Talk:Partial function

message: If a partial function associates with every element in its codomain precisely one element of its domain, then it is called a total function, or simply a function.
This doesn't sound right to me.  I think the words codomain and domain should be swapped.  The function described above sounds like a 1:1 function, not an arbitrary total function.
We need a better definition of "total function" Pizza Puzzle
----------------------------
Partial Function
from-set contains members not in the domain, i.e. domain is subset of the from-set. 
Total Function
Domain is the entire from-set
-Mikael-
-----------------------------
alrighty smartyhairsplitterses. What is to domain of a partial function what range is to codomain? (whaddya call {x|f(x) exists}?) Kwantus 2005 June 28 17:23 (UTC)
:In my usage, {x|f(x) exists} is the domain of the partial function f.  What I don't know a name for is the set X, when f is a partial function from X to Y.  This needs to be clearly explained--someone following the link here from Uniformization (set theory) would be confused.  --Trovatore 19:12, 6 September 2005 (UTC)
thread_title: 2005

message: I am confused by the grammar in this explanation. It would be nice if somebody could explain the concept in a clearer fashion, perhaps something that would not confuse a non-mathematician. If nobody does so by this weekend, I'll get around to it if I have time. Suggestions?
--Kooky 02:19, 20 July 2005 (UTC)
replies: text: Kooky: Your prayers have been answered (I hope).  I'll get right on it. Vonkje 16:21, 1 August 2005 (UTC)

text: Okay, I reworked the intro, but I'm keeping the to-be-cleaned-up tag 'cuz the rest of this article needs work, namely: 1) The section on Discussion and examples was kept from the original article, although it makes those annoying types of plays on mathwords that had turned me off to Mathematics as a lower-division undergraduate.  2) A modern formal definition like the kind from the MathWorld site will be needed in a new section titled: Formal definition.  This would be a nice couple of things for you to do if you are interested.  Vonkje 20:47, 1 August 2005 (UTC)
thread_title:  Cleanup 

message: I noticed that total function redirects to partial function.  While the article in question does explain what a total function is, I think it is quite confusing to have total function simply redirect here.  I'm not sure that creating a page called "total function" would be advisable, perhaps we could rename this to something like Partial vs Total functions, although I think that wording is too cumbersome.  Anyone have any suggestions? Grokmoo 17:59, 24 January 2006 (UTC)
replies: text: I agree it's a little peculiar, but there's plenty of precedent for this sort of thing. I don't think it's worth a whole article. Changing the redirect so it points to function (mathematics) would be defensible; I don't have a strong preference one way or the other for which article it should point to. --Trovatore 20:48, 24 January 2006 (UTC)

text: I agree too. It's just impossible to come up with what a total function should be, with this article. It gave me nothing and was useless, except for knowing that a total function seems to be the “opposite” of a partial function. But on what dimension, and what the center point of inversion for this opposition is, stays absolutely in the dark.So: What the hell is a total function now???— 94.220.250.151 (talk) 08:53, 3 November 2009 (UTC)
replies: text: It's not the opposite of a partial function.  All total functions are partial functions, but not vice versa.  A partial function f:A\to B is a total function just in case the domain of f\, is all of A\,; that is, for every x\in A, f(x) is defined. --Trovatore (talk) 09:43, 3 November 2009 (UTC)
thread_title:  Total Function 

message: This section moved to Talk:function (mathematics)#Merge? where the mergeto template points. CMummert 12:32, 14 October 2006 (UTC)
thread_title:  Merge 

message: In the lead:
:"However, not every element of the domain has to be associated with an element of the codomain."
This implies that a total function is a special kind of partial function!!! Why? Are you sure? I would write:
:"However, some element of the domain is not associated with an element of the codomain."
Paolo.dL 17:26, 7 September 2007 (UTC)
replies: text: A total function is a special kind of partial function; this is completely standard. This sort of thing is quite usual in mathematics -- typically, when you loosen a restriction, you just loosen it; you don't require that an object in the new category actually fail the old definition. Similarly, a subset doesn't have to omit an element of the larger set; it's just permitted to. --Trovatore 17:43, 7 September 2007 (UTC)
replies: text: This reminds me of the so-called "red herring principle" in mathematics--in general, a red herring need not be either red or a herring. A partial function may be total, and hence need not be "partial", and in the original definition of 'function', all functions must be total. This is reminiscent of such definitions as 'manifold with boundary', 'non-associative algebra' and 'nonunital ring'. Formally, a partial function is nothing but a 'span' in the category of sets, where we impose the requirement that the 'left' arrow (function) be an injection. If we were to combine partial and 'multivalued functions' (left-total relations) to form 'partial multivalued functions', we would just have spans in Set. This becomes of interest in the theory of (meromorphic functions defined on) Riemann surfaces, which need not be defined everywhere on the Riemann surface which surjectively maps to the complex plane. So in some sense, complex analysis is the study of 'analytic spans' in an appropriate category of complex manifolds. 
--Daviddwd (talk) 20:05, 14 September 2014 (UTC)
thread_title:  The whole is a part of itself 

