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I'm not a mathematician, but there's a bit of this proof that seems wrong. It goes
"Suppose first that f (c) > u."
I don't see how we can suppose this since by definition
c = sup {x in [a, b] : f(x) ≤ u}.
which I understand as meaning that c is the largest number of those real numbers x such that f(x) is less or equal to u. If so, how could f(c) be greater than u??
My mistakem, which I realised on Friday evening when I looked up the definition of supremum. Wondering why I didn't get this from Wikipedia, I looked at the definition there supremum, and saw a non-mathematical definition precedes it, which could cause confusion. DB. ________________
There is indeed an error in the proof, which I copied below, as I shall explain.
The most flagrant error is found in the step "Pick ε = f(c) − u": "ε", in the ε-δ definition of continuity, is any positive number, however small; if we start out with the assumption that f(c)≠u, we may not also assume that f(c) − u can be made as small as we like.
Another one is really an exercise in futility: proving that f(c) > u is contradictory: since c=sup(S)∈S, it immediately follows from the definition of S that f(c)≤u.
Toolnut (
talk) 21:24, 15 October 2011 (UTC)
I have considerably revised it before reposting, taking Gandalf61's concerns into account. Does anyone have any rebuttal to my first observation of the original treatment, the "flagrant error"? Toolnut ( talk) 02:34, 17 October 2011 (UTC)
Why don't you write what you are trying to prove? Write the negation of the definition of a limit. That is, write this:
Convince me that this holds for the function . Note that this is an existential statement: you prove it by exhibiting one example of ε. Sławomir Biały ( talk) 02:09, 19 October 2011 (UTC)
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)
I agree, and I changed this
Grokmoo 13:29, 21 October 2005 (UTC)
Hmmm...if it only involves continuity, shouldn't it REALLY be called a result of TOPOLOGY???
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)
Note that the statement with (a, b) is actually a STRONGER statement. In any case, there's no point in including a and b as endpoints, since it's impossible for them to work (plug them in and try).
It can if you like, but this case it trivial, so I wouldn't worry about it. The standard statement is for (a, b).
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).
I reorganized this article and fixed the proof.
Grokmoo 13:29, 21 October 2005 (UTC)
Would it be useful to add an intuitively understandable example to this page? Something like: If person A is climbing a mountain from 6 to 7 am, and person B is coming down the mountain during the same time interval, then there has to be some time t in that time interval when they are both at exactly the same altitude?
The example of the "without lifting the pencil" is quite intuitive and very nice.
I agree. However, opening the page and being faced with a set of formulae isn't very nice. It would be better if someone put the following paragraph at the beginning, or in the introduction:
""This captures an intuitive property of continuous functions: given f continuous on [1, 2], if f (1) = 3 and f (2) = 5 then f must be equal to 4 somewhere between 1 and 2. It represents the idea that the graph of a continuous function can be drawn without lifting your pencil from the paper.""
Khedron 00:41, 23 December 2006 (UTC)
Too bad that this idea of "not lifting the pen from the paper" is quite wrong. For example, consider the Vitali Cantor Function. It is a continuous (even holder continuous) function but it is quite impossible to draw (or even, to think of). I think that this kind of "generally true" statements should be avoided when writing mathematics. Francesco di Plinio (francesco.diplinio@libero.it) —Preceding unsigned comment added by 151.57.122.59 ( talk) 16:53, 27 September 2007 (UTC)
"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).
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)
It reads a little better, but I still think that it is not explicit enough to give a thorough understanding as to how the two are related. Super-c-sharp ( talk) 19:00, 10 September 2008 (UTC)
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)
...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)
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)
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)
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)
The graph of the function has 3 places where f(x) = u. The choice of c in the graph is the the middle one. There is a region to the right of this where f(x)≤ u. In other words the location of c in the graph ought to be the right most of the three since c is the lowest number that is greater than or equal to every member of S. Right? - Brian Minchau —Preceding unsigned comment added by Brianjamesminchau ( talk • contribs) 04:52, 2 December 2010 (UTC)
The last I touched Mathematics was around 30 years ago, while earning my Electrical Engineering degree. So my claim that I can remember something fuzzy here is well placed. Currently I work in education sector and I am responsible for IT service delivery.
My suggestion to the community of the wiki is that we should make this more interesting for our current generation of students. This all is purely mathematical and while it will interest many it won't draw in the student who is needs that little bit extra to make things clear for him or the pure theory will not be of much interest to a person who cannot see the practical application of these great findings.
How can we make all this interactive and interesting? Let us start by providing some real life working examples, let us show how this theorem is applied in real world. —Preceding unsigned comment added by 60.241.168.218 ( talk) 21:05, 6 April 2011 (UTC)
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)
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)
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).
