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{{Short description|Order-preserving mathematical function}}
{{redirect|Monotonicity|information on monotonicity as it pertains to [[voting systems]]|monotonicity criterion|information on monotonicity as it pertains to logical systems|Monotonicity of entailment}}
{{Redirect|Monotonic|other uses|Monotone (disambiguation)}}

[[Image:Monotonicity example1.svg|right|thumb|Figure 1. A monotonically non-decreasing function]]
[[Image:Monotonicity example2.svg|right|thumb|Figure 2. A monotonically non-increasing function]]
[[Image:Monotonicity example3.svg|right|thumb|Figure 3. A function that is ''not'' monotonic]]

In [[mathematics]], a '''monotonic function''' (or '''monotone function''') is a [[function (mathematics)|function]] between [[List of order structures in mathematics|ordered sets]] that preserves or reverses the given [[order relation|order]].<ref>{{Cite book|title=Oxford Concise Dictionary of Mathematics|last1=Clapham|first1=Christopher|last2=Nicholson|first2=James|publisher=Oxford University Press|year=2014|edition=5th}}</ref><ref name=":1">{{Cite web|url=http://mathworld.wolfram.com/MonotonicFunction.html|title=Monotonic Function|last=Stover|first=Christopher|website=Wolfram MathWorld|language=en|access-date=2018-01-29}}</ref><ref name=":0">{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Monotone_function|title=Monotone function|website=Encyclopedia of Mathematics|language=en|access-date=2018-01-29}}</ref> This concept first arose in [[calculus]], and was later generalized to the more abstract setting of [[order theory]].

== In calculus and analysis ==
In [[calculus]], a function <math>f</math> defined on a [[subset]] of the [[real numbers]] with real values is called  ''monotonic'' if it is either entirely non-decreasing, or entirely non-increasing.<ref name=":1" /> That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease.

A function is termed ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'')<ref name=":0" /> if for all <math>x</math> and <math>y</math> such that <math>x \leq y</math> one has  <math>f\!\left(x\right) \leq f\!\left(y\right)</math>, so <math>f</math> preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'')<ref name=":0" /> if, whenever <math>x \leq y</math>, then <math>f\!\left(x\right) \geq f\!\left(y\right)</math>, so it ''reverses'' the order (see Figure 2).

If the order <math>\leq</math> in the definition of monotonicity is replaced by the strict order <math><</math>, one obtains a stronger requirement. A function with this property is called ''strictly increasing'' (also ''increasing'').<ref name=":0" /><ref name=":2">{{Cite book|last=Spivak|first=Michael|title=Calculus|publisher=Publish or Perish, Inc.|year=1994|isbn=0-914098-89-6|location=Houston, Texas |pages=192}}</ref> Again, by inverting the order symbol, one finds a corresponding concept called ''strictly decreasing'' (also ''decreasing'').<ref name=":0" /><ref name=":2" /> A function with either property is called ''strictly monotone''. Functions that are strictly monotone are [[one-to-one function|one-to-one]] (because for <math>x</math> not equal to <math>y</math>, either <math>x < y</math> or <math>x > y</math> and so, by monotonicity, either <math>f\!\left(x\right) < f\!\left(y\right)</math> or <math>f\!\left(x\right) > f\!\left(y\right)</math>, thus <math>f\!\left(x\right) \neq f\!\left(y\right)</math>.)

To avoid ambiguity, the terms ''weakly monotone'', ''weakly increasing'' and ''weakly decreasing'' are often used to refer to non-strict monotonicity.

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 non-monotonic function shown in 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.

A function <math>f</math> is said to be ''absolutely monotonic'' over an interval <math>\left(a, b\right)</math> if the derivatives of all orders of <math>f</math> are [[nonnegative]] or all [[nonpositive]] at all points on the interval.

=== Inverse of function ===

All strictly monotonic functions are [[Inverse function|invertible]] because they are guaranteed to have a one-to-one mapping from their range to their domain.