message: Thanks for your comment. Ok, I can understand the usefullness of calling a set a subset of itself. But in this case the use of the word "partial" to describe a total function seems crazy to me, because this terminology has a huge drawback: if what I suspect is confirmed (see this discussion), according to the modern formal and strict definition of the word "function", non-total functions are not functions! Thus, some "partial functions" are functions, but most of them are not! This is equivalent to a definition such that A+B is allowed to be a subset of A, with B \notin A (A=true functions, B=non-total functions, which by strict definition are not true functions, A+B=partial functions)! The whole is part of a proper part of it! That's too loose to be acceptable.
Alternative terminology 1. Imagine how simple the strict definition of the word function would become if this counter-intuitive terminology were abandoned and a more intuitive definition of the expression "partial function" were adopted:
* Only total and single-valued relations are functions (this is the currently adopted standard definition).
* There are two kinds of relations:
:# total relations (also called functions, when single-valued).
:# partial (=non-total!) relations (this is the change). 
* Partial (=non-total!) relations are not true functions. Thus, strictly speaking, the expressions "partial function" (="non-total function") is a misnomer.
(Similarly, "multivalued" or "multiple-valued" should mean "non-single-valued", and "multivalued function" should be replaced by "multivalued relation")
Isn't that much simpler and nicer? A question: does your counter-intuitive terminology include the expression "proper partial function" (analogous to "proper subset"), meaning "non-total function"? By the way, of course, "crazy" terminology is profuse in my field as well :-) Paolo.dL 15:38, 9 September 2007 (UTC)
replies: text: Yes, both "partial function" and "multivalued function" are misnomers, since neither of these types of objects must be a function. But the terminology is nevertheless common in mathematics. The term "proper partial function" is not common; the only way to express that is to say that the partial function isn't total. The word "domain" also has multiple, mutually contradictory, meanings in teh context of partial functions - see domain (mathematics).  — Carl (CBM · talk) 18:33, 9 September 2007 (UTC)
thread_title:  The whole is part of a proper part of it 

message: This is the most conservative alternative (see below for the rationale):
* Only total and single-valued relations are functions (this is the currently adopted standard definition).
* There are two kinds of relations:
:# Total relations (also called functions, when single-valued).
:# Non-total relations (which are not functions)
* There's no need to use the word "partial", or the expression "partial function"; hence, there's no misnomer.
Similarly, "multivalued functions" could be replaced by "single-valued relation" or "non-single-valued relation". 
(Note that, according to the current definitions, some partial functions are total, and some multivalued functions are single-valued)
Paolo.dL 12:13, 12 September 2007 (UTC)
replies: text: This is completely unacceptable, being not used in any published mathematical works.  — Arthur Rubin |  (talk) 14:56, 12 September 2007 (UTC)

text: The burden of proof is on you to name a work in which it's used.  All I can say for sure is that it's not standard.  — Arthur Rubin |  (talk) 09:07, 13 September 2007 (UTC)
thread_title:  Alternative terminology 1b 

message: As Arthur Rubin pointed out, the above presented alternative terminologies are just hypothetical. They are not necessarily used in the literature. The purpose of this discussion was to critically compare them with the standard terminology, as described in Partial function and Function (mathematics).
Carl and Trovatore, thank you very much for your enlightening contributions. Trovatore, by explaining the reason why the standard strict definition of function is necessary, you convinced me that the hypothetical "alternative terminology 1b" is better than 1 and 2. You also showed that the inconsistency in standard terminology is a "weak" inconsistency :-). However, in this case, full consistency is possible and easy. My opinion on this is irrelevant: the readers are free to choose what they prefer. For sure, now they are better informed.
Carl, when possible, I prefer full terminological consistency because I believe it facilitates the learning process and makes it stabler (I see it as a service for students). Yes, we are not here to change the world, but this discussion will provide good food for thought for the readers. They will be able to compare the definitions of partial function and function with possible alternative ones. At least, the text in this section will help them to better understand the standard definitions, and what's more important, the rationale behind them. 
With kind regards, Paolo.dL 11:46, 13 September 2007 (UTC)
thread_title:  Tentative conclusions 

