Math 425: Homework 1
Mary Radclie
due 13 Jan 2014
In Rudin: Page 114, problems 1, 2, 5, 8, 9, 11, 14
1. Let f be dened for all real x, with |f (x) f (y)| (x y)2 for all real x
and y. Prove that f is constant.
Proof. Notice that
(x y)2
f (x) f (y)
= |x y|,
Math 425: Homework 2
Mary Radclie
due 22 Jan 2014
In Rudin: Page 114, problems 22, 27, 28
Also, complete the following.
1. Let
f (x) =
e1/x
0
2
x=0
x=0
Prove that f has derivatives of all orders at x = 0, but that the power
(n)
series n=0 f n!(0) xn is eq
Name:
Math 425/575: Midterm
7 February 2014
Turn o and put away your cell phone.
No notes or books are permitted during this exam.
No calculators or any other devices are permitted during this exam.
Read each question carefully, answer each question compl
Math 425: Homework 4 Solutions
Mary Radclie
due 5 Feb 2014
In Rudin: Page 239
5. Prove that to every A L(Rn , R1 ) corresponds a unique y Rn such that
Ax = x y. Prove also that A = y 2 .
Proof. Let yi R be such that Aei = yi . Then for any x =
Rn , we hav
Math 425: Homework 7
Mary Radclie
due 5 Mar 2014
In Rudin: Page 138: 4, 5, 8, 11, 12
Also, complete the following:
1. (a) Let f : [0, 1] R have the property that for all > 0, cfw_x | |f (x)| >
is nite. Show, for all continuous, monotonically increasing f
Math 425: Homework 6
Mary Radclie
due 26 Feb 2014
In Rudin: Page 239
20. Take n = m = 1 in the implicit function theorem, and interpret the
theorem (as well as its proof) graphically.
Solution. Here we have a continuously dierentiable function f : E
R2 R
Math 425: Homework 2
Mary Radclie
due 22 Jan 2014
In Rudin: Page 114, problems 22, 27, 28
22. Suppose f is a real function on (, ). Call x a xed point of f if
f (x) = x.
(a) If f is dierentiable and f (t) = 1 for all t, prove that f has at most
one xed po
Math 425: Homework 7
Mary Radclie
due 5 Mar 2014
In Rudin:
4. if f (x) = 0 for all irrational x, f (x) = 1 for all rational x, prove that
f R on [a, b] for any [a, b].
/
Proof. Note that for any partition P = cfw_x0 , x1 , . . . , xn , we have that
n
Mi =
Math 425: Homework 5
Mary Radclie
due 19 Feb 2014
In Rudin: Page 239, problems 16, 17, 21, 23
Also, complete the following:
1. (a) Let f = (f1 , f2 , . . . , fn ) be a vector valued function dened on Rn , and
suppose f C 1 (Rn ). Let g1 , g2 , . . . , gn
Math 425: Homework 3
Mary Radclie
due 29 Jan 2014
In Rudin: Page 114
18. Suppose f is a real function on [a, b], n is a positive integer, and f (n1)
exists for every t [a, b]. Let , , P be as in Taylors theorem. Dene
Q(t) = f (t)f () for t [a, b], t = . D
Math 425: Homework 6
Mary Radclie
due 26 Feb 2014
In Rudin: Page 239, problems 20, 24, 25, 26, 29
Also, complete the following:
1. Suppose that f : R3 R, and a = (a1 , a2 , a3 ) has f (a) = 0. Suppose moreover that f C 1 (B) for some neighborhood B of a,
Math 425: Homework 1
Mary Radclie
due 13 Jan 2014
In Rudin: Page 114, problems 1, 2, 5, 8, 9, 11, 14
Also, complete the following.
1. Let f be a dierentiable function on (a, b), where f is bounded. Prove
that limxa f (x) exists.
2. Here we devise an analo
Math 425: Homework 4
Mary Radclie
due 5 Feb 2014
In Rudin: Page 239, problems 5, 6, 7, 10, 13
Also, complete the following:
1. Let A be an m n matrix. Prove that A
i,j
a2
ij
1/2
.
2. (a) Let fi : (a, b) R be a real-valued dierentiable function for 1 i
k
Math 425: Homework 5 Solutions
Mary Radclie
due 19 Feb 2014
In Rudin:
16. Show that the continuity of f at the point a is needed in the inverse
function theorem, even in the case n = 1: if f (t) = t + 2t2 sin 1 for
t
t = 0, and f (0) = 0, then f (0) = 1,
Math 425: Homework 2
Mary Radclie
due 29 Jan 2014
In Rudin: Page 114, problems 18, 25; Page 196, problems 4, 5, 6b
Also, complete the following.
1. Let f and g be analytic at 0, with radii of convergence Rf and Rg , respectively.
(a) Show that the product