@TheGreatDuck I guess you could use the Koch snowflake as a motivating example for there being functions which are everywhere continuous and nowhere differentiable.
@TheGreatDuck Not all continuous functions--the Weierstrass function is a good example. It's everywhere continuous but nowhere differentiable, so you can't make a Taylor series at any point.
@TheGreatDuck Yep. Every analytic function must be smooth. If not, then there is some final derivative that makes sense, and the Taylor series can't be constructed past that degree.
@TheGreatDuck This gets weirder for complex functions: a function $f: \Bbb C \rightarrow \Bbb C$ is analytic if and only if it is differentiable (only once!) on an open set.
well, no, if you get far enough in algebra, there's something called primary decomposition in commutative rings generalizing prime factorization in $\Bbb Z$.
Artin is sort of unique in integrating linear algebra and tying into all sorts of different things in mathematics (e.g., representation theory, Hopf map, Riemann surfaces).
Fargle, I was supposed to send you algebra exercises, but most come from my book. I can, however, send exams as a start for you, if you want. Remind me later.
and it's kind of cool, because it's on a computer, and it adjusts the questions depending on how well you're doing, so if you get one right, it'll give you a harder one.
there's a reading one, a math one, and a science one, I think.
it's the only good standardized test i've ever taken.
Me too. I always found it rather hard to read in one go though, even if it's supposed to be a minimalist satirical tragicomedy. The setup/environment/repetitive nature of the story somehow gives a terrible sense of claustrophobia.
@Fargle Oh yeah I loved it
Have you read Imagination Dead Imagine by Beckett? It's a 2 page stream of consciousness monologue
