He lives no longer.

i keep forgetting

@TedShifrin I forgot that continuity is defined for points, not intervals? Is this right?

A function is continuous on an interval (or any set) if it is continuous at each point of that interval (or set).

And the set of real numbers is also an interval? I think it is.

Sure.

Wait. Then $\tan x$ is discontinuous in the set of real numbers.

We only talk about points of the domain, remember? At least, that's how any sensible math person will talk about it.

We have agreed that calculus books lead to nonsense.

Another question. Since $\tan x$ is "discontinuous" in the set of real numbers, we restricted it to the set of values of $x$ such that $x$ is not a multiple of $\pi/2$ for it to be continuous. Is my way of thinking wrong, but got the correct result? Or is it still wrong?

You keep ignoring what I say.

So I'm not playing this game any more.

I get what you say. I just want to cross from my part to your part with certainty. I mean, understanding why this is that and things like that

when people describe a functional as proper does this mean it is not identically $\infty$?

If you're taking a beginning calculus course, you have to do what your teacher/book says. Otherwise, what I say is the way mathematicians talk about it. Your universe is the domain of the function.

@soupless, what is 'discontinuous'? How do you define it?

@DanielAdams I've never heard of that. Proper has other meanings in mathematics.

@soupless A function is discontinuous if at least one of the conditions here is not satisfied

I cant seem to find where the authors of this book define a proper functional

but they just use that term throughout

Koro: Read above for the whole conversation.

@Daniel Ordinarily a functional will map to $\Bbb R$. They must define it early on.

@TedShifrin what would you guess they mean

Professor Ted: can you please give me a hint on how to count homomorphisms from $S_n$ to $\mathbb Z_m$? I thought about something but I think the idea is vague. By $S_n$, I mean group of permutations on n symbols.

yes I know what they mean by functional, but not a proper functional.

Proper is used in mathematics for a mapping between topological spaces if the preimage of every compact set is compact. Intuitively, this means that the function "goes to infinity" as you "go to infinity" in the domain.

mm yes I just saw that online, guess thats what they mean.

cheers

@Koro Do you know how to generate $S_n$ with just a few things?

yes, $S_3=\langle (12), (123)\rangle$

(if you weren't here I would have written < instead of $\langle$.)

:)

OK, so does that help you answer the question for $n=3$?

Sorry, it's 2am here. Need to sleep now. Bye!

Night.

But for the case when m is divisible by 3, $f(123)$ could be equal to an order 3 element in $Z_m$, this is what's bothering me.

But $(123)=(13)(12)$.

I agree. But I meant to say that: $(123)$ has order 3 and since f is a homomorphism $f(123)$ should have an order that divides order of (123).

If you agree, what must $f$ do to it?

Oh, so you want to do $f((12))f((13))=f((123))$.

Right.

Due to abelian nature, order of $f(12)f(13)$ is lcm (2,2)=2.

You already argued that $f((ab))=0$.

So order of f(123) is $2$ which must divide 3 which is a contradiction.

Well, OK, different contradiction.

So only trivial homomorphism exists. Right?

Right. When $m$ is odd.

Well, you still have multiples of $3$ left, I suppose.

But you shouldn't.

yeah, but I realize now that I need not consider those. Only parity of m.

OK. So, what if $m$ is even?

I actually don't know what you know about $S_n$.

I know its definition and cyclic decompositions etc.

Do you know about $A_n$ and its properties?

$A_n$ is group of even permutations of $S_n$ and that it is normal in $S_n$ and has index 2.

I do not know how you are supposed to do your exercise. There are various important facts that I know that you do not.

The crucial thing to think about is the fact that $\Bbb Z_m$ is abelian and $S_n$ is very much not abelian.

yes, I am familiar with that.

So if m is even, then we saw above that order of f(123) is $1$ so $f(123)=0$. Now, $f(12)$ has order either 1 or 2. So two possible homomorphisms.

@Koro I should have said order of f(123) is either 1 or 2.

this came up in another problem you were doing. be sure you nail down why both of those possibilties are possible, i.e., are actually implemented by homomorphisms.

just knowing the order has to be something doesn't get you there.

it might be helpful to identify the commutator subgroup of S_n. i'm not sure if this counts as a spoiler or just a rumination.

