Talk:Monotonic function

message: I'm not sure about the following statement in the paragraph 'computer science': "...  In some heuristic algorithms, such as A*, the algorithm can be considered optimal if it is monotonic.[2]"
The A star algorithm can just as well be considered optimal if the heuristic function doesn't satisfy the monotonicity restriction. The only practical relevance is that redundant path deletion can be done more efficient. A star without any extension related to this matter can't guaranty that the next selected path is indeed the best path to the last node on the respective path. If the chosen heuristic function 'is' monotonic, we can assume that the next selected path is indeed the best path to the last node on the selected path. It's a matter of efficiency, not optimality.  — Preceding unsigned comment added by Gheylen (talk • contribs) 16:14, 12 April 2012 (UTC)
thread_title:  [Monotonicity in computer science] 

message: if u r going to have nondecreasing redirect here, then you should better define nondecreasing. inpired by the shape of the graph? can you show the graph then??--Jaysscholar 23:21, 12 October 2005 (UTC)
thread_title:  nondecreasing 

message: does this definition mean that the equation's slope is always positive or negative (no points of inflection)?  The definition is unclear (similarly so in my calculus textbook, unfortunately...). --anon
: There can be points of inflection. Positive slope is about first derivative, and points of inflection are about second derivative, so they are not mutually exclusive. Example: 
::f(x)=x^3+(x-1)^3.
:First derivative is always > 0, while second derivative is 6(2x-1) which is 0 at x=1/2, so there are points of inflection. Oleg Alexandrov (talk) 19:12, 21 February 2006 (UTC)
thread_title:  clarification 

message: Should Isotone really redirect here? I'm looking at a different definition in Bhatia's "Matrix Analysis" - if x is majorized by y, then f(x) is submajorized by f(y). For instance, the entropy function (from Rn+ to R+) is isotone. A5 18:59, 6 March 2006 (UTC)
thread_title:  Isotone 

message: I noticed by chance while reading the article that it doesn't have any citations.  References will be needed. Especially handy would be references to proofs of some of the statements in the article, such as the section that implies that all monotone functions have both left and right limits at every point in their domain, and that only having jump discontinuities implies only having a countable number of discontinuities.  (I'm sure the statements are true, but it would be informative to reference proofs to that effect.) Dugwiki 18:07, 31 October 2006 (UTC)
thread_title:  Article needs references 

message: Ok, for the heck of it I was trying to prove that a monotonic function can only have countably many discontinuities, but I'm stuck. :/  I'm sure it's true, but I haven't figured out how to formally prove it.  Anybody have a nice little proof of it, just out of curiosity? Dugwiki 22:56, 1 November 2006 (UTC)
replies: text: If you have access to a copy of Riesz and Nagy's book 'Functional Analysis', it is worth reading - they discus lots of ideas around this area, with some excellent proofs. Madmath789 20:51, 2 November 2006 (UTC)
replies: text: Thanks. I don't have access to that book, unfortunately, but I'll keep that in mind if I'm at Borders. Dugwiki 21:59, 2 November 2006 (UTC)

text: Oh, also, if you can pinpoint the proofs in that book, you might want to include them in the article as footnotes for references.  The article is currently uncited. Dugwiki 22:01, 2 November 2006 (UTC)
thread_title:  Proof of only countable discontinuities? 

message: This page should have some pictures of graphs. —Ben FrantzDale 15:46, 25 May 2007 (UTC)
replies: text: Graphs have been added. --pfctdayelise (talk) 14:16, 2 August 2008 (UTC)
thread_title:  Diagram 

message: It is claimed in the current version of the article that, if f has limits from the right and from the left at every point of its domain, then f has a limit at infinity (either ∞ or −∞) of either a real number, ∞, or −∞.  This is false, as the example f(x) = sin(x) shows.  Correction and a general cleanup of all the claims made in this article is needed. Sullivan.t.j (talk) 00:14, 30 December 2007 (UTC)
replies: text: Good point. I don't know who added that, but I reworded the statement to say that those properties are true for monotonic functions rather than following one from another. That's enough for the purposes of the article anyway. Have you seen other mistakes? Oleg Alexandrov (talk) 00:46, 30 December 2007 (UTC)
thread_title: Another error

message: Should the article include, a monotonic function can be discontinuous at a countable dense set? It's very interesting because I can't visualize it :O  —Preceding unsigned comment added by Standard Oil (talk • contribs) 14:01, 31 May 2009 (UTC)
thread_title:  More interesting stuff 

message: Apologies if this is an unusual route but I figured I could highlight a possible error:
The terms "non-decreasing" and "non-increasing" should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing". For example, the function of figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing.
To me it makes a whole lot more sense if it says decreasing and increasing, but I am not a mathematician. I hope one of you could verify? Thanks! :)  — Preceding unsigned comment added by Beyonix (talk • contribs) 21:07, 2 December 2011 (UTC)
replies: text: That is correct.  They are different concepts with similar names.  Zaslav (talk) 21:48, 6 March 2016 (UTC)
thread_title:  Possible error? 

