Timeline for If $f$ is infinitely differentiable then $f$ coincides with a polynomial
Current License: CC BY-SA 3.0
52 events
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| Dec 11, 2023 at 8:50 | comment | added | smyrlis | This result is due to two Catalonian Mathematicians, Balaguer and Corominas - F. Sunyer i Balaguer, & E. Corominas, Sur des conditions pour qu’une fonction infiniment dérivable soit un polynôme, Comptes Rendues Acad. Sci. Paris, 238 (1954), 558-559. | |
| May 13, 2023 at 17:14 | answer | added | aleph2 | timeline score: 0 | |
| Apr 19, 2022 at 16:21 | comment | added | Liviu Nicolaescu | What a coincidence! I've just presented this result to my honors students. The earliest I could track the result is a 1954 Comptes Rendues note by some spaniards; see Example 17.2.24 of these lecture notes www3.nd.edu/~lnicolae/Hon_Calc_Lectures.pdf | |
| Jan 26, 2019 at 21:13 | review | Suggested edits | |||
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| Jul 23, 2018 at 13:16 | comment | added | Jarek Kuben | This has been an interesting discussion. The conclusion seems to be that Polish mathematicians consider this problem much less difficult than the rest of the world. Of course this isn't surprising since functional analysis has a long and rich tradition in Poland. | |
| S Jul 23, 2018 at 11:06 | history | suggested | Ali Taghavi |
I add a tag.
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| Jul 23, 2018 at 10:36 | review | Suggested edits | |||
| S Jul 23, 2018 at 11:06 | |||||
| Mar 30, 2018 at 22:12 | comment | added | Piotr Hajlasz | @TheMathemagician I hope you did not get offended by my joke :) Thank you for your nice comments. I indeed put a lot of effort to write useful notes and make the available to everyone. Also I am a new user of Mathoverflow and I do my best to write clear and detailed answers like the one below. I want it to be a website with answers ready to use. | |
| Mar 30, 2018 at 22:05 | comment | added | The Mathemagician | @PiotrHajlasz You'll be waiting a long time, Piotr. The Gottwald HS in Warsaw is one of the premier schools in Europe. Maybe I should have said 10. My point was this is NOT a standard question for the average first year university student. You clearly were a superior student and consequently attended superior courses.(By the way, I'm a HUGE fan of your online handwritten lecture notes. You're clearly a terrific teacher. Your students are lucky to have you in Pittsburgh.) | |
| Mar 30, 2018 at 18:52 | comment | added | Piotr Hajlasz | @TheMathemagician Get ready! This was a problem in a problem book written by Michal Krych for the Gottwald High School in Warsaw. I did solve the problem in the 10th grade and many other students did solve it. You are asking for a regularly offered course at a university so high school should count. Three more answers and I will look for your YouTube chanel! If you do not appear naked I will vote down all your questions and answers :) | |
| Mar 30, 2018 at 14:34 | answer | added | Piotr Hajlasz | timeline score: 33 | |
| Aug 11, 2017 at 14:39 | comment | added | LSpice | @FrankScience, no, I misunderstood you, not the other way around. I misread it as the totally elementary statement "a differentiable function is continuous." Sorry! | |
| Aug 11, 2017 at 6:53 | comment | added | user20948 | @LSpice I don't know whether I misunderstood your response. What I said is that, given a differentiable function (on $\mathbb R$, say) $f$, then the derivative $f'$ (not $f$) should be continuous at some point $x_0\in\mathbb R$. I don't think that it's tautological. | |
| Aug 11, 2017 at 1:56 | comment | added | LSpice | @S.Carnahan, the question you reference gives the stronger hypothesis suggested by @ZenHarper. | |
| Aug 11, 2017 at 1:55 | comment | added | LSpice | @FrankScience, I suppose that one can derive any true statement from anything, but is there a sense other than this formal one in which the continuity of a differentiable function is a consequence of Baire's category theorem? | |
| Feb 19, 2016 at 9:28 | comment | added | user20948 | @TheMathemagician It's not too far away from a homework for first year calculus or analysis, which doesn't mean that it's easy, but only that it could be written in elementary language within a considerable length. In my university, the fact is the derivative of a differentiable function is continuous at a point, is left as a homework exercise, which is also, in fact, a consequence of Baire's category theorem. | |
| Feb 24, 2014 at 15:33 | answer | added | user45639 | timeline score: 5 | |
| Feb 2, 2013 at 10:46 | answer | added | Sungjin Kim | timeline score: 2 | |
| Jul 5, 2012 at 16:55 | comment | added | Margaret Friedland | My memory failed me on this problem. The only thing I am now sure about is that it was assigned as a freshman homework when I was a student at Jagiellonian U. in Krakow, Poland, a little more than 20 years ago (no, I did not solve it then). Rudin was used as a supplementary text, but I must have seen it printed somewhere else (or was it added in the Polish translation?). Proving various properties of function spaces via Baire category was a Polish specialty in 1930's, but, as indicated by @juan, this particular theorem apparently has different origin. Sorry for the confusion. | |
| Jul 3, 2012 at 9:42 | history | edited | C.S. | CC BY-SA 3.0 |
$ added for "f"
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| Mar 28, 2012 at 17:55 | comment | added | Todd Trimble | Margaret, I looked through Rudin's Principles of Mathematical Analysis (3rd edition) but didn't find this exercise. What page is it on? Also, where do you come from (that this exercise is assigned to freshmen)? | |
| Mar 28, 2012 at 8:55 | history | edited | C.S. |
I don't think baire-category is a tag.