message: I donnot understand in section "subtraction of natural numbers" this assertion :
Thus he shows that any partially computable function f(x) can be completed by setting 
g(x) = 
\begin{cases} 
  f(x)  & \mbox{where }f(x)\mbox{ is defined} \\
  0, & \mbox{otherwise}. 
\end{cases}
which seems to me ton contradict basic and well known facts (halting problem) in computability theory.  —Preceding unsigned comment added by 81.57.95.102 (talk) 10:25, 5 September 2009 (UTC)
replies: text: I removed that paragraph from the article just now. Thank you for pointing it out. — Carl (CBM · talk) 11:32, 5 September 2009 (UTC)
thread_title:  Completion of partial recursive functions ? 

message: It seems to me that the reciprocal function f(x) = 1/x is one of the more basic "partial functions" over the reals (since it is not defined for x = 0), so that should be reflected in the intro or examples. It's also a partial function over the rationals and over complex numbers for the same reason. 72.19.93.38 (talk) 04:07, 20 September 2011 (UTC)
thread_title:  Reciprocal? 

message: Fully agree with the merge. Total function only exists because of partial functions and it is a pretty tiny article. Dmcq (talk) 22:49, 10 March 2012 (UTC)
replies: text: Total Function was created recently as a separate article, but total function already existed as a redirect to this article. Isheden (talk) 22:59, 10 March 2012 (UTC)
replies: text:  Initially,  was a redirect to Function (mathematics), as it should be, but one user decided to increase the confusion. The new article should be merged to Function (mathematics)#Types of functions, not to "partial function". Incnis Mrsi (talk) 09:27, 11 March 2012 (UTC)
replies: text: I believe Total function should redirect to partial function rather than to function (mathematics). Just because a total function is a function does not mean that is the best redirect, a redirect to partial function where both can be dealt with is better because it deals with the main reason a person would be looking up total function, i.e. why is the word 'total' there? Most times they'd already have a good idea of what a function is. Dmcq (talk) 11:09, 11 March 2012 (UTC)
replies: text: Moreover, since "total function" is just a synonym for function, it is not a "type of function". A partial function is a generalization of the concept of a function, so partial functions should be discussed only towards the end of the article "function" in a section on extensions. I also think that single-valued function should redirect to multi-valued function, because it is natural to discuss these contrasting concepts together. Isheden (talk) 12:22, 11 March 2012 (UTC)

text:  Please, read with attention what I said. The new article  should be merged to Function (mathematics)#Types of functions (or, maybe even better to Function (mathematics)#Definition),  not to "partial function". This implies that "total function" should redirect to a section of "function (mathematics)" where the sense of a qualifier "total" will be explained. Contrary, the article partial function should be focused not to contrasting its definition to total functions, but to uses of partial functions in computation theory, algebra, function theory and applied mathematics. Another acceptable option would be merger to something like total, partial and multi-valued functions devoted to differences in definitions. Incnis Mrsi (talk) 12:31, 11 March 2012 (UTC)
replies: text: Well, I think we understood what you said the first time, so there is no need to reiterate the argument. We all agree that total function is really the same as a function, but the qualifier "total" is used to distinguish it from a partial function. It would be strange to mention total functions in Function (mathematics)#Definition when partial functions are only mentioned as a generalization at the end of the article. What you could do is redirect to that generalizations section at the end of the article, but then it is not a natural redirect anyway. I think this situation is similar to an example on WP:Merging that both "flammable" and "non-flammable" can be explained in an article on "flammability" to avoid excessive overlap. I see no problem with the article also covering the applications you mentioned. Isheden (talk) 15:01, 11 March 2012 (UTC)
replies: text:  Since Isheden reiterates the reference to "flammability", it is the time to explain the difference between cases. Flammability is a property of a chemical substance, not a definition of some notion at its own right. In almost any case where "X is flammable" and "X is non-flammable" can occur, an article either states explicitly or it is clear from the context that X is a substance. One can easily browse to such links as chemical compound, or to the article on a concrete substance, for explanations. But if I wrote "ƒ is a total function from X to Y", then there would be no sources of information about ƒ but the link "total function" and possibly links to its (concrete) domain and codomain (which would explain nothing about the notion of "function"). If one says "ƒ is a total function", the link may not be diverted to an article starting with definition of a partial function. Incnis Mrsi (talk) 17:10, 11 March 2012 (UTC)
replies: text: Why not? If the link is to "total function" then the point is to emphasize the totality (vis-a-vis partiality) of the function. If they just meant "function" then they should not link to "total function", which I do think should redirect to this article. But at the same time the lede of this article ought to have total function in bold if it is the target of that redirect. — Carl (CBM · talk) 19:16, 11 March 2012 (UTC)
replies: text: Incidentally, it has total function in bold already. Isheden (talk) 19:48, 11 March 2012 (UTC)