However, functions that are only weakly monotone are not invertible because they are constant on some interval (and therefore are not one-to-one).

A function may be strictly monotonic over a limited a range of values and thus have an inverse on that range even though it is not strictly monotonic everywhere. For example, if <math>y = g(x)</math> is strictly increasing on the range <math>[a, b]</math>, then it has an inverse <math>x = h(y)</math> on the range <math>[g(a), g(b)]</math>.

The term ''monotonic'' is sometimes used in place of ''strictly monotonic'', so a source may state that all monotonic functions are invertible when they really mean that all strictly monotonic functions are invertible.{{cn|reason=Give an example for such a source.|date=August 2022}}

=== Monotonic transformation ===

The term ''monotonic transformation'' (or ''monotone transformation'') may also cause confusion because it refers to a transformation by a strictly increasing function. This is the case in economics with respect to the ordinal properties of a [[utility function]] being preserved across a monotonic transform (see also [[monotone preferences]]).<ref>See the section on Cardinal Versus Ordinal Utility in {{harvtxt|Simon|Blume|1994}}.</ref> In this context, the term "monotonic transformation" refers to a positive monotonic transformation and is intended to distinguish it from a "negative monotonic transformation," which reverses the order of the numbers.<ref>{{cite book |last=Varian |first=Hal R. |title=Intermediate Microeconomics |edition=8th |year=2010 |publisher=W. W. Norton & Company |page=56 |isbn=9780393934243}}</ref>

=== Some basic applications and results ===
[[File:Monotonic dense jumps svg.svg|thumb|550px|Monotonic function with a dense set of jump discontinuities (several sections shown)]]
[[File:Growth equations.png|550px|thumb|Plots of 6 monotonic growth functions]]
The following properties are true for a monotonic function <math>f\colon \mathbb{R} \to \mathbb{R}</math>:
*<math>f</math> has [[limit of a function|limits]] from the right and from the left at every point of its [[Domain of a function|domain]];
*<math>f</math> has a limit at positive or negative infinity (<math>\pm\infty</math>) of either a real number, <math>\infty</math>, or <math>-\infty</math>.
*<math>f</math> can only have [[jump discontinuities]];
*<math>f</math> can only have [[countably]] many [[Discontinuities of monotone functions|discontinuities]] in its domain. The discontinuities, however, do not necessarily consist of isolated points and may even be dense in an interval (''a'', ''b''). For example, for any [[summable sequence]] <math display>(a_i)</math> of positive numbers and any enumeration <math>(q_i)</math> of the [[rational number]]s, the monotonically increasing function <math display=block>f(x)=\sum_{q_i\leq x} a_i</math> is continuous exactly at every irrational number (cf. picture). It is the [[cumulative distribution function]] of the [[discrete measure]] on the rational numbers, where <math>a_i</math> is the weight of <math>q_i</math>.
*If <math>f</math> is [[differentiable]] at <math>x^*\in\Bbb R</math> and <math>f'(x^*)>0</math>, then there is a non-degenerate [[interval (mathematics)| interval]] ''I'' such that <math>x^*\in I</math> and <math>f</math> is increasing on ''I''.  As a partial converse, if ''f'' is differentiable and increasing on an interval, ''I'', then its derivative is positive at every point in ''I''.

These properties are the reason why monotonic functions are useful in technical work in [[mathematical analysis|analysis]]. Other important properties of these functions include:
*if <math>f</math> is a monotonic function defined on an [[interval (mathematics)|interval]] <math>I</math>, then <math>f</math> is [[derivative|differentiable]] [[almost everywhere]] on <math>I</math>; i.e. the set of numbers <math>x</math> in <math>I</math> such that <math>f</math> is not differentiable in <math>x</math> has [[Lebesgue measure|Lebesgue]] [[measure zero]]. In addition, this result cannot be improved to countable: see [[Cantor function]].
*if this set is countable, then <math>f</math> is absolutely continuous
*if <math>f</math> is a monotonic function defined on an interval <math>\left[a, b\right]</math>, then <math>f</math> is [[Riemann integral|Riemann integrable]].

