You might enjoy "Counterexamples in analysis" [1], it shows, for example, a two argument function for which the order of integration matters, while the geometric interpretation would suggest otherwise. A must have!

[1] www.amazon.com/dp/0486428753

Yeah, that brings up another of those peculiar things about math, and especially, dealing with math culture. It's hard to talk about a main idea without getting mired in weird exceptions.

There are indeed precise conditions that guarantee that a function can (a) be integrated and (b) the integral can be differentiated, and there are subtly different definitions of "integral" that lead to different sets of extremely weird functions being integrable or not. I can't remember any of the details despite having come across them many times.

In software culture, it seems much easier to talk in broad strokes when appropriate and in precise details when appropriate. You might get an argument, but it's usually about something relevant.

Duh. It just hit me. Math is mostly edge cases. Consequently, the urge to be have perfect logical validity (same as the urge to write a bugless program) requires you to spend most of your attention on the edge cases.

That urge is stronger in mathematicians than in software engineers.