text: I'm sure we are both keen to help the reader who clicks on total function understand what it is. To my understanding, you believe that the best way is to redirect to function (mathematics), where it is mentioned that "total function" is synonymous to "function". I believe that the best way is to redirect to partial function, which defines a "total function" already in the lead in terms of a partial function for which the two sets X and X' coincide. I acknowledge your point of view, however. Isheden (talk) 19:46, 11 March 2012 (UTC)
replies: text:  The thing several users (Dmcq, Isheden, CBM, Habitmelon) talk about is not redirecting. It actually is a merger. If so, then for which reason the title of the resulting article is "partial function"? It must be something like "total and partial functions", because there is no special word for the total/partial opposition. Incnis Mrsi (talk) 07:04, 12 March 2012 (UTC)
replies: text: There are basically two questions: What to do with the material in Total Function and to what total function should redirect. Several users answer the second question with an article on partial functions. There is no need to include "total function" in the title, since a total function can be viewed as a special case (possibly degenerate) of a partial function. Isheden (talk) 08:28, 12 March 2012 (UTC)
replies: text:  Good point. Let us go further and merge "function (mathematics)" to "partial function". There is no need to have two separate articles since a function can be viewed as a special case (possibly degenerate) of a partial function. Incnis Mrsi (talk) 08:33, 12 March 2012 (UTC)
replies: text:  I think we both agree that "function" is notable enough for an own article and that "partial function" as a generalization of function deserves an own article. But what is your answer to the two questions mentioned above? Isheden (talk) 08:46, 12 March 2012 (UTC)

text: As to the article title: There is a common name for this, see Totality. If this is a concern, the article could be renamed to for example "Totality (mathematics)". Isheden (talk) 09:09, 12 March 2012 (UTC)
replies: text:  Totality (mathematics)? An exciting thought. Could you combine total/partial functions with total/partial order such that the resulting article would not violate WP:OR? Incnis Mrsi (talk) 10:35, 12 March 2012 (UTC)
replies: text:  I think that "partial function" is a better title. "Totality" is not recognizable - someone could guess it is about partial functions, maybe, but most people would have no idea. "Partial function" on the other hand is perfectly clear. Moreover the main topic is partiality, not totality. — Carl (CBM · talk) 11:07, 12 March 2012 (UTC)

text: I'd prefer to see total function redirect to this page, because of the same argument Dmcq gave. — Carl (CBM · talk) 12:27, 11 March 2012 (UTC)

text: I take full responsibility for "increasing the confusion", after reading this discussion I wholeheartedly agree that Total function should redirect to Partial function.  — Preceding unsigned comment added by Habitmelon (talk • contribs) 19:53, 11 March 2012 (UTC)
replies: text: I think "increasing the confusion" referred to the redirect from total function to this article. However, it would be nice if you as the creator of Total Function would move any nonredundant material there to either partial function or function (mathematics). Isheden (talk) 08:32, 12 March 2012 (UTC)

text: Well where I don't feel strongly I defer to the person who actually goes and does the work! Dmcq (talk) 12:00, 20 March 2012 (UTC)
thread_title:  Merge Total function into this article 

message: A partial function is said to be injective or surjective when the total function given by the restriction of the partial function to its domain of definition is. 
is what?  Slashdottir (talk) 15:21, 24 February 2016 (UTC)
replies: text: "... is injective or surjective (respectively)."  Not a great sentence, I agree.  Anyone have a better re-wording than the one I gave here? --Trovatore (talk) 18:55, 24 February 2016 (UTC)
thread_title:  is what? 