Well, since the issue is $\Bbb Z_m$ and not just a general abelian group, maybe one can bypass that, but that's what I had thought of.

Can't we say just as we do in linear algebra, that mapping on generators (bases in LA) determines homomorphism (linear map in LA)?

It doesn't though

@leslietownes sure, I'll do that. Thanks :).

Now, I understand my earlier discussion with Jakobian and @love_sodam on this question.

If T is a surjective linear map between Banach spaces, then from Michael selection theorem there is continuous f such that T o f = Id

Does there exist a continuous linear map like this?

Thanks a lot @TedShifrin.

koro for context if you are defining a homomorphism from Z_2 + Z_2 to S_3. (1,0) and (0,1) both have to go to elements of order dividing 2. but there is no homomorphism f that sends (1,0) to (12) and (0,1) to (23), although this is a well defined map on the set of generators that sends each generator to elements of the right order. the images of the generators have to commute. those candidate elements do not commute.

when the target is a cyclic group things simplify considerably but you are using properties specific to cyclic groups in this case.

and it is worthwhile pinning down how, and where.

hey any suggestions for measure theory books? i've been told the best is axler's book

monoidal i like bartle's book. 'integration' is in the title.

thanks

i would dis-recommend rudin's principles of mathematical analysis measure theory chapter, or royden.

folland is OK.

why would you disrecommend royden? i've been told it's really good. I still didn't get my copy yet though

first off it's a poorly printed book. it fell apart in my hands although it was new. the glue didn't hold it together.

secondly the index sucks. about half of the topics aren't in the index, and some of the page numbers are wrong for the topics that are in the index.

also, it leaves some stuff i consider important in exercises.

it's not that bad as a general real analysis book but for measure theory specifically there are better options.

it's better than rudin though. just steer clear of rudin.

I went through Royden when I was studying for my real analysis qual ... I am not a measure theory fan, regardless.

the index of that book is a disaster. it must have pre-dated automated indexing.

maybe Ted should write a measure theory book

maybe they fixed it in a subsequent printing. if they haven't, i created my own index for royden. if you do use it, i can send it to you.

Well, even with automated indexing, it depends on the human. I didn't put enough index entries in any of my books, I think.

@monoidal Never. I'm very much not interested and very much not qualified.

just key theorems and definitions, entirely absent from the index. not niche stuff. stuff that sections were named after.

jost's riemannian geometry book, first edition, is another book like that. complete car crash in the index.

I rarely put theorems in the index.

what about terry tao's book on measure theory

is it any good?

well, entertain for a moment that an index can be bad, and then imagine it being worse, and that's royden.

i've never noticed the quality of indexing, outside of that book, and jost.

tao's book is too new for me.

that's not an aesthetic judgment, i mean, it didn't exist the last time i read books involving measure theory, so i don't have an opinion.

@leslietownes: did that proof make sense? Not constructive in any way.

@TedShifrin do you put rare theorems in the index?

@robjohn I know not rare theorems.

Medium-rare, maybe.

you only put well-done theorems in your books.

I abhor anything well done.

robjohn: the heine borel one? i liked it.

sous vide theorems. air fried theorems.

@leslie: yeah. Its only intuitive at all when you've been dealing with that stuff a lot.

@leslietownes deep fried lemmas

rob: i think it's like a law of nature, in any analysis class, there will be a moment where things come to a screeching halt for something. whether it is open cover definition of compactness, or something else. it's this key difficulty that can be pushed around and put in different clothes but does not go away.

koerner's companion to analysis treats this pretty well. i think he would call that a 'lion-hunting' proof.

I like using the sequential criterion for compactness in $\Bbb R^n$ for a first course (e.g., my multivariable math course). The abstraction in topological spaces can come later.

draw a circle around where the lions aren't, invert

Körner's Fourier book is masterful.

that's one that i'm most upset about the USPS losing. it's not cheap to replace.

i may have found an infringing copy somewhere.

it's an offsite backup

I can leave you mine in my will (in which I have no arrangements made for math books, cookbooks, or anything else practical).

yes. it's my vhs recording of a licensed broadcast of it. which is legal according to the supreme court.

olivia has been running around the house as if chased by unseen forces for about 2 hours now. no sign of running out of energy.

she's hopping up on windowsills, meowing. running up and down stairs. meowing. abruptly changing direction as if being confronted by ghosts.

