Reddit mentions of Counterexamples in Analysis (Dover Books on Mathematics)

Yes. This is a classic question and the typical answer is

f(x) = x^2 sin(1/x) if x != 0

f(x) = 0 if x = 0

The proof that f is continuous, and f' exists but is not continuous is left as an exercise for the reader. :-)

The book Counterexamples in Analysis has this and more. Having this book handy will do wonders for you and your class and I highly recommend it. Thank god Dover got hold of the copyright and re-printed it, it is a great book and the original is hard to find.

Counterexamples in Analysis is a wonderful menagerie of mathematical oddities—it's full of pathological examples. It's the most fun math book I know of.

http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753 click to look inside.

There is an excellent series of Counterexamples in ... books which might be relevant to this thread:

counterexamples in...

• calculus
  
• analysis
  
• topology
  
• probability & statistics
  
• several complex variables
  
  Or just run a search on amazon for "Counterexamples in".

Here are some suggestions :

https://www.coursera.org/course/maththink

https://www.coursera.org/course/intrologic

Also, this is a great book :

http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ref=sr_1_5?ie=UTF8&qid=1346855198&sr=8-5&keywords=history+of+mathematics

It covers everything from number theory to calculus in sort of brief sections, and not just the history. Its pretty accessible from what I've read of it so far.


EDIT : I read what you are taking and my recommendations are a bit lower level for you probably. The history of math book is still pretty good, as it gives you an idea what people were thinking when they discovered/invented certain things.

For you, I would suggest :

http://www.amazon.com/Principles-Mathematical-Analysis-Third-Edition/dp/007054235X/ref=sr_1_1?ie=UTF8&qid=1346860077&sr=8-1&keywords=rudin

http://www.amazon.com/Invitation-Linear-Operators-Matrices-Bounded/dp/0415267994/ref=sr_1_4?ie=UTF8&qid=1346860052&sr=8-4&keywords=from+matrix+to+bounded+linear+operators

http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753/ref=sr_1_5?ie=UTF8&qid=1346860077&sr=8-5&keywords=rudin

http://www.amazon.com/DIV-Grad-Curl-All-That/dp/0393969975

http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536/ref=sr_1_2?s=books&ie=UTF8&qid=1346860356&sr=1-2&keywords=chaos+and+dynamics

http://www.amazon.com/Numerical-Analysis-Richard-L-Burden/dp/0534392008/ref=sr_1_5?s=books&ie=UTF8&qid=1346860179&sr=1-5&keywords=numerical+analysis

This is from my background. I don't have a strong grasp of topology and haven't done much with abstract algebra (or algebraic _____) so I would probably recommend listening to someone else there. My background is mostly in graduate numerical analysis / functional analysis. The Furata book is expensive, but a worthy read to bridge the link between linear algebra and functional analysis. You may want to read a real analysis book first however.

One thing to note is that topology is used in some real analysis proofs. After going through a real analysis book you may also want to read some measure theory, but I don't have an excellent recommendation there as the books I've used were all hard to understand for me.

I wouldn't call it a "branch" exactly, but pathological functions are pretty much the definition of "weird." Things like Weierstrass functions, the Cantor function, the Conway base 13 function. There's a good book with a lot of this stuff in it called Counterexamples in Analysis. There's another one on topology I haven't read yet.

Counterexamples in topology

Counterexamples in analysis

Take your pick from this book: https://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753

Sorry, I went on vacation and totally blanked about posting these for you!

Anyway, here are some books

Linear Algebra Done Right (Undergraduate Texts in Mathematics) https://www.amazon.com/dp/3319110799/ref=cm_sw_r_cp_api_1L8Byb5M5W9D3

This one is actually for analysis but depending on your appetite, it might help greatly with the proof side of your class. You can buy it here: Counterexamples in Analysis (Dover Books on Mathematics) https://www.amazon.com/dp/0486428753/ref=cm_sw_r_cp_api_GS8BybQWYBFXX

But there's also a PDF hosted here: http://www.kryakin.org/am2/_Olmsted.pdf