An important application of monotonic functions is in [[probability theory]]. If <math>X</math> is a [[random variable]], its [[cumulative distribution function]] <math>F_X\!\left(x\right) = \text{Prob}\!\left(X \leq x\right)</math> is a monotonically increasing function.

A function is ''[[unimodal function|unimodal]]'' if it is monotonically increasing up to some point (the ''[[Mode (statistics)|mode]]'') and then monotonically decreasing.

When <math>f</math> is a ''strictly monotonic'' function, then <math>f</math> is [[injective]] on its domain, and if <math>T</math> is the [[range of a function|range]] of <math>f</math>, then there is an [[inverse function]] on <math>T</math> for <math>f</math>. In contrast, each constant function is monotonic, but not injective,<ref>if its domain has more than one element</ref> and hence cannot have an inverse.

The graphic shows six monotonic functions. Their simplest forms are shown in the plot area and the expressions used to create them are shown on the ''y''-axis.

==In topology==
{{anchor|Monotone function (topology)|Monotonic function (topology)}}

A map <math>f: X \to Y</math> is said to be ''monotone'' if each of its [[Fiber (mathematics)#Fiber in naive set theory|fibers]] is [[Connected (topology)|connected]]; that is, for each element <math>y \in Y,</math> the (possibly empty) set <math>f^{-1}(y)</math> is a connected [[Subspace topology|subspace]] of <math>X.</math>

== In functional analysis ==

In [[functional analysis]] on a [[topological vector space]] <math>X</math>, a (possibly non-linear) operator <math>T: X \rightarrow X^*</math> is said to be a ''monotone operator'' if

<math display="block">(Tu - Tv, u - v) \geq 0 \quad \forall u,v \in X.</math>
[[Kachurovskii's theorem]] shows that [[convex function]]s on [[Banach space]]s have monotonic operators as their derivatives.

A subset <math>G</math> of <math>X \times X^*</math> is said to be a ''monotone set'' if for every pair <math>[u_1, w_1]</math> and <math>[u_2, w_2]</math> in <math>G</math>,

<math display="block">(w_1 - w_2, u_1 - u_2) \geq 0.</math>
<math>G</math> is said to be ''maximal monotone'' if it is maximal among all monotone sets in the sense of set inclusion.  The graph of a monotone operator <math>G(T)</math> is a monotone set. A monotone operator is said to be ''maximal monotone'' if its graph is a ''maximal monotone set''.

== In order theory ==
{{anchor|Monotone function (order theory)}}
Order theory deals with arbitrary [[partially ordered set]]s and [[preorder|preordered sets]] as a generalization of real numbers. The above definition of monotonicity is relevant in these cases as well. However, the terms "increasing" and "decreasing" are avoided, since their conventional pictorial representation does not apply to orders that are not [[total order|total]]. Furthermore, the [[strict order|strict]] relations <math><</math> and <math>></math> are of little use in many non-total orders and hence no additional terminology is introduced for them.

Letting <math>\leq</math> denote the partial order relation of any partially ordered set, a ''monotone'' function, also called ''isotone'', or ''{{visible anchor|order-preserving}}'', satisfies the property

<math display="block">x \leq y \implies f(x) \leq f(y)</math>

for all {{mvar|x}} and {{mvar|y}} in its domain. The composite of two monotone mappings is also monotone.

The [[duality (order theory)|dual]] notion is often called ''antitone'', ''anti-monotone'', or ''order-reversing''. Hence, an antitone function {{mvar|f}} satisfies the property

<math display="block">x \leq y \implies f(y) \leq f(x),</math>

for all {{mvar|x}} and {{mvar|y}} in its domain.

A [[constant function]] is both monotone and antitone; conversely, if {{mvar|f}} is both monotone and antitone, and if the domain of {{mvar|f}} is a [[lattice (order)|lattice]], then {{mvar|f}} must be constant.