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)
This
level-5 vital article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
I'm not a mathematician, but there's a bit of this proof that seems wrong. It goes
"Suppose first that f (c) > u."
I don't see how we can suppose this since by definition
c = sup {x in [a, b] : f(x) ≤ u}.
which I understand as meaning that c is the largest number of those real numbers x such that f(x) is less or equal to u. If so, how could f(c) be greater than u??
My mistakem, which I realised on Friday evening when I looked up the definition of supremum. Wondering why I didn't get this from Wikipedia, I looked at the definition there supremum, and saw a non-mathematical definition precedes it, which could cause confusion. DB. ________________
There is indeed an error in the proof, which I copied below, as I shall explain.
The most flagrant error is found in the step "Pick ε = f(c) − u": "ε", in the ε-δ definition of continuity, is any positive number, however small; if we start out with the assumption that f(c)≠u, we may not also assume that f(c) − u can be made as small as we like.
Another one is really an exercise in futility: proving that f(c) > u is contradictory: since c=sup(S)∈S, it immediately follows from the definition of S that f(c)≤u.
Toolnut (
talk) 21:24, 15 October 2011 (UTC)
I have considerably revised it before reposting, taking Gandalf61's concerns into account. Does anyone have any rebuttal to my first observation of the original treatment, the "flagrant error"? Toolnut ( talk) 02:34, 17 October 2011 (UTC)
Why don't you write what you are trying to prove? Write the negation of the definition of a limit. That is, write this:
Convince me that this holds for the function . Note that this is an existential statement: you prove it by exhibiting one example of ε. Sławomir Biały ( talk) 02:09, 19 October 2011 (UTC)
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)
I agree, and I changed this
Grokmoo 13:29, 21 October 2005 (UTC)
Hmmm...if it only involves continuity, shouldn't it REALLY be called a result of TOPOLOGY???
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)
Note that the statement with (a, b) is actually a STRONGER statement. In any case, there's no point in including a and b as endpoints, since it's impossible for them to work (plug them in and try).
It can if you like, but this case it trivial, so I wouldn't worry about it. The standard statement is for (a, b).
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).
I reorganized this article and fixed the proof.
Grokmoo 13:29, 21 October 2005 (UTC)
Would it be useful to add an intuitively understandable example to this page? Something like: If person A is climbing a mountain from 6 to 7 am, and person B is coming down the mountain during the same time interval, then there has to be some time t in that time interval when they are both at exactly the same altitude?
The example of the "without lifting the pencil" is quite intuitive and very nice.
I agree. However, opening the page and being faced with a set of formulae isn't very nice. It would be better if someone put the following paragraph at the beginning, or in the introduction:
""This captures an intuitive property of continuous functions: given f continuous on [1, 2], if f (1) = 3 and f (2) = 5 then f must be equal to 4 somewhere between 1 and 2. It represents the idea that the graph of a continuous function can be drawn without lifting your pencil from the paper.""
Khedron 00:41, 23 December 2006 (UTC)
Too bad that this idea of "not lifting the pen from the paper" is quite wrong. For example, consider the Vitali Cantor Function. It is a continuous (even holder continuous) function but it is quite impossible to draw (or even, to think of). I think that this kind of "generally true" statements should be avoided when writing mathematics. Francesco di Plinio (francesco.diplinio@libero.it) —Preceding unsigned comment added by 151.57.122.59 ( talk) 16:53, 27 September 2007 (UTC)
"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).
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)
It reads a little better, but I still think that it is not explicit enough to give a thorough understanding as to how the two are related. Super-c-sharp ( talk) 19:00, 10 September 2008 (UTC)
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)
...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)
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)
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)
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)
The graph of the function has 3 places where f(x) = u. The choice of c in the graph is the the middle one. There is a region to the right of this where f(x)≤ u. In other words the location of c in the graph ought to be the right most of the three since c is the lowest number that is greater than or equal to every member of S. Right? - Brian Minchau —Preceding unsigned comment added by Brianjamesminchau ( talk • contribs) 04:52, 2 December 2010 (UTC)
The last I touched Mathematics was around 30 years ago, while earning my Electrical Engineering degree. So my claim that I can remember something fuzzy here is well placed. Currently I work in education sector and I am responsible for IT service delivery.
My suggestion to the community of the wiki is that we should make this more interesting for our current generation of students. This all is purely mathematical and while it will interest many it won't draw in the student who is needs that little bit extra to make things clear for him or the pure theory will not be of much interest to a person who cannot see the practical application of these great findings.
How can we make all this interactive and interesting? Let us start by providing some real life working examples, let us show how this theorem is applied in real world. —Preceding unsigned comment added by 60.241.168.218 ( talk) 21:05, 6 April 2011 (UTC)
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)
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)
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).
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)