Talk:Inverse function theorem

Dimensions

It should be stated clearly that the theorem only applies when domain and target have the same dimensions. — Preceding unsigned comment added by 134.58.253.57 (talk) 17:01, 25 August 2015 (UTC)

New example, references

I messed up this article more than a year ago because I confused it with something else. Apparently, nobody noticed. In case anyone was misled, I apologize. I realized my error recently and have made tremendous revisions based on two great references. Teply 02:50, 17 May 2007 (UTC)

Relationship to Implicit function theorem

How is this related to the implicit function theorem? If that relationship is substantial, I think it should be mentioned on both pages. Dfeuer 17:12, 7 October 2007 (UTC)

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For infinite dimensional Banach spaces, the frechet differential need not be invertible even if it is onto this theorem can still be proved by using quotient spaces.

I don't see how. If f is locally C^1 diffeomorfism than f'(x) must be invertible. Scineram (talk) 14:43, 5 May 2010 (UTC)

Merger proposal

In my opinion, the material in inverse functions and differentiation ought to be merged into this article. This article would be far easier to understand if it began with a thorough exposition of the single-variable case, and I don't see any reason for a separate article on the single-variable version of the theorem. Jim (talk) 06:30, 17 October 2009 (UTC)

Fine by me. Teply (talk) 05:46, 18 October 2009 (UTC)

Yes I agree initially, though I wonder if it wouldn't be more suitable to merge inverse functions and differentiation into the article Inverse function. This is because Inverse function already has more "starter material" and because it focus on explaining the concept. It could for instance be added to the section Inverse function#Inverses and derivatives which is a bit short.

In turn, we should make it more clear in this article that the Inverse function article is the place to go for explanations of the basics. EverGreg (talk) 08:35, 18 October 2009 (UTC)

It's perhaps logical to merge the two into one, but I don't think the general reader appreciates it. Somebody looking for information on the Inverse Function Theorem probably has picked up on the term in a pretty advanced context. Conversely, people who are looking for the mere basics of inverses and differentiation will only become confused with all the lingo required in the present article.

In short, there is a too wide gap in the required background for the two articles to be merged in my oppinion. (My reason for wanting to keep them separated is thus the same as Jim's reason to merge.) YohanN7 (talk) 22:04, 22 October 2009 (UTC)

There has been a trend of making Wikipedia too technical. It defeats the purpose of spreading knowledge. If having two articles on slightly related subjects improves the presentation why not have both of them ? The aim of Wikipedia is not to produce a perfect Bible but a usable textbook. So I vote for not deleting this article. nirax (talk) 16:03, 10 January 2010 (UTC)

Bad idea to merge two things of different levels, even if there is a nontrivial overlap between the names of the two things.

To differentiate an inverse function is a low level knowledge as far as you know, that it exists at all. But the existence itself is a definitely higher level knowledge.

Therefore the target population of the two things are not the same. The first case is an entry level western BSc Calculus student, while the second case is a western MSc or a student of second semester in the former eastern higher education.

prohlep (talk) 09:17, 31 October 2009 (UTC)

Is that right?

This article says

${\bigl (}f^{-1}{\bigr )}'(b)={\frac {1}{f'(a)}}$

while Inverse functions and differentiation says

$\left[f^{-1}\right]'(a)={\frac {1}{f'\left(f^{-1}(a)\right)}}$

Aren't these mutually contradictory or am I just having a math fail? — Preceding unsigned comment added by 141.209.173.212 (talk) 21:00, 27 March 2012 (UTC)

this is fine - the labels are just different: replace a->b in the second form to get

$\left[f^{-1}\right]'(b)={\frac {1}{f'\left(f^{-1}(b)\right)}}$

where

$f^{-1}(b)=a$

regards, Falktan (talk) 17:55, 16 July 2012 (UTC)

To answer your query, Falktan is correct, but it precisely highlights the confusion that we should try to avoid. I also prefer the representation used on Inverse functions and differentiation, which preserves the input variable $a$ on both sides of the relation. It would also enable a consistent definition across the two pages. To that end, in the current version of the Inverse function theorem page we should either include $f^{-1}(b)=a$ explicitly or modify the expression to the Inverse functions and differentiation representation. My first edit was reverted, I do not want to cause conflict, and I am unsure of how to proceed. Could editor Wcherowi please inform me/comment below?

Regards, MJASmith (talk) 16:50, 26 October 2019 (UTC)

As I mentioned in my edit summary, part of the problem with your previous edit was the introduction of non-conventional notation. Your intention, as indicated above, is fine and can easily be dealt with without introducing confusing notation. I've modified the presentation to incorporate your concern, but there are certainly other ways to do this. --Bill Cherowitzo (talk) 19:33, 26 October 2019 (UTC)

Way to distinguish this article from Inverse function rule

@D.Lazard

This article has a similar title to Inverse function rule which can be confusing in my opinion, so I was thinking of inserting something so people who end up on the wrong page won't be confused. How should I do this? SentientObject (talk) 18:34, 26 April 2025 (UTC)

The inverse function rule is a corollary of the inverse function theorem. When topics are so related WP:RELATED says Disambiguation hatnotes [...] are not intended to link to topics that are simply related to each other, or to a specific aspect of a general topic. We are exactly in this case, since the inverse function rule is a part of the inverse function theorem. This is the reason for which the link to inverse function rule must appear in the first paragraph of the lead. I do not imagine a better way to explain the difference between the two articles.

By the way, the link to formula is not needed and distracting, since a reader who searches for this topic must know what is a formula. The same is true for other links that I have removed D.Lazard (talk) 20:23, 26 April 2025 (UTC)

Formal inverse function theorem

This is more of a note to myself but the article should mention the formal inverse function theorem, the theorem for formal power series. Maybe already covered somewhere in Wikipedia? —- Taku (talk) 09:32, 14 September 2025 (UTC)