MATH 4650 - Analysis I

Class information:

Homework:

  • Hw 1 and solutions
     
  • Hw 2 and solutions
     
  • Hw 3 and solutions
     
  • 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)]
     
  • Hw 5 and solutions
     
  • 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!):