Maybe the Santa Annas?

Screech goes through spurts like that every day for several hours.

the wind does seem to feed into it.

usually livvy has maybe 30 min in the morning and 30 min at night or early morning.

I took two of our cats to the vet on Monday, because the daughter had a rash on her tummy, and the mom was there to comfort her (and to augment the bill another $600 for blood tests). The daughter got some antibiotics and ended up having so much energy afterwards that she was running around the house all night. She just never did that before,

my office looks out on the first floor. it's a drop of maybe 20 ft. when livvy is really on a tear, she jumps onto the ledge and looks down. it's unnerving.

robjohn: note to self, olivia does not need antibiotics.

Gotta love people who shoot their mouths off without knowledge. (First comment.)

sigh. i remember hearing this from fellow TAs in a calculus class where df was expressly defined in the book. they assumed the book was just writing BS and didn't read it. and i guess didn't generally know about d as its own thing.

@TedShifrin you, too, can learn from Wikipedia ;-)

These idiots don't even click through to check profiles before they condescend.

From his profile, I'm guessing he's an undergraduate student who knows more algebra.

@leslietownes I am wondering if Tiger (the daughter) has had a problem for a while and the antibiotics cleared it up. She has not been running around the last couple of days, so maybe it was just an "oh joy, I feel so good" run.

I'm glad she's feeling better, @robjohn. I can't even manage to trim Screech's claws.

@TedShifrin it has been a long time since we have trimmed our cats' claws. They trimmed Tiger's and Panther's while we were there, which was a nice surprise.

Well, the vet had such trouble with Screech when she got spayed that now they expect me to sedate her any time I bring her in. I am contemplating trying a new vet, although I paid for all the shots to be done (eventually) at the first vet's.

i can trim livvy's claws when we are 1 hour into a 2-hour movie and she has been on my lap the whole time. this happens at most once a week. zero times a week when i am too tired to watch a movie on fridays. 30 minutes of watching something is not enough lap time to sedate her.

The commenter is also ignoring Riemann Stieltjes integrals where $\int f'(x)\,\mathrm{d}x=\int\,\mathrm{d}f$

Well, the d notation in Riemann-Stieltjes integration one could argue is "just notation."

i don't like geometry, but df is definitely a thing. d is definitely a thing. i don't like it, but there it is.

Differential forms are not geometry.

okay, i don't like differential forms either.

Read about Kähler differentials :P

Pure algebra.

You don't — by your own admission — like much other than symbol-pushing, and differential forms are very good for that.

that is how i survived two semesters of algebraic topology.

i used a spectral sequence once. i forget why. i learned nothing.

I have a spectral sequence in my thesis :D

In an appendix.

"by your own admission" is very lawyerly. i like to think i am influencing the room

Algebraic topology can (see Spanier) be very algebraic and formal, but in modern days (see Hatcher) it's become more geometric.

I write well, but I'm far from lawyerly. :)

making an argument from what someone else said is very lawyerly. anything the other side said in an adversarial dispute is gold. i did this successfully last week. a judge agreed with me. it made me feel big.

But that won't stop know-it-alls from knowing it all.

it really won't. although in this case it did.

Criminality and overthrow of the government by the president are just commonplace now.

the bad guys were arguing in one forum that two cases were completely different, and in another that they were exactly the same. they had, let's say, nuanced reasons for why this was consistent. which involved them knowing all.

Hmm, let's put their statements side by side and let the judge see.

that's what i did. in front of the judge, i said "the other side believes these cases to be different. it acknowledged this when it said in those other cases:" and then a bullet pointed list of things they'd said in the other proceeding.

that was enough.

cries but I want my cake and to eat it too.

normally when i do that someone more senior than me asks me to finish the argument by putting one or more lines of commentary below that. but this time everyone agreed the list was enough.

Well, you needed a summary "I rest my case."

well the next line was basically us asking the judge to do something. but there was no connector between the list of statements and the desired outcome.

That know-it-all is continuing to say that his incorrect statement is fine because it's written to an OP who is just a calculus student.

some calculus books rigorously define df. stewart basically defines df as a linear functional on the tangent space. i may not like it but he does.

he doesn't say "functional" or "tangent space", but he gives what is the actual definition.