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| Mar 27, 2012 at 17:24 | answer | added | juan | timeline score: 33 | |
| Nov 2, 2011 at 7:57 | history | edited | C.S. | CC BY-SA 3.0 |
deleted 7 characters in body
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| Nov 1, 2011 at 19:54 | history | edited | C.S. | CC BY-SA 3.0 |
added 11 characters in body
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| Nov 1, 2011 at 17:03 | comment | added | C.S. | @Margaret: Thanks, I would like to know the history of this problem as well. I will be thankful if you could provide me with the history. | |
| Nov 1, 2011 at 13:59 | comment | added | Margaret Friedland | This is stated as an exercise in Rudin's "Principles", and yes, it gets assigned as freshman's homework (at least where I come from). Originally it was formulated and solved in 1930's by Banach, if I remember correctly. I will try to dig up a more precise reference. | |
| Nov 1, 2011 at 11:20 | history | edited | Emil Jeřábek | CC BY-SA 3.0 |
fix capitalization
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| Jul 31, 2011 at 2:39 | answer | added | Richard Hevener | timeline score: 7 | |
| Jun 12, 2011 at 14:51 | comment | added | Todd Trimble | Olivier: you are referring to the notion of "perfect set", and there are no nonempty countable perfect sets. The argument is essentially a Baire category argument. See pirate.shu.edu/~wachsmut/ira/topo/proofs/pfctuncb.html | |
| Jun 12, 2011 at 3:13 | history | edited | C.S. | CC BY-SA 3.0 |
added 1 characters in body
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| May 8, 2011 at 4:59 | comment | added | Olivier Bégassat | nevermind, I think I have one. | |
| May 8, 2011 at 4:56 | comment | added | Olivier Bégassat | in cooking up my own solution to this classic problem which I've already encountered elsewhere, I came to wonder wether there are any closed, denumerable sets of reals that have no isolated points... Anyone know the answer to this? | |
| May 8, 2011 at 2:12 | answer | added | Gerald Edgar | timeline score: 39 | |
| May 8, 2011 at 1:58 | comment | added | S. Carnahan♦ | Also asked here: mathoverflow.net/questions/64246 | |
| May 8, 2011 at 0:39 | history | edited | Andrey Rekalo | CC BY-SA 3.0 |
Tag added
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| Sep 6, 2010 at 14:51 | vote | accept | C.S. | ||
| Aug 1, 2010 at 17:39 | comment | added | Michael Hardy | @Qiaochu, You're quite right; I was hasty. | |
| Aug 1, 2010 at 13:16 | comment | added | Anweshi | I agree with Andrew L.'s opinion(but not the more extreme part of it). If such hard questions are given as homework for a first year calculus course, then there will be complaints about the instructor, and indeed about the department. It is my modest contention that anyone who criticizes a question as homework should be able to substantiate it by giving a short solution in the comments. This doesn't take much effort. What I am preaching is just a variant of "All right, but let the one who has never sinned throw the first stone!". Before closing a question as homework, first solve it. | |
| Aug 1, 2010 at 1:55 | answer | added | Andrea Ferretti | timeline score: 16 | |
| Aug 1, 2010 at 1:06 | comment | added | Yemon Choi | Like Zen, I have seen a version of this question before set as an "exercise" - the tricky part, which I never solved on my own, is what to do once you've done the "obvious" Baire category part. Here I say "obvious" in the context of it being one of several BaireCat flavoured exercises in a batch, not "so obvious that everyone should have thought of it" | |
| Jul 31, 2010 at 23:52 | comment | added | Victor Protsak | In view of Ryan's and Zen's comments, can the author, please, indicate the origin of the question? | |
| Jul 31, 2010 at 23:32 | comment | added | Zen Harper | This is basically a double-starred exercise in the book "Linear Analysis" by Bela Bollobas (second edition), and presumably uses the Baire Category Theorem. Since it is double-starred, it is probably very hard!! Solutions are not given, and even single starred questions in that book can be close to research level. However, the version in that book has $f$ on the whole real line, and $f^{(m)}(x) = 0$ for ALL $m>n$. So are you sure your question is correct, since it's assuming a lot less but coming to roughly the same conclusion? | |
| Jul 31, 2010 at 23:20 | answer | added | Andrey Gogolev | timeline score: 185 | |
| Jul 31, 2010 at 22:39 | comment | added | Qiaochu Yuan | @Ryn: no, this is a classic little problem. @Michael: the problem is correct as stated. | |
| Jul 31, 2010 at 22:38 | comment | added | Michael Hardy | ......OK, maybe this is subtler than I thought........ | |
| Jul 31, 2010 at 22:35 | answer | added | fosco | timeline score: 7 | |
| Jul 31, 2010 at 22:34 | comment | added | Michael Hardy | Chandru, my suspicion is that instead of "for each $x \in [0,1]$, there is an integer $n \in \mathbb{N}$, it said there is an integer $n \in \mathbb{N}$ such that for each $x \in [0,1]$. | |
| Jul 31, 2010 at 22:10 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
copyediting, improved formatting
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| Jul 31, 2010 at 22:05 | comment | added | Andrey Gogolev | This is a jewel, I will try to recall the solution. | |
| Jul 31, 2010 at 21:51 | comment | added | Ryan Budney | This seems like a homework problem in a 1st year course on calculus. | |
| Jul 31, 2010 at 21:37 | history | asked | C.S. | CC BY-SA 2.5 |