Monotone functions are central in order theory. They appear in most articles on the subject and examples from special applications are found in these places. Some notable special monotone functions are [[order embedding]]s (functions for which <math>x \leq y</math> [[if and only if]] <math>f(x) \leq f(y))</math> and [[order isomorphism]]s ([[surjective]] order embeddings).

== In the context of search algorithms ==
In the context of [[search algorithm]]s monotonicity (also called consistency) is a condition applied to [[heuristic function]]s.  A heuristic <math>h(n)</math> is monotonic if, for every node {{mvar|n}} and every successor {{mvar|n'}} of {{mvar|n}} generated by any action {{mvar|a}}, the estimated cost of reaching the goal from {{mvar|n}} is no greater than the step cost of getting to {{mvar|n'}} plus the estimated cost of reaching the goal from {{mvar|n'}},

<math display="block">h(n) \leq c\left(n, a, n'\right) + h\left(n'\right) .</math>

This is a form of [[triangle inequality]], with {{mvar|n}}, {{mvar|n'}}, and the goal {{mvar|G<sub>n</sub>}} closest to {{mvar|n}}.  Because every monotonic heuristic is also [[admissible heuristic|admissible]], monotonicity is a stricter requirement than admissibility.  Some [[heuristic algorithm]]s such as [[A* search algorithm|A*]] can be proven [[asymptotically optimal algorithm|optimal]] provided that the heuristic they use is monotonic.<ref>Conditions for optimality:  Admissibility and consistency pg. 94–95 {{Harv|Russell|Norvig|2010}}.</ref>

== In Boolean functions ==
{| style="float:right;"
| [[File:Hasse3 x impl y and z.svg|thumb|With the nonmonotonic function "if {{mvar|a}} then both {{mvar|b}} and {{mvar|c}}", {{color|#c00000|false}} nodes appear above {{color|#00c000|true}} nodes.]]
|}
{| style="float:right;"
| [[File:Hasse3 ge2.svg|thumb|Hasse diagram of the monotonic function "at least two of {{mvar|a}}, {{mvar|b}}, {{mvar|c}} hold". Colors indicate function output values.]]
|}
In [[Boolean algebra]], a monotonic function is one such that for all {{mvar|a<sub>''i''</sub>}} and {{mvar|b<sub>''i''</sub>}} in {{math|{0,1}<nowiki />}}, if {{math|''a''<sub>1</sub> ≤ ''b''<sub>1</sub>}}, {{math|''a''<sub>2</sub> ≤ ''b''<sub>2</sub>}}, ..., {{math|''a''<sub>''n''</sub> ≤ ''b''<sub>''n''</sub>}} (i.e. the Cartesian product {{math|{0, 1}<sup>n</sup>}} is ordered [[coordinatewise order|coordinatewise]]), then {{math|f(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>) ≤ f(''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub>)}}. In other words, a Boolean function is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. Graphically, this means that an {{mvar|n}}-ary Boolean function is monotonic when its representation as an [[hypercube|{{mvar|n}}-cube]] labelled with truth values has no upward edge from ''true'' to ''false''.  (This labelled [[Hasse diagram]] is the [[Duality (mathematics)#Dimension-reversing dualities|dual]] of the function's labelled [[Venn diagram]], which is the more common representation for {{math|''n'' ≤ 3}}.)

The monotonic Boolean functions are precisely those that can be defined by an expression combining the inputs (which may appear more than once) using only the operators ''[[logical conjunction|and]]'' and ''[[logical disjunction|or]]'' (in particular ''[[negation|not]]'' is forbidden). For instance "at least two of {{mvar|a}}, {{mvar|b}}, {{mvar|c}} hold" is a monotonic function of {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, since it can be written for instance as (({{mvar|a}} and {{mvar|b}}) or ({{mvar|a}} and {{mvar|c}}) or ({{mvar|b}} and {{mvar|c}})).