Yeah, I remember reading in Thomas that $dx$ can be any real number (which is using $\Bbb R^* = \Bbb R$) and that $dy = f'(x)\,dx$, blah blah blah.

I doubt Thomas ever learned differential forms, however. His multivariable calculus was a mess.

He was nominally a classical number theorist.

I don't use that language at all in a regular calculus course. I talk about linear approximations and $\Delta y_{\text{tan}}$.

I might mention in passing that a math major might eventually learn how to make it all right.

which is basically what stewart does. he says, let's write the "variable" of this function as delta x, or dx, i forget which.

I wish people would explain, though, that separation of variables is just the chain rule, not just symbol-pushing.

I've seen lots of students (both in teaching and on MSE) who were truly baffled by separation of variables and the formalism.

i agree with that. symbol pushing is not helpful for anything related to DEs. other than maybe motivating the result.

Well, sure, the symbol-pushing is efficient and, I might agree, a powerful technique, but once they should see it's nothing mysterious.

Physics classes use it everywhere.

Welcome back!

someone asked a question about semigroups of operators a while back and the answer was the chain rule. the OP didn't seem to understand that the chain rule actually proved it, it wasn't just some notational thing.

@robjohn To whom?

amWhy just dropped in

Oh! Howdy, @amWhy.

@leslie I wonder if in some way our two lines of discussion might really coalesce there.

amWhy will be able to chat in 12 days. she is apparently in chat jail for crimes against the state.

Are you her registered attorney, @leslie?

I didn't realize that in "chat jail" one could eavesdrop but not chat. Weird.

no, just a disinterested observer who can do nothing. you know, like the UN.

I just found out that the little number appearing on my little spinning top meant that I had flags to vote on. ... How do you know about chat jail if you're a disinterested observer?

it was displayed on the profile.

Ah. How investigative of you.

One can eavesdrop without even logging in.

I was wondering why they were so quiet.

Lots of my friends getting today's Wordle in 2 moves, but I can't do better than 3.

i got it in 3 although if i'd altered the order in which i play my usual openers, maybe 2.

@user4539917 Martin Sleziak will often appear when called upon ;-)

@user4539917 I only know how to do that if I've been pinged and can click from the ping notification; but that often is to an old page.

@robjohn True dat.

I haven't yet adopted a standard opener. Just different words with two vowels.

I've been trying different pairs. So I guess I have at least $\binom52$ vowel combinations, not to mention locations and choices of consonants.

I keep getting to the point where there are various choices with 4 letters nailed down, and only today did I make the right guess at that juncture (actually with only 3 nailed down this time).

@robjohn Professor Sleziak is a masterful eavesdropper

@user4539917 indeed

:-)

Being silent has never been my strong suit.

@TedShifrin does your strong suit have a big S on the chest?

Yes, for spades.

You've got the gift of the gab; and I mean that in the most respectful way, sir.

LOL ... there's no respectful interpretation of that one.

:-)

OK, lunchtime for this bonzo.

cya

@TedShifrin that sounds like a good idea.

I might make a k'sa'd'ya

(fewer letters)

So it turns out that my intuition was right, and $\sum_{i=1}^k a_iz_i^n$ doesn't converge as $n\to\infty$ for $z_i\in S^1\setminus\{1\}$ such that $z_i\neq z_j$ when $i\neq j$.

And $a_i\neq 0$

@Jakobian I wouldn't expect it would with all the circling about the circle that the terms do.

they circle about the circle like northern shovelers in a duck pond.

@leslietownes as opposed to uninterested?

yes, disinterested. thank you for noticing the difference.

i'm interested in anything.

coolio

@robjohn the function I asked you about yesterday does exist

the professor solved the question in class

hope you can read portuguese

:-)

(-:

@Derivative I thought the first $n$ derivatives ($0$ through $n-1$) vanished at $0$. Did I misread that?

they do, don't they? I might have made a mistake

Oh, his solution has that. I need to think more about my Schwarz proof.

oh right, the indexing is wrong

I can see there is a familiar blaschke factor even though I don't understand the written language.

Yeah, but I don’t believe this. I think it contradicts the Schwarz lemma when $n=1$.

Right, we don’t.