Modern Analysis, Homework 1
Due July 11, 2011
1. [20] (3 points) Rudin, Ch 7, #1: Prove that every uniformly convergent sequence of
bounded functions is uniformly bounded.
2. Rudin, Ch 7, #2: If cfw_fn and cfw_gn converge uniformly on a set E ,
(a) [20]
Modern Analysis, Homework 3
1. Recall that a function f on an interval [a, b] is of bounded variation if
n
|f (xi ) f (xi+1 )| < +.
sup
nN
x0 =a<x1 <xn =b
i=0
In such a case, we write f BV[a, b]. Given f BV[a, b], for x [a, b], dene
n
Tf (x) =
|f (xi ) f
Modern Analysis, Exam 1
Possible solutions
1. (a) Suppose that fn does not converge uniformly to f on K . Then there is > 0 such
that for all n N, there exists mn N and a point, call it xmn K , such that
|fmn (xmn ) f (xmn )| .
Clearly we may assume that
Intro to Modern Analysis
Summer MMXI Final Exam
Name
DIES LVNAE AD VII KAL AVG MMDCCLXIV
Do all problems, in any order.
Explain all answers. An unjuified response alone may not receive full credit.
No notes, texts, or calculators may be used on this exam.
Student Information
Intro to modern analysis II, Math S4062Q
July 5, 2011
Name:
Major(s):
Email Address:
School:
I would like to make a course webpage with the information above listed. The
idea is to help students work together in this class. Do you agre
Modern Analysis, Homework 5
1. We saw that the implicit function theorem follows from the inverse function theorem.
Show that the inverse function theorem follows from the implicit function theorem.
2. Let A, B Rn . Set
d(A, B ) = inf cfw_|x y | ; x A, y
Modern Analysis, Homework 4
1. Suppose that f : R2 R2 is dened by f (x, y ) = (x2 + xy, y ). Prove directly from the
denition that f is dierentiable at each point of R2 .
2. Rudin, Ch 9 # 14: Dene f (0, 0) = 0 and
f (x, y ) =
x3
if (x, y ) = (0, 0).
x2 +
Modern Analysis, Homework 4
Possible Solutions
1. For h, k R not both zero, we have
1
2x + y x
f (x + h, +k ) f (x, y )
2
0
1
+k
h2
h
k
|h| + |k |
|h + k |
|h|
= |h|
2 + k2
h
h2 + k 2
(1)
But |h| + |k | 2 h2 + k 2 for any pair h, k R (square both side
Modern Analysis, Homework 2
Due July 18, 2011
1. [20] (3 points) Rudin, Ch 7, #18: Let cfw_fm be a uniformly bounded sequence of
functions which are Riemann-integrable on [a, b], and for each x [a, b], put
x
fn (t) dt.
F n ( x) =
a
Prove that a there exi
Modern Analysis, Homework 2
Possible solutions
1. Since cfw_fm is uniformly bounded on [a, b], there exists C R such that for all m N
x
x
we have supx[a,b] |f (x)| M. This implies that |Fn (x)| = a fn (t) dt a |fn (t)| dt
C (b a), and the sequence cfw_F
Modern Analysis, Homework 1
Possible solutions
1. Consider = 1, and let f = limn fn . By uniform convergence, there exists N N
such that n > N implies fn f u 1. Since each fn is bounded, this shows that f
is bounded and that fn u f u + 1 if n > N . Set
M
Intro to Modern Analysis II, Math S4061X
Section 2
Summer 2011 Exam 1
Name:
July 25, 2011
Do all problems, in any order.
No notes, texts, or calculators may be used on this exam.
Problem
1
2
3
4
5
TOTAL
Possible Points
Points Earned
10
10
10
5
3
38
1. Let