The number of such functions on {{mvar|n}} variables is known as the [[Dedekind number]] of {{mvar|n}}.

[[SAT solving]], generally an [[NP-hard]] task, can be achieved efficiently when all involved functions and predicates are monotonic and Boolean.<ref>{{cite conference | doi=10.1609/aaai.v29i1.9755 | url=https://ojs.aaai.org/index.php/AAAI/article/view/9755 |first1=Sam |last1=Bayless |first2=Noah |last2=Bayless |first3=Holger H. |last3=Hoos |first4=Alan J. |last4=Hu | title=SAT Modulo Monotonic Theories | editor= | conference=Proc. 29th AAAI Conf. on Artificial Intelligence | publisher=AAAI Press | series= | volume= | pages=3702&ndash;3709 | year=2015 | doi-access=free | arxiv=1406.0043 |url-status=live |archive-url=https://web.archive.org/web/20231211023402/https://ojs.aaai.org/index.php/AAAI/article/view/9755 |archive-date= Dec 11, 2023 }} 
</ref>

== See also ==
* [[Monotone cubic interpolation]]
* [[Pseudo-monotone operator]]
* [[Spearman's rank correlation coefficient]] - measure of monotonicity in a set of data
* [[Total monotonicity]]
* [[Cyclical monotonicity]]
* [[Operator monotone function]]
* [[Monotone set function]]
* [[Absolutely and completely monotonic functions and sequences]]

== Notes ==
{{reflist}}

== Bibliography ==
*{{cite book
 | last = Bartle
 | first = Robert G.
 | title = The elements of real analysis
 | edition = second
 | year = 1976
}}
*{{cite book
 | last = Grätzer
 | first = George
 | title = Lattice theory: first concepts and distributive lattices
 | year = 1971
 | publisher = W. H. Freeman
 | isbn = 0-7167-0442-0
}}
*{{cite book
 | last       = Pemberton
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 |author2=Rau, Nicholas
 | title      = Mathematics for economists: an introductory textbook
 | publisher  = Manchester University Press
 | year       = 2001
 | isbn       = 0-7190-3341-1
}}
* {{cite book
|author1=Renardy, Michael  |author2=Rogers, Robert C.
 |name-list-style=amp |    title = An introduction to partial differential equations
|   series = Texts in Applied Mathematics 13
|  edition = Second
|publisher = Springer-Verlag
| location = New York
|     year = 2004
|    pages = 356
|       isbn = 0-387-00444-0
}}
*{{cite book
 |author1=Riesz, Frigyes  |author2=Béla Szőkefalvi-Nagy
  |name-list-style=amp | title = Functional Analysis
 | year = 1990
| publisher = Courier Dover Publications
| isbn = 978-0-486-66289-3
}}
*{{cite book
 | last1=Russell |first1=Stuart J. |last2=Norvig |first2=Peter 
 | title = Artificial Intelligence: A Modern Approach
 | year = 2010
 | edition = 3rd
 | publisher = Prentice Hall
 | publication-place = Upper Saddle River, New Jersey
 | isbn = 978-0-13-604259-4
}}
*{{cite book
 | last1 = Simon |first1=Carl P. |first2=Lawrence |last2=Blume 
 | title = Mathematics for Economists
 | edition = first
 | date = April 1994
 |publisher=Norton | isbn = 978-0-393-95733-4
}} (Definition 9.31)

==External links==

* {{springer|title=Monotone function|id=p/m064830}}
* [http://demonstrations.wolfram.com/ConvergenceOfAMonotonicSequence/ Convergence of a Monotonic Sequence] by Anik Debnath and Thomas Roxlo (The Harker School), [[Wolfram Demonstrations Project]].
* {{MathWorld |title=Monotonic Function |id=MonotonicFunction}}

{{Order theory}}

[[Category:Functional analysis]]
[[Category:Order theory]]
[[Category:Real analysis]]
[[Category:Types of functions]]