@futurebird Of the few I understood, 43 comes closest to an ambiguity that annoys me. What does sin²x mean?
I've seen it used for both. If it's the first answer to 43 then it must also be my first answer below, mustn't it?
My sister (independently) asked one of her mathematical physics lecturers about it and he said it was obvious from the context. Super!
Also, what does x/y/z mean? (x/y)/z or x/(y/z)? One of my computer science lecturers said it was formally undefined (by a vote at some mathematical congress or something) but I've not been able to find a reference.
You can do all the things to it that you're allowed to do with continuous functions as long as you are only looking at an interval that doesn't cross zero?
It's difficult I seem to remember to talk about continuity without starting "continuous in x if for ever epsilon there is a small intervall around x , etc etc".
The function is "smooth" everywhere where it's defined. One has to admit this.
@futurebird@petealexharris its tail towards infinity is interesting too as it is a classic example of a function that isn’t integrable from one to infinity. Even around zero, In a complex function context, it is infinitely differentiable with a well behaved pole at zero. Compare it to e^(1/z) or sin(1/z).
@futurebird There are situations when I find it useful to define continuous as "continuous where defined", but I would never expect continuous to mean that out of context.
@futurebird Well, it's clearly continuing to make people upset.
Plus, according to the very reliable lists of facts I received via e-mail in the 90s, 1/x is continuous when measured by Chuck Norris, or Ninjas, or specifically Ninja Turtles.
@futurebird I loved to fill that out, because in university I got too confused by some sources using different or even conflicting notations. Especially if a professor introduced it one way, and then in the homework there was a single task which used a different notation, that was never shown before.
@futurebird Maybe a third of these I haven’t encountered, and the lion’s share of the rest I have no strong opinion about, but there’s about 10% of them that I am livid about the “alternatives” even being included. It was an illuminating experience regardless, because I discovered some conventions I follow without even realizing I was.
@futurebird
oh, and the fifth octant is buried a hop, a skip, and a jump away from the fifth elephant, and it is a terrible cross between an octopus and an ant, and probably the source of the dungeon dimensions.
@futurebird
most of these, I can accept either way, so long as a given book or document spells out which convention they're following, and sticks to it consistently. But people who think f(x)=3 is "increasing" are not just wrong, they're spreading misinformation, and probably cackling with gleeful evil as they do so.
@futurebird Continuity is a point-wise property. "Is f(x)=1/x is continuous?" doesn't contain the information needed to answer the question, because "Where are you askin'?" what any proper Real Analyst would ask next.
There are more restrictive continuity properties, like Uniform Continuity, that are global, rather than point-wise, but that is a different discussion.
@futurebird Not a mathematician here, merely math-adjacent in school (chemistry with a relatively large extra dose of physics), but I recognize enough of these to grimace at more than a few of them, and I damn near lost it laughing at 100 (I'm on team "third answer" for that one).