Talk:Intermediate value theorem

message: The IVT for integration is certainly a result of calculus, but the original IVT involves neither differentiation nor integration but only continuity, so shouldn't it be referred to in the first sentence as a result of analysis? --131.111.249.207 17:31, 2 Jun 2005 (UTC)
thread_title:  Calculus? 

message: Uncommon though the formation may be, I think the theorem can say f(x) = c for x in [a, b] not just (a, b). [article] says so too. -- Taku 01:07, 13 October 2005 (UTC)
thread_title:  Closed interval 

message: The proof right below the intermediate value theorem that there exists some x such that f(x) = x is wrong.  This statement is in fact not generally true, even for f(x) continuous.  There are additional requirements, for example, if the domain of f is all the reals and the function is bounded (above and below).
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I reorganized this article and fixed the proof.
Grokmoo 13:29, 21 October 2005 (UTC)
thread_title:  Proof that f(x) = x for some x 

message: "Suppose first that f (c) > u ... whenever | x - c | < δ" I think it should be  c - δ < x < c (left neighbourhood)
and
c < x < c + δ (right neighbourhood) for f(c) < u
so we can omit absolut function. In first whenever c - δ < x < c  -->  f(x) - f(c) < 0   so |f(x) - f(c)| = -f(x) + f(c) —The preceding unsigned comment was added by 149.156.124.14 (talk) 15:48, 23 January 2007 (UTC)
thread_title:  left and right neighbourhood  

message: The explanation reads a little funny in this section.  It states that one can multiply (b-a) by "some function value f(c)" and you will get the area under the curve.  However the c refers back to the c found in the function using the mean value theorem not just any c value in between a and b chosen on a whim.  Although the section refers to the mean value theorem, it does not explicitly state the connection and I think it is hardly enough to explain the derivation fully enough for a newer calculus student to comprehend the equation in its totality.
Super-c-sharp (talk) 20:21, 9 September 2008 (UTC)
replies: text: This has now been done. Xantharius (talk) 22:50, 9 September 2008 (UTC)
thread_title:  Intermediate value theorem of integration 

message: Funny fact: in paragraph 35 of Euler's work Introduction to Analysis of the Infinite (i have John D, Blanton's translation in my hand):
If the polynomial functien Z takes the value A when z= a and takes the value B when z=b, then there is a value of z between a and b for which the function Z takes any value between A and B.
So there already was a notion of the intermediate value theorem before Bolzano: euler lived in 1707-1783  —Preceding unsigned comment added by 132.229.215.173 (talk) 14:28, 27 January 2009 (UTC)
thread_title:  History 

message: ...from those who have a much better grasp of the foundations of analysis that I.  I have just put some stuff about how Bolzano's analytic proof of the IVT introduced the idea of analytic proof that went on, through Bolzano's writings on logic, to become of fundamental importance in proof theory.  I have, no doubt, made several errors, and what I have not commented on is why Bolzano thought the idea of analytic proof to be so important.
I have the idea that Bolzano's concern was that analysis is more fundamental than geometry, and so should not depend upon it, but this is too much like guesswork for me to write up.  I'd be grateful if anyone can help me out with this! — Charles Stewart (talk) 14:15, 3 March 2009 (UTC)
thread_title:  [[Analytic proof]]: an appeal for help 

message: In addition to the alternative proof link non-standard calculus possibly add refference to the Constructivist analysis article.  There is a is a long section on IVT in that page.  The Constructivist analysis IVT section forward references to this page.  A link to that discussion might be usefull in this article.
However, I believe the that the the discussion of 'classical analysis IVT' on that page is not in close aggrement with this page.
--138.162.0.45 (talk) 17:36, 21 May 2009 (UTC)
thread_title:  Suggest adding another non-standard IVT reference 

message: Just wanted to know why some people apply the capital letter for a theorem name, while some don't. Is there any standard which applies for this case ?  —Preceding unsigned comment added by 129.240.64.201 (talk) 14:51, 17 November 2009 (UTC)
thread_title:  Capital letter for the theorem name 