message: I have to say I am not really familiar with this sense of the term partial function.  User:D.Lazard, I have to ask — is it possible you're "thinking in French" here?
Even if this sense is in fact current in English, I am not convinced by the restriction of the context from  to .  I think the sense used in this article is overwhelmingly the most common meaning in mathematics generally, including analysis, and we should go back to saying "in mathematics". --Trovatore (talk) 20:56, 20 April 2018 (UTC)
replies: text: About the restriction to , I note first that all sources of this article belong to these fields, and none is related to analysis. Secondly, in analysis the term partial function is not commonly used in the sense of the article, as phrases like "a function of a real variable" mean "a function whose domain is contained in the reals and contains a non-empty open interval". In other words, most functions of analysis are partial in the sense of this article, and a special term is rarely useful. Therefore the distinction between partial and total functions is totally useless in analysis (I must exclude the theory of analytic functions, where one use entire function and meromorphic function in place of total and partial). If I am wrong, you must provide a reliable source showing that the meaning of this article is commonly used in analysis. On the other hand, I agree that the meaning suggested by my hatnote is not commonly used, but it is used by some authors: a Scholar Google search on "partial function" + "partial derivative" provides 269 hits, including a book by Olver. I have not access to this book, but the hits for which I have access generally use the meaning suggested in my hatnote. Finally, the common use of "partial derivative" makes plausible that some users may be confused by the two meanings of "partial", making the hatnote necessary. D.Lazard (talk) 07:58, 21 April 2018 (UTC)
replies: text: Well, let me put it this way:  I think the sense you're calling "abstract algebra" is really not so much algebra, but (elementary) set theory.  Since analysis these days (indeed most mathematics) is generally phrased in set-theoretic terms (even when it doesn't have much to do with the preoccupations of research set theorists), it seems appropriate to me to say "in mathematics".  Analysts might not have a lot of day-to-day use for the term, but they're accustomed to expressing things in elementary set-theoretic terms and will not be confused by the usage. --Trovatore (talk) 15:37, 21 April 2018 (UTC)
thread_title:  Multi-argument function with some arguments fixed 

message: The particulars of this article are covered in the theory of binary relations and heterogeneous relations. Improvements can rely on several relation articles. — Rgdboer (talk) 23:28, 20 June 2018 (UTC)
thread_title:  Relations 

message: The current definition does not include a total function in the set of partial functions. That is contrary to commonly accepted form.  — Preceding unsigned comment added by 82.35.207.91 (talk) 09:21, 7 March 2019 (UTC)
replies: text: I agree, and I have fixed that before reading this post. Also the lead was self contradictory, as saying that the term is mainly used when the domain is not known, which means that it is used when one does not know whether the function is total. D.Lazard (talk) 10:37, 7 March 2019 (UTC)
thread_title:  A total function is in the set of partial functions 

message: De description (definition?) in the intro is not complete. The partial function should not relate any element in X outside S to an element of Y. Madyno (talk) 08:09, 3 April 2022 (UTC)
replies: text: "Domain of definition" may have two meanings. The one of the article differs from yours. I have added a clarification to the article. D.Lazard (talk) 13:04, 10 April 2022 (UTC)
replies: text: I don't take Madyno as offering a different definition, I take Madyno as not understanding that "the domain of f is S" means that f is defined on S and only on S.  Separately: I am not entirely convinced by the parenthetical you added, but I am confident that it does not belong in the second sentence of the article.  I feel like the section "Basic concepts" is the correct home of a discussion of this subtlety.  I will add something or other about "domain of definition" there, and I invite you to expand on it.  --JBL (talk) 14:42, 10 April 2022 (UTC)

text: Really it looks to me like a bunch of details in the lead should be moved into the body. --JBL (talk) 14:44, 10 April 2022 (UTC)
thread_title:  Intro 

message: Rather odd, that the sidebar has the title 'function', and shows a variety of unexplained "examples", of which the connection with the topic is unclear. Madyno (talk) 12:23, 10 April 2022 (UTC)
replies: text: I agree. I have moved the sidebar to the "See also" section. D.Lazard (talk) 13:05, 10 April 2022 (UTC)
thread_title: Sidebar