message: It seems obvious that an absolutely monotonic function is 1-1, but is a 1-1 function in R^1 an absolutely monotonic function?  I.e. are the two synonyms?  — Preceding unsigned comment added by 99.149.190.128 (talk) 20:59, 19 June 2012 (UTC)
thread_title:  Absolutely Monotonic = 1-1 function? 

message: I contest this statement. Imagine the following function: f:R → R with f(x) = x for x < 0 and f(x) = x+1 for x ≥ 0. This function is injective, but not surjective. -- David N. Jansen.  — Preceding unsigned comment added by 131.174.142.225 (talk) 12:59, 11 September 2012 (UTC)
thread_title:  Strictly monotonic implies bijectivity? 

message: It needs to be clarified what is meant by a function "between" ordered sets. What is the domain and what is the codomain? Also, the article could use some inline citations. I find it surprising that "monotonic" doesn't refer equally to both monotonically increasing and monotonically decreasing. Isheden (talk) 08:25, 29 July 2012 (UTC)
thread_title:  Sloppy explanation 

message: Your definitions imply that a constant function is increasing, non-increasing, decreasing, and non-decreasing all at the same time. The definitions of increasing and decreasing conflict with calculus books I have consulted, e.g. Salas and Hille, was well as with common sense and English.  — Preceding unsigned comment added by 98.213.32.21 (talk) 01:40, 25 April 2013 (UTC)
replies: text: Could you provide precise citations from calculus textbooks including page number? At present, there are no in-line citations that support the definitions in the article. Isheden (talk) 12:16, 25 April 2013 (UTC)
replies: text: From Salas and Hille, Calculus of One and Several Variables, 1982, Section 4.2 "Increasing and Decreasing Functions", Definition 4.2.1, p. 142:  "A function f is said to (i) increase on the interval I iff for every two numbers x1, x2 in I x1 < x2 implies f(x1) < f(x2).  (ii) decrease on the interval I iff for every two numbers x1, x2 in I x1 < x2 implies f(x1) > f(x2)."  Also Section 12.1 "Sequences of Real Numbers", p. 491:  "A sequence {an} is said to be (i) increasing iff an < an+1 for each positive integer n, (ii) nondecreasing iff an ≤ an+1 for each positive integer n, (iii) decreasing iff an > an+1 for each positive integer n, (iv) nonincreasing iff an ≥ an+1 for each positive integer n."  See also James Stewart, Calculus Concepts and Contexts, 4th Edition, p. 560,  There are certainly sufficient references to support these definitions, and wikipedia has an opportunity to define these in a way that makes sense. If you want to talk about a function that increases or stays the same, we already have a perfectly good term in nondecreasing, so there is no reason to define increasing the same way.  We all know what it means to increase.  Constant functions don't increase.  Non-increasing functions don't increase.  To state otherwise is to perpetuate unnecessary confusion.  — Preceding unsigned comment added by 98.213.32.21 (talk) 22:12, 1 May 2013 (UTC)
replies: text: Perhaps we should consider only the definitions of monotone functions at this stage and discuss sequences later (although the terminology is analogous). There are four terms for monotone functions with clear definitions: non-increasing, non-decreasing, strictly increasing, and strictly decreasing. I think the article should use this terminology to avoid confusion. After introducing these terms, it should be mentioned that some references define increasing to mean strictly increasing, whereas others define increasing to mean non-decreasing. Isheden (talk) 05:34, 2 May 2013 (UTC)

