Outer Measure
Classication Theorem
Classication of Riemann Integrable Functions
Wojciech Komornicki
October 28, 2013
Wojciech Komornicki
Classication of Riemann Integrable Functions
Outer Measure
Classication Theorem
The measure of a set
How do we determi
Homework 8
Wojciech Komornicki
October 17, 2013
p 115: 15. Suppose a R1 , f is a twicedierentiable real function on (a, ), and M0 , M1 , M2 are the
least upper bounds of f (x), f (x), f (x), respectively, on (a, )[. Prove that
2
M1 4M0 M2 .
Hint: I
Homework 11
p 140 13.
Wojciech Komornicki
November 14, 2013
Dene
x+1
f (x) =
sin(t2 ) dt.
x
(a) Prove f (t) < 1/x if x > 0.
Hint: Put t2 = u and integerate by parts to show that f (x) is equal to
cos(x2 ) cos (x + 1)2
2x
2(x + 1)
(x+1)2
x2
cos u
du
4u3/
Homework 9
Wojciech Komornicki
October 18, 2013
p 138 1. Suppose increases on [a, b], a x0 b, is continuous at x0 , f (x0 ) = 1, and f (x) = 0 if
x = x0 . Prove that f R() and that f d = 0.
Proof: Let P = cfw_y0 , y1 , y2 , . . . , yn be a partition of [
Homework 12
Wojciech Komornicki
November 14, 2013
p 165 2. If cfw_fn and cfw_gn converge uniformly on a set E, prove that cfw_fn + gn converges uniformly on E.
If in addition cfw_fn and cfw_gn are sequences of bounded functions, show that cfw_fn gn
Homework 13
p 196 4.
Wojciech Komornicki
November 24, 2013
Prove the following limit relations:
bx 1
= log b (b > 0)
x0
x
Solution Recall that, by denition
(a) lim
bx = E(xL(b)
Also we know that
E(h) 1
= 1.
h
lim
h0
Hence
bx 1
E(xL(b) 1
= lim
x0
x0
x
x
E(
21.
Suppose K and F are disjoint sets in a metric space X where K is compact and F is closed. Prove that there
exists > 0 such that d(p, q) > if p K and q F .
Proof: For x X, let
F (x) = inf d(x, z).
zF
Then
F x = 0 if and only if x F .
Since F is closed,
Homework 6
p 98: 1.
Wojciech Komornicki
4 October 2013
Suppose f is a real function edned on R1 which satises
lim [f (x + h) f (x h)] = 0
h0
for every x R1 . Does this imply that f is continuous? the converse true?
Solution
x if x = 0
1
if x = 0
No. Con
Homework 2
Solutions
10.
Wojciech Komornicki
13 September 2013
Suppose z = a + bi, w = u + iv and
a=
w + u
2
1/2
b=
,
w u
2
1/2
.
Prove that z 2 = w if v 0 and that (z)2 = w if v 0. Conclude that every complex number (with
one exception!) has two comp
Homework 3
p 23: 16.
Wojciech Komornicki
May 18, 2014
Suppose k 3, x, y Rk , x y = d > 0, and r > 0. Prove
(a) If 2r > d, there are innitely many z Rk such that
z x = z y = r.
(1)
Proof: Let x, y Rk and suppose that 2r > d. There exists an invertibl
Homework 4
p 78: 1.
Wojciech Komornicki
May 18, 2014
Prove that the convergence of cfw_sn implies the convergence of cfw_sn . Is the converse true?
Proof: Let s R such that lim sn = s.
n
Let > 0 and let N N such that if n N , sn s < .
Then if n N ,

Homework 1
1.
Wojciech Komornicki
13 September 2013
If r is a rational (r = 0) and x is irrational, prove that r + x and rx are irrational.
Proof: First suppose that s = r + x is rational. Then x = s r. Since Q is eld, s r Q.
Next suppose that s = rx is
Math 5910: Final Examination
17 December 2013
1. Complete the following sentences:
a) A set X together with a function d : X X R is called a metric space if d satises the
following:
a)
b)
c)
d)
For all x, y X, d(x, y) 0.
If x, y X, d(x, y) = 0 if and only
Math 5910: Exam 2
4 November 2013
1. Complete the following sentences:
a) Let X and Y be metric spaces. A function f : X Y is said to be continuous if for every
x X and every > 0 there exists > 0 such that d(f (x), f (y) < whenever d(x, y) < .
b) Let X an
Math 5910: Exam I
Solutions
27 September 2013
1. Complete the following sentences:
a) A eld is a set F with two operations called addition and multiplication which satisfy the following:
(A) Axioms for addition
(A1) If x F and y F , then their sum x + y i