Math Self-Study Progress Report 5

Chapter 5 is about differentiation.  It's a relatively short chapter but I struggled quite a bit with the last section 5.4, an optional section talking about continuous but nowhere differentiable functions.  It was definitely an eye-opener.

Because Gemini Pro is going to be unavailable to me, I have been using Gemini Flash (and it's now Hydrogen) with High Thinking mode.  As I mentioned last time, it's still very capable, and proved to be indispensable when I was struggling with various exercises.  It's also extremely good at catching a very subtle mistake I made.  It would be an understatement to say that I couldn't survive without AI.

Here's a summary:

• The author started the chapter by posing an interesting question: can derivative functions necessarily be continuous?  I never thought about that.  These are the kind of things that I encountered from time to time during my self-study where I knew I don't have the talent to be a successful mathematician.
• I studied the Mean Value Theorem in secondary school but the Generalized Mean Value Theorem was new to me.  It is required to prove L'Hospital's Rule.  No wonder L'Hospital's Rule was never proved in secondary school.
• I finally got to see the proof of L'Hospital's Rule.  I actually asked AI for a proof different from the book.  I also read the proof in another book, Elementary Analysis by Kenneth Ross.  Ross said that the proof in his book came from the famous "Baby Rudin" book.  To be honest, I'm still digesting it and it felt slick.
• That the $\infty/\infty$ case only requires the denominator to be $\infty$ was new to me.
• Also, if $\lim f'(x)/g'(x)$ conforms to the forms required by L'Hospital's Rule, and $\lim f'(x)/g'(x)=\infty$, then $\lim f(x)/g(x)=\infty$, was new to me.
• That L'Hospital's Rule doesn't even require $f(a)$, $g(a)$, $f'(a)$ and $g'(a)$ to be defined was completely new to me.
• The last section, A Continuous Nowhere-Differentiable Function, was a total surprise.  The author used the Takagi/Weierstrass sawtooth function as the example but Weierstrass himself actually accomplished it with an infinite sum of cosine functions.  It was highly non-intuitive that putting enough cosine functions together would make something nowhere-differentiable.  It also makes me rethink what I've learned in college regarding Fourier series.  I'm looking forward to Chapter 8 when Abbott will hopefully touch upon the subject of Fourier Analysis.

The next chapter, chapter 6, is the longest chapter in the book.  It's about sequences and series of functions.  At this pace, I hope I can finish chapter 7 by February next year at the latest.  At that point, I would have finished the core of the book as all the materials in chapter 8 are optional.  I'll likely continue with chapter 8 but go through it selectively based on my own interest.