous functions
from X to R, and let c be a constant. Show that the functions
cf(x), f(x) + g(x), and f(x)g(x) are continuous.

2) Prove that polynomials in one variable are continuous.
Prove that all polynomials are continuous.

3) Prove that an alternating infinite series converges if the absolute
values of its terms go to zero monotonically.

4) Prove the rearrangement theorem: Let c be a real number.
If an infinite series Σ_{n ε N} a_n is convergent but not absolutely convergent,
then there is a bijection f : N → N such that c = Σ_{n ε N} a_f(n)

5) Prove that if the complement in R of a closed set C has at least 3 components,
then C is disconnected.

6) Define a function f: [-4,∞) → R² as follows.
When x is in [-4,-1], f(x) = (0, -(x+3)).
When x is in [-1,0], f(x) = (x+1, -2).
When x is in [0,2π], f(x) = (1,x/π -2).
Finally, when x is in [2π, ∞), f(x) = (2π/x, sin(x)).
It helps to draw a picture.
(Note that f is injective.)
Prove that f is continuous.
The image of function f, with its subspace metric, is called the pseudo-circle.

7) Let X = [-4,∞) and let f be the function from 6). Consider f as map from X to f(X).
Show that f: X → f(X) is not open. (A function is open if f(U) is open whenever U is.)



HW −3 (never due):

a) The pseudo-circle is a closed subset of the plane.
Show that its complement is disconnected.



HW 5 (due April 21): 10.1, 10.2, 10.3, 10.34, and?
1) Let f be continuous on a compact interval I. Show that f is Lebesque integrable.

2) A function F is essentially bounded if there is a constant M such that
the set of x such that F(x) > M has measure zero.
Give an example of an integrable function F that is not essentially bounded.

3) Prove or disprove the following statement: Let F be a function on a compact
interval I whose restriction to any interval J ⊂ I is not essentially bounded.
Then F is not integrable.



HW –4 (never due): 10.4,