Continuity (calculus & limits) (Part 2 Rules for Building Complex Functions) #4.3.2.2c2 #4.3.2.2b9

This is the final video in the series on limits in one variable. In this video we prove some theorems that allow us to build complicated functions from simple continuous functions.

This video is part of a series on Chapter 2 of "Baby Rudin," "Basic Topology." We need to develop these concepts in order to prove three theorems on which all of calculus are based: the Intermediate Value Theorem, the Extreme Value Theorem, and the Uniform Continuity Theorem.

In the coming weeks we will be discussing, topology, metric spaces, connected sets and compact sets, and the proof of L'Hôpital's Rule. This series will be the most challenging thing we have done yet!

If you enjoy these videos, please consider supporting the channel on patreon at https://patreon.com/greg55666 . Even your $1 pledge could make all the difference!

For the next several weeks we will be developing multivariable calculus, so we can acquire Green's Theorem, which is required for complex integration. Complex analysis is central to the proof of Fermat's Last Theorem.

With the first videos on this channel, we managed to keep the math very simple, almost exclusively nothing more advanced than algebra. Now, to complete our brief tour of elliptic curves, we have to take an ENORMOUS leap forward in the level of the mathematics involved.

I want to make sure we are all on the same page before we begin learning the more advanced stuff in earnest. I am going to assume we are all familiar with high school math and college math through first or second year calculus. I'm going to introduce complex numbers and complex integration in some detail after examining multivariable calculus.

This series will culminate with the Weierstrass Equation, which is the thing that connects elliptic curves to complex numbers, and thus allows us to connect them to modular forms, which is what Wiles's proof is all about.

Here is the outline of this series on complex numbers:
0. Introduction
1. The Real Number System
2. Multivariable Calculus (we are here)
3. Complex Numbers
4. Complex Functions
5. Exponential and Trigonometric Functions
6. Complex Integration
7. Cauchy's Integral Theorem
8. Cauchy's Integral Formula
9. Laurent Series
10. Complex Residues
11. Lattices and Doubly Periodic Functions
12. Lattices and Tori and Groups
13. The Weierstrass p-Function
14. The Weierstrass Equation: Complex Functions and Elliptic Curves

Please leave any questions, comments, or suggestions in the comments below!

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Credits:
Music: "Home" and "St. Jarna" by Depeche Mode

Bibliography:
"Calculus, Early Transcendentals," by James Stewart