# MATH 4650 - Analysis I

Class information:

Homework:

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• Hw 4 and solutions
- Note: In problems 1--6, every point in the domain of each function is a limit point.   So to check continuity we just have to check that the limit of f(x) at c equal f(c).   This is what I checked in the solutions.
- Typo: On problem
5 it says to set delta = epsilon / [ squareroot(a/2) + squareroot(a)].     This should be multiplication, not division.   It should be this: epsilon * [squareroot(a/2) * squareroot(a)]

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• Hw 6 and solutions (problem 7 solution is below)
- Problem 7 solution: part 1, part 2, part 3
(typo: it should say "for all n >= N" in the 2nd picture right after "where |a_n - L| < epsilon")
- Typo: Homework 6, problem 5(d) is false as it is stated.   The solution I wrote for 5(d) is another way to answer 5(c).  Instead, here is a revised problem to replace 5(d) with a solution written out: part 1, part 2, part 3.

• Hw 7 and solutions

Test Solutions:

Past Tests:

My Notes (these are a little messy, but will give you an idea of what we will cover in class):

• Part 1 - Real numbers, completeness axiom, and absolute value
• Part 2 - Limits of sequences
• Part 3 - Limits of functions
• Part 4 - Continuity (pages 41 and 42 will be a handout on the intermediate value theorem)
• Part 5 - Open and closed subsets of R (pages 47 and 48 will be handouts)
• Part 6 - Compactness (pages 56 and 57 will be handouts)
• Part 7 - Uniform continuity
• Part 8 - (if time we will sketch this for fun) Todd Kemp's notes on constructing R from Q using Cauchy sequence

Fall 2018 Student Notes (Thank you Cynthia!):