Talk:Non-analytic smooth function

message: Section 1.4 about the transition function seems to be missing some details / explanation as to its relevance. There is nothing to explain its relationship to the previous section and, even after re-reading several times, I personally cannot figure out what it has to do with the topic or why it is relevant in any way, to this page. For reference, I have a basic applied-maths background at the undergrad level and landed on this page looking for some clarification. Perhaps I'm not the target-audience, and it would be clear to others, but personally, I would appreciate clarification, if possible, as it certainly looks interesting. 69.168.163.21 (talk) 18:09, 28 June 2019 (UTC)
thread_title: What is section 1.4 About?

message: I just wonder if formalizing the proof is really a good thing, or rather a liability. I sort of liked the older approach, which was driving the point home while keeping the discussion informal. Is the new proof a bit less approachable for non-mathematicians? Oleg Alexandrov 17:50, 18 Apr 2005 (UTC)
replies: text: Sorry for the delay in reply. Maybe you're right (I was somehow afraid when I saw how it grew big...) But in fact I wanted to have the formula written down for further reference (I didn't remember that it was on x^3n, e.g.). Could we avoid complete annihilation by moving it to somewhere else (subpage /proof ?). — MFH: Talk 16:30, 21 Apr 2005 (UTC)
replies: text: Ugh - three problems with that. No need to divide the page; the use of subpages A/X has been deprecated for several years. What was the third? Doesn't really matter ... might have been that the copy-pasted page makes little sense. Charles Matthews 17:03, 21 Apr 2005 (UTC)

text: Actually we have a smooth function page as it is, and I think this page was only made to link to from list of mathematical examples. That's OK - the result is very important. But no need to chop everything so fine. Why not put proofs on smooth function, and leave this less formal - Oleg's point is good. Charles Matthews 17:06, 21 Apr 2005 (UTC)
thread_title: How it is ill-behaved : added formula for n-th derivative with proof

message: By Stone-Weierstrass, there must be some sequence of polynomials in x^n converging uniformly to the non-analytic smooth function on [-1, 1]. What does this sequence look like? (IT can't approach a Taylor polynomial).
I forgot to log in; username nnn9245129.2.56.113 18:04, 16 February 2007 (UTC)
replies: text: That sequence will look absolutely and completely different than Taylor's polynomial. That two functions are close to each other says nothing about whether their first derivatives are close to each other, same for higher order derivatives. And the Taylor polynomial is about all derivatives. Oleg Alexandrov (talk) 03:22, 17 February 2007 (UTC)
thread_title: correlation with Stone-Weierstrass?

message: I arrived here purely by chance and am not a mathematician. The opening sentence says that there are two important types of function. The second sentence states that "One can prove that...". Does this mean that it is possible to prove, or that one of the two function types can be used to prove? I assume it is the former, but the wording is unclear enough that I wish to raise the question.  —Preceding unsigned comment added by 87.113.135.124 (talk) 17:38, 30 June 2009 (UTC)
thread_title: Requesting a clarification

message: I have noticed that the proof considers affinely extended real numbers and not real numbers.
(I refer to the non-analytic function in the article (the one on top).)
exp(-1/x) for x > 0 can lead to strange things when considering just real numbers. (non necessarily correct)
:I have been interchanging \lim x \downarrow 0 and 1 / \infty. There is a subtle difference. This is the difference between arbitrary small and positive (i.e. negligible) and something that is more difficult to imagine. 
Why?
This is a set-theoretic issue.
Let +0 be 1 / infinity and let +0 be an element of the real set. (Which is not the case......)
When considering this page:
https://en.wikipedia.org/wiki/Real_number
It seems that there is a strict lexicographical ordering on the reals.
(By quickly looking at it.)
if not a < b and not b < a then a >= b and b >= a meaning that a == b.
I guess that a + (+0) > a.
Thus if you are very strict, a + (+0) <> a.
In calculus, the == operator is overridden for practical purposes.
In the couter example:
first -1 / x is computed.
Let's take x = +0
We get -1 / x == negative infinity, which is not a real number.
Now suppose that affinely extended real numbers are used.
Then exp(negative infinity) = 0, according to mathworld.
Using affinely extended real numbers seems to make things really complicated.
I think that exp(-1 /x) is already non analytic for x = +0.
My understanding of non-analytic smooth functions is that the constants of the power series do not shrink fast enough. As i (term index) gets larger.
As I was told, and what is illustrated on Wikipedia, is that the natrual logarithm seems to diverge in the domain x >= 2. However, this part of the function seems to be smooth.
Perhaps the natural logarithm is a better example.  — Preceding unsigned comment added by Wouterbu (talk • contribs) 03:45, 30 April 2013 (UTC)
replies: text: why do you need to consider x=0+ ? the definition only requires to compute exp(-1/x) for x > 0 ; if x > 0 then 1 / x is a real positive number and exp(-1/x) is a real positive number . No need to use affinely extended real numbers anywhere. Mennucc (talk) 09:20, 30 April 2013 (UTC)

text: You should read  http://en.wikipedia.org/wiki/Archimedean_property where it says "In the axiomatic theory of real numbers, the non-existence of nonzero infinitesimal real numbers is implied by the least upper bound property". Usually in calculus +0 does not exists. +0 exists in "non standard analysis" but that is quite esoteric, never taught in calculus 101 . Moreover IEEE standard is not a good reference to understand how the mathematical real line works. Mennucc (talk) 11:59, 30 April 2013 (UTC)
thread_title: Doubts about first proof (Which seem to be unjustified) 

message: In the detailed proof of smoothness drop section, the limits use sea-arrows and not normal arrows? Should it not be like this:
\lim_{x \to 0}
and not:
\lim_{x \searrow 0}
What is the purpose of this? --MrMineHeads (talk) 01:01, 6 January 2019 (UTC)
replies: text: It means limit from above, that is, it's an alternate notation for \lim_{x \to 0^+}.  It might be nice to actually mention this in the article, though!  --JBL (talk) 15:34, 6 January 2019 (UTC)

text: I have now added a link to one-sided limit to clarify.  --JBL (talk) 17:51, 6 January 2019 (UTC)
replies: text:  Alright thanks for the clarification. --MrMineHeads (talk) 23:52, 6 January 2019 (UTC)
thread_title: What is with the searrows?

message: Is this edit correct?  (And separately, that section is in really bad need of cleanup.  Is there a source for that construction?!) --JBL (talk) 23:44, 26 May 2020 (UTC)
thread_title:  Is the last edit right? 

message: In the section A smooth function which is nowhere real analytic, a caption to an illustration of an approximation to the smooth but nowhere analytic function reads:
"Approximation of the smooth-everywhere, but nowhere-analytic function mentioned here. This partial sum is taken from k=20 to 2500."
But it is not at all clear that this is the same k used in the definition of the function:
"Define for all x \in \mathbb{R}
:F(x):=\sum_{k\in \mathbb{N}} e^{-\sqrt{2^k}}\cos(2^k x)\ ."
Especially since 2(2500)  >  2(10150)  >  10(10149)  >  googolplex seems well beyond current computational ability.
If it is not, then I hope someone knowledgeable about this subject will fix this ambiguity. 
In any case I hope this will be clarified. (If it is not the same k in the caption as in the text, maybe the picture caption should describe its index by using the same k as in the text.)  — Preceding unsigned comment added by 2601:200:c082:2ea0:d8ea:383e:f1c6:7024 (talk) 01:31, 30 March 2024 (UTC)
thread_title:  Is this the same k or not?