message: Franklin.vp reverted a change I made on the page concerning Bolzano's proof of the IVT.  The current text states that Bolzano's proof was unrigorous, and provides as reference the famous article by Grabiner.  However, Grabiner's position is that the proof is rigorous.  We should either make a note of this on the page, or provide alternative sources for any claim that Bolzano's and Cauchy's proofs were unrigorous.  Tkuvho (talk) 15:36, 31 January 2010 (UTC)
replies: text: Your description of my motivation is spurious. Tkuvho (talk) 17:34, 31 January 2010 (UTC)
replies: text: * It is probably because I am not describing your motivation. How, can I if I don't even know you. What I am saying is that the question about qualifying the proof as rigorous is only important as to whether or not consider Bolzano the first giving a proof which is what really matters (especially because poor Bolzano worked lonely and forgotten and probably was not the SOB that was Cauchy). Cauchy is also mentioned for giving not only a proof but also a formalization of the whole framework in which the theorem is stated. definition of limits, function, ...  franklin  17:49, 31 January 2010 (UTC)
replies: text: My original edit was motivated by the fact that the page attributes to Grabiner a view she did not state, namely that Bolzano's proof was unrigorous.  This error has still not been corrected.  As far as the rigor of Cauchy's framework, Grattan-Guinness has shown that Bolzano's framework was, in fact, superior (albeit less influential).  Tkuvho (talk) 11:07, 1 February 2010 (UTC)
replies: text: * I think there is no place in the article right now saying that Grabiner stated that Bolzano's is unrigorous. Where is it that you find such a statement? Grattan-Guinness also said that Cauchy plagiarized Bolzano, a statement that is now disregarded. I don't have Grattan-Guinness' work can you bring/expose what are the reasons why Grattan-Guinness says that? I doubt very much that a comparison of the two proves according to rigorousness can be established. I was reading this work by Grabiner and I don't find it as having the position of Bolzano's being more rigorous. Actually is saying that Cauchy's proof was "crystal clear" and Bolzano's was involved. For example Bolzano proves first that if f(a)<g(a) and f(b)>g(b) then f(x)=f(x) for some x, after proving that bounded monotonic sequences have a limit to then derive IVT as a consequence. Cauchy approaches directly f(a)f(b)<0 implies f(x)=0 for some x using a nested interval argument. The one given in the references of the WP article doesn't have such perspective either (or I didn't see it). Again, comparison about rigorousness seems to be nonsense, the important is that both proves work, that Bolzano did it first, that Cauchy's is clear and simple and its done within a bigger formalization of the concepts of analysis.   franklin  13:08, 1 February 2010 (UTC)
replies: text: I am sorry, I misspoke.  I meant to say that Freudenthal said that Bolzano's framework was superior to Cauchy.  The consensus among scholars is that Bolzano's proof is no less rigorous than Cauchy's.  The current version of the article implies, contrary to what you have clearly outlined above, that Bolzano did not have a proof.  He certainly did, as everyone knows, including yourself, to judge by your comment.  Tkuvho (talk) 13:41, 1 February 2010 (UTC)

text: * Oh my. Crazy me. I noticed now, it says "stated" instead of "proved". Actually the theorem was stated much earlier and used. That section also needs being populated with more info. The story is richer than just a line.  franklin  14:05, 1 February 2010 (UTC)
replies: text: I see that there was a bit of a misunderstanding.  Thanks for your cooperation!  Tkuvho (talk) 14:19, 1 February 2010 (UTC)
thread_title:   Franklin.vp's edit 

message: Perhaps the section "implications of theorem in real world" should be removed, or rewritten to concentrate on the "wobbly table" example.  The point about there being, on any great circle and any physical quantity, a pair of antipodal points for which that quantity takes the same value on each point of the pair, is a cute mathematical consequence but while it is a mathematical abstraction of a fact about the real world, I don't see how it has much relevance; it seems like an essentially *mathematical* property of (an abstract description of) the real world, not really the kind of implication that will convince anybody of the theorem's relevance.  It's unlikely anyone motivated by anything practical would ever bother to actually *find* this pair of antipodal points, whereas the wobbly table application describes something that someone might actually do for reasons other than illustrating a mathematical fact.  I suppose the section title could be changed and the example kept roughly the same...certainly the point about antipodal pairs is an interesting mathematical consequence, so one wouldn't want it removed from the article entirely.  MorphismOfDoom (talk) 03:18, 12 September 2012 (UTC)
replies: text: I think the formulation of the section is nonsense. It is a purely mathematical theorem, it does not have any implications in the “real world”. It has implications for some physical models assuming the world to be euclidean space and temperature a continuous function. --22:52, 24 September 2012 (UTC)
replies: text: Actually, the proof in this section is incorrect. Here is my refutation:  The angle between the two points of equal value is not fixed at 180 degrees along the great circle, which would be a non-physical geometric constraint.  The angle is less than or equal to 180 degrees.  So this could be restated that along any great circle, there will be two points within 180 degrees of each other with the same value (of course within measurement errors).  And the maximum and minimum values along the great circle each are degenerate cases with zero degrees of separation.  So the use of antipodal is incorrect -- the two points are *at most* antipodal, which is kind of a gee wiz answer anyway.  Maybe you could expand that to the table -- you would have to rotate it *at most* +-180 degrees to find a stable orientation.96.88.65.241 (talk) 18:05, 11 August 2016 (UTC)
thread_title:  The main "Real world implication" is just more mathematics. 

message: It appears from the graph that y=u could lead to the false conclusion that the red line is the distance from a to b.
I see that the graph is trying to indicate that there are many points on f(x) which would satisfy the condition of y=u.
Just to make it clear, why not put a red dot "u" on the y axis?
116.55.65.151 (talk) 12:34, 7 January 2013 (UTC)
thread_title:  Graph might be made clearer 

message: Is there a particular reason for the fact that this article uses plain Wiki-markup for formulas (including inline single variables)?. That makes the article hard to read because the formulas don't stand out from the rest of the text, and the Wiki-markup is also harder to understand and modify than LaTeX code. Mario Castelán Castro (talk) 20:37, 7 February 2015 (UTC)
thread_title:  Change all formulas to <nowiki><math></math></nowiki> 

message: AVM2019 wrote, "The placement and phrasing of this remark may suggest that the classical proof is somehow "intuitive" and not rigorous, which is not the case." referring to the remark after the proof after the "relation to completeness".  
My own feeling is that the remark is not confusing, but I'm not a beginner, so maybe we should ask an undergrad.Irchans (talk) 19:39, 18 January 2023 (UTC)
thread_title:  "rigorous footing"  Clarification Request