Talk:Darboux's theorem (analysis)

Wrong from the very beginning

The statement of the theorem is wrong. Since the very beginning. In the original source, [1], it is correct. --Fioravante Patrone en (talk) 01:36, 11 February 2008 (UTC)

No, it was correct. The problem was the derivative symbol that was not clearly visible. I have added a thin space. --Fioravante Patrone en (talk) 10:12, 7 May 2008 (UTC)

Propose merge

The article Darboux function should be merged with this one since the content has significant overlap. McKay (talk) 08:53, 21 April 2009 (UTC)

In Proof: functions are not in general increasing/decreasing.

Hi, just a small point. Since derivatives are not necessarily continuous we can't say that functions in general are decreasing or increasing in a neighbourhood of a point regardless of the value of the derivative there. You can create examples easily enough by considering functions that oscillate 'uncontrollably' out of respect of my lecturer I won't give out his examples here but they do exist.

What can be said in the context of your proof of Darboux theorem is that in a right neighbourhood of a there exists a y such that f(a)<f(y)(since as the function was differentiable the limit f(a)- f(x)/x-a existed thus was positive in the right neighbourhood) thus the max is to the right of a and similar for b.

just to labour the point saying the function increases in a right-neighbourhood of a says that if x, y belong to that neighbourhood and x lies between a and y then f(x)<f(y). However all that is implied by the derivative taking a positive value at a is that f(a)<f(x) and f(a)<f(y).

Hope this is in the right section this time

regards 122.110.191.137 (talk) 16:28, 28 October 2009 (UTC) CWC

Fermat's Theorem

The proof of the theorem mention's Fermat's theorem but I don't think it is quite Fermat's theorem although it is related. Pratyush Sarkar (talk) 02:06, 23 July 2013 (UTC)

You are absolutely correct. The first mention of Fermat's theorem was not correct. I removed it. It's true that $\phi '(a)\neq 0$, so Fermat's theorem tells us a local maximum cannot occur at $a$. But we are trying to argue that the absolute maximum on $[a,b]$ cannot occur at $a$.

Here is the reason the absolute maximum of $\phi$ on $[a,b]$ cannot occur at $a$. If the absolute maximum did occur at $a$, then $\phi (a)\geq \phi (t)$ for all $a\leq t\leq b$. This implies $(\phi (a)-\phi (t))/(a-t)\leq 0$ for all $a\leq t\leq b$, and letting $t\rightarrow a$ gives $\phi '(a)\leq 0$. But this contradicts that $\phi '(a)=f'(a)-y>y-y=0$. LoonetteTheClown (talk) 15:10, 4 September 2016 (UTC)

Lars Olsen

The article says the proof based on Fermat's theorem and its corollary is due to Lars Olsen. But I have read exactly the same proof from much older text. — Preceding unsigned comment added by Mscdancer (talk • contribs) 13:21, 24 November 2015 (UTC)

The article currently says "Another proof based solely on the mean value theorem and the intermediate value theorem is due to Lars Olsen," and cites Olsen, Lars: A New Proof of Darboux's Theorem, Vol. 111, No. 8 (Oct., 2004) (pp. 713–715), The American Mathematical Monthly. The proof being referred to is not due to Lars Olsen. Despite the title "A New Proof of Darboux's Theorem," Olsen's proof is not new. The proof appears, for example, in Mathematical Analysis (2e) by Tom M. Apostol. See Theorem 5.16 of that book. LoonetteTheClown (talk) 13:33, 4 September 2016 (UTC)

Assessment comment

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Hi, just a small point. Since derivatives are not necessarily continuous we can't say that functions in general are decreasing or increasing in a neighbourhood of a point regardless of the value of the derivative there. You can create examples easily enough by considering functions that oscillate 'uncontrollably' out of respect of my lecturer I won't give out his examples here but they do exist. What can be said in the context of your proof of Darboux theorem is that in a right neighbourhood of a there exists a y such that f(a)<f(y)(since as the function was differentiable the limit f(a)- f(x)/x-a existed thus was positive in the right neighbourhood) thus the max is to the right of a and similar for b. just to labour the point saying the function increases in a right-neighbourhood of a says that if x, y belong to that neighbourhood and x lies between a and y then f(x)<f(y). However all that is implied by the derivative taking a positive value at a is that f(a)<f(x) and f(a)<f(y). anyway hope i used the discussion feature correctly hear goes 122.110.191.137 (talk) 16:16, 28 October 2009 (UTC) CWC == Wrong implementation of Ferma's theorem == First usage of Ferma's theorem is wrong because you can use it on (a, b) and not [a, b]. for example for function f(x) = x^2 on [-2, 2] f has maximums on -2 and 2 (f(2) = f(-2) = 4) and derivation of f on neither -2 nor 2 is 0. — Preceding unsigned comment added by 2.178.30.192 (talk) 17:00, 3 November 2012 (UTC)

Last edited at 17:01, 3 November 2012 (UTC). Substituted at 01:58, 5 May 2016 (UTC)

imbecilic punctuation

An amazingly extreme case of inattentiveness to what one is doing was this:

$x\mapsto {\begin{cases}\sin(1/x)&{\text{for }}x\neq 0\\0&{\text{for }}x=0\end{cases}}$.

Note the location of the period at the end of the sentence in the display above. I changed it to this:

$x\mapsto {\begin{cases}\sin(1/x)&{\text{for }}x\neq 0,\\0&{\text{for }}x=0.\end{cases}}$

— Preceding unsigned comment added by 2601:445:437F:FE66:900:C28F:955A:E198 (talk) 14 August 2017

Implications

Why does nobody write what the implications of this theorem are? In the Intro it says "But even when ƒ′ is not continuous, Darboux's theorem places a severe restriction on what it can be.". Exactly that is the question: What severe restrictions? This is one of the most important parts of the theorem and nobody seems to talk about it? — Preceding unsigned comment added by 213.55.184.241 (talk) 18:01, 21 January 2018 (UTC)

Incorrect Proof at beginning.

The statement of the theorem is that there exists a c in the open interval (a,b). Yet the very first line of the proof states that one can choose c to be either a or b. This is wrong. One must either relax the condition and let c be in the closed interval [a,b] or prove the existence of such a c in the open interval correctly. — Preceding unsigned comment added by 128.163.239.158 (talk) 14:32, 24 October 2018

Nope. The first sentence of the proof shows the property of the function, which is then used to apply the extreme value theorem for $c$ being between $a$ and $b$. It does not extend the theorem. --CiaPan (talk) 16:37, 24 October 2018 (UTC)

Definition of intermediate value property of a function.

To define the IVP, I think, is necessary to have a connected domain. In the cited work (Ciesielski) is for instance the real line. Otherwise we have also [a,b] or a generic interval I. — Preceding unsigned comment added by 82.136.67.75 (talk) 17:42, 29 May 2019 (UTC)

missing reference and link

After the sentence beginning "In particular, the derivative of the function" should come the sentence "This function is used in the construction of Volterra's function.", with a link. Kontribuanto (talk) 06:51, 3 March 2024 (UTC)