Midterm 2 Math 405 November 18, 2013
Show all work in a clear, concise and legible style.
Each problem is worth 25 points.
1. Let f (x) be a bounded monotone increasing continuous function on [a, b). Show that f
extends to a continuous on [a,b] in the fol
Strictly monotone functions and the inverse function theorem
We have seen that for a monotone function f : (a, b) R, the left and right hand limits
y0 = lim f (x)
x x
0
+
and y0 = lim f (x)
xx+
0
both exist for all x0 (a, b). This implies any discontinuit
Practice Midterm 1 Math 405 Fall 2013
1. (10 pts each) True or false; justify as much as you can.
a. Two uncountable sets have the same cardinality
b. The numbers of the form 3k , k, n N are dense in R+ .
n
c. For any a 0, (1 + a)n 1 + na for all n N.
d.
Midterm 1 Math 405 October 7, 2013
Show all work in a clear, concise and legible style.
1. (10 pts each) True or false; justify as much as you can.
a. The set of all sequences consisting of zeroes and ones is countable.
b. A sequence is convergent if and
HW1: due in section Friday September 13
p.13: 1,2,4,6
p.37: 2,3,4,5
9. The Babylonians seem to have known the following recursive algorithm for nding
S, S > 0. Start with any x0 > 0 and dene
1
S
xn+1 := (xn +
).
2
xn
a. Show that x2 +1 S and therefore xn+
Midterm 2 Math 405 November 18, 2013
Show all work in a clear, concise and legible style.
Each problem is worth 25 points.
1. Let f (x) be a bounded monotone increasing continuous function on [a, b). Show that f
extends to a continuous on [a,b] in the fol
Practice Midterm 2
1. (10 pts each) True or false; justify as much as you can.
a. If f (x), g (x) are continuous functions on [0,1] which agree at every rational, then
f = g on [0,1].
b. If |f (x)| is continuous at x0 then f (x) is continuous at x0 .
c. I
Department of Mathematics
Johns Hopkins University
110.405 Real Analysis I
Course Syllabus
The following list of topics is considered the core content for the course 110.405 Real
Analysis I, and is the first course in a two semester course series along wi
Midterm 1 Solutions
1. (10 pts each) True or false; justify as much as you can.
a. The set S of all sequences consisting of zeroes and ones is countable.
False by Cantors diagonalization argument. If the set (say S) was countable, i.e S =
cfw_b1 , b2 , .
Practice Midterm 2
1. (10 pts each) True or false; justify as much as you can.
a. If f (x), g (x) are continuous functions on [0,1] which agree at every rational, then
f = g on [0,1].
True. Given any > 0 and y (0, 1) choose > 0 so that |(f (x) g (x) (f (y
NUMBERS
Michael E. Taylor
Contents
1.
2.
3.
4.
5.
6.
7.
8.
Peano arithmetic
The integers
Prime factorization and the fundamental theorem of arithmetic
The rational numbers
Sequences
The real numbers
Irrational numbers
Cardinal numbers
1
2
Introduction
The
A Short Math Guide for L TEX
Michael Downes American Mathematical Society
Version 1.09 (2002-03-22), currently available at http:/www.ams.org/tex/short-math-guide.html
A 1. Introduction This is a concise summary of recommended features in L TEX and a coup