text: I checked a few calculus textbooks and it seems they often  define "increasing function" to mean "strictly increasing function". Perhaps that should be mentioned in the article including a reference. In any case an in-line citation would be needed to support the terminology in the article. Isheden (talk) 12:38, 25 April 2013 (UTC)
replies: text: Your reference Bartle, Elements of Real Analysis, 1976, Section IV.20 "Local Properties of Continuous Functions", p. 145, problem 20.R contains your definition of increasing.  "A function f defined on an interval I contained in R is said to be increasing on I if x ≤ x', x, x' in I imply that f(x) ≤ f(x').  It is said to be strictly increasing on I if x < x' x,x' in I imply that f(x) < f(x').  Similar definitions can be given for decreasing and strictly decreasing."  Note though that a) this is an exercise, and b) he doesn't use the terms nondecreasing and nonincreasing.  It would make sense that when these latter terms are used in the same reference as increasing and decreasing that all four terms are defined to have different meanings as does Salas and Hille that I quoted above.  — Preceding unsigned comment added by 98.213.32.21 (talk) 22:40, 1 May 2013 (UTC)
replies: text: Here are a few other references that define increasing function to mean either non-decreasing or strictly increasing:       Isheden (talk) 05:53, 2 May 2013 (UTC)
thread_title:  A Function Can Be Both Increasing and Non-increasing? 

message: Would it be possible to simplify the first sentence of the article to make it easier to understand for calculus students? E.g. "In mathematics, a real-valued function of a real variable is called monotonic (or monotone) if it is either increasing or decreasing." Or would it be better to change the present redirects Increasing function and Decreasing function to short articles that provide the basic definitions without mentioning ordered sets? Isheden (talk) 12:51, 25 April 2013 (UTC)
thread_title:  First sentence of the lead 

message: Like many science/technology Wiki pages, this one is way too complicated to be useful to non-specialists. Could someone do a Simple English version, getting across the basic ideas, e.g. that "monotonically increasing" means "always going up", or whatever it does mean, in simple layman's terms.  — Preceding unsigned comment added by 86.159.201.166 (talk) 11:52, 17 September 2014 (UTC)
thread_title:  Too sophisticated 

message: The definition stated in the introduction at this time is incorrect.  "Isotone" means monotone increasing.  "Antitone" means monotone decreasing.  "Monotone" means either isotone or antitone.  "Monotone" of a function of several variables can mean monotone in each variable, thus, not necessarily monotone as a whole since it could be isotone in one variable and antitone in another variable.  It is true that "monotone" is too often used loosely (and incorrectly) to mean "monotone increasing" but that does not make increase the correct standard definition.  I will edit the article if I don't get good arguments not to do so.  I await replies.  Zaslav (talk) 21:53, 6 March 2016 (UTC)
thread_title:  Monotone, isotone, antitone 

message: I contributed 50 to Wikipedia because it was made technically easy to do so. 178.38.132.48 (talk) 21:11, 5 December 2017 (UTC)
thread_title:  Contribution 

message: The section on monotonicity in functional analysis introduces this notation: (Tu - Tv, u - v) \geq 0 \quad \forall u,v \in X. Having almost no experience in functional analysis, in what sense does it make to say that an (apparent) ordered pair (a, b) can be greater than or equal to 0? (a, b) >= 0? What does this mean? In this case, clearly it does not seem to mean that both a >= 0 and b >= 0. I can only guess that it means both a and b have the same sign, or are both zero. Or, perhaps, a*b >= 0. The main point is that this notation appears out of nowhere and is not explained. Also a little later on here: (w_1 - w_2, u_1 - u_2) \geq 0. 24.57.106.253 (talk) 14:45, 12 May 2018 (UTC)
replies: text: In this context (a,b) is used to denote either the inner product of a and b if they are elements of a Hilbert space, or to denote a(b) the application of a on </math> if b is an element of a topological vector space, and a is an element of the dual of that space, i.e. a (usually continuous) linear functional on that space. Cactus0192837465 (talk) 13:58, 29 January 2019 (UTC)
thread_title:  Introduces unexplained notation, presumably from functional analysis 

message: Order-reflecting, which is linked in Order_embedding, redirects to order-preserving. The term order-reflecting however does not appear in this article, an in Order_embedding it does not mean the same as order-preserving, but the converse: if f(x) < f(y) then x < y. - 17:01, 4 June 2021‎ 2001:9e8:e256:6900:468a:5bff:fe9a:57e7
:I fixed the redirect Order-reflecting, it now points to Order theory#Functions between orders. - Jochen Burghardt (talk) 15:44, 4 June 2021 (UTC)
thread_title:  Order-reflecting 

message: "Decreasing sequence" redirects here but this page has nothing to say about decreasing sequences. 159.196.132.1 (talk) 07:59, 2 January 2024 (UTC)
replies: text: No, Decreasing sequence redirects to sequence. - Jochen Burghardt (talk) 08:38, 2 January 2024 (UTC)
thread_title:  Sequences