Math 104 – Introduction to Analysis

Lecture notes on Weierstrass Function: Notes

Homework

Midterms and Exams

Challenge problems

For any interested students, here is a two-part challenge problem. The first part is to prove the Banach fixed-point theorem in metric spaces. The second part is to use the Banach fixed-point theorem to prove existence and uniqueness for a special class of ordinary differential equations.

• Part I: Banach's fixed-point thorem
• Part II: Existence/uniqueness for differential equations

The challenge problems will count as extra credit towards the homework component of the course grade, and are due by the last day of class.

Writing mathematical proofs

A mathematical proof is an argument convincing other people that a mathematical statment is true. In mathematical and scientific writings, it is standard that equations should occur naturally within full sentences and flow with the surrounding text. It is strongly recommended that students practice this style of writing in their homework. The homework solutions posted on this website will adhere to this style, and can be used as a guide. Some good resources include:

• Michael Hutchings' introduction to mathematical arguments: This is a very good resource on writing mathematical proofs.
• George Bergman's notes on sets, logic and mathematical language: This is a great resource for information on set and logic notation, and it also discusses the universal quantifier "for all" and the existential quantifier "there exists" that are used in this course.

LaTeX

A great way to write mathematics electronically is to use the software package LaTeX. LaTeX is widely used for the publication of scientific documents in many fields, including mathematics, physics, computer science and engineering. Some good resources for getting started with LaTeX are

• The not so short introduction to LaTeX by Tobias Oetiker.
• Wikibook on LaTeX.
• UC Berkeley LaTeX course from 2011.

The homework assignments, solutions, midterms, and finals for this course are all written in LaTeX, and I will post the LaTeX files for all homework assignments throughout the term.

Many students have asked what software packages and text editor I use with LaTeX. For quite a while now, I have been using the command line editor vim, along with vim-LaTeX, and a PDF-viewer that automatically updates when the pdf file changes (e.g., Skim on Mac, or Evince in Linux). I have used this setup on both Mac OSX and Linux (Ubuntu and Slackware) operating systems.