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In mathematics, a monotonic function (or monotone function) is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
between ordered sets that preserves or reverses the given order. This concept first arose in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, and was later generalized to the more abstract setting of
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article int ...
.


In calculus and analysis

In
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, a function f defined on a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of the
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
with real values is called ''monotonic'' if and only if it is either entirely non-increasing, or entirely non-decreasing. 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 called ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\right), so it ''reverses'' the order (see Figure 2). If the order \leq in the definition of monotonicity is replaced by the strict order <, one obtains a stronger requirement. A function with this property is called ''strictly increasing'' (also ''increasing''). Again, by inverting the order symbol, one finds a corresponding concept called ''strictly decreasing'' (also ''decreasing''). A function with either property is called ''strictly monotone''. Functions that are strictly monotone are one-to-one (because for x not equal to y, either x < y or x > y and so, by monotonicity, either f\!\left(x\right) < f\!\left(y\right) or f\!\left(x\right) > f\!\left(y\right), thus f\!\left(x\right) \neq f\!\left(y\right).) 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 f\!\left(x\right) is said to be ''absolutely monotonic'' over an interval \left(a, b\right) if the derivatives of all orders of f are
nonnegative In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or i ...
or all
nonpositive In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
at all points on the interval.


Inverse of function

All strictly monotonic functions are
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
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 y = g(x) is strictly increasing on the range
, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math>, then it has an inverse x = h(y) on the range (a), g(b)/math>. Note that 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.


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 As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophe ...
being preserved across a monotonic transform (see also
monotone preferences In economics, an agent's preferences are said to be weakly monotonic if, given a consumption bundle x, the agent prefers all consumption bundles y that have more of all goods. That is, y \gg x implies y\succ x. An agent's preferences are said to ...
). 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.


Some basic applications and results

The following properties are true for a monotonic function f\colon \mathbb \to \mathbb: *f has
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
from the right and from the left at every point of its
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function ** Domain of holomorphy of a function * ...
; *f has a limit at positive or negative infinity (\pm\infty) of either a real number, \infty, or -\infty. *f can only have jump discontinuities; *f can only have
countably In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
many 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 In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mat ...
(a_i) of positive numbers and any enumeration (q_i) of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s, the monotonically increasing function f(x)=\sum_ a_i is continuous exactly at every irrational number (cf. picture). It is the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
of the
discrete measure In mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if it is concentrated on an at most countable set. The support need not be a discrete set. Geometri ...
on the rational numbers, where a_i is the weight of q_i. These properties are the reason why monotonic functions are useful in technical work in
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
. Other important properties of these functions include: *if f is a monotonic function defined on an interval I, then f is
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
on I; i.e. the set of numbers x in I such that f is not differentiable in x has
Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
measure zero In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null ...
. In addition, this result cannot be improved to countable: see
Cantor function In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. ...
. *if this set is countable, then f is absolutely continuous *if f is a monotonic function defined on an interval \left , b\right/math>, then f is
Riemann integrable In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of G� ...
. An important application of monotonic functions is in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
. If X is a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
, its
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
F_X\!\left(x\right) = \text\!\left(X \leq x\right) is a monotonically increasing function. A function is ''
unimodal In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. Unimodal probability distribution In statistics, a unimodal ...
'' if it is monotonically increasing up to some point (the ''
mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is ...
'') and then monotonically decreasing. When f is a ''strictly monotonic'' function, then f is
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposit ...
on its domain, and if T is the
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
of f, then there is an
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
on T for f. In contrast, each constant function is monotonic, but not injective, and hence cannot have an inverse.


In topology

A map f: X \to Y is said to be ''monotone'' if each of its
fibers Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
is
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
; that is, for each element y \in Y, the (possibly empty) set f^(y) is a connected subspace of X.


In functional analysis

In
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on ...
on a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...
X, a (possibly non-linear) operator T: X \rightarrow X^* is said to be a ''monotone operator'' if :(Tu - Tv, u - v) \geq 0 \quad \forall u,v \in X. Kachurovskii's theorem shows that convex functions on
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
s have monotonic operators as their derivatives. A subset G of X \times X^* is said to be a ''monotone set'' if for every pair
_1, w_1 Onekama ( ) is a village in Manistee County in the U.S. state of Michigan. The population was 411 at the 2010 census. The village is located on the shores of Portage Lake and is surrounded by Onekama Township. The town's name is derived from "O ...
/math> and
_2, w_2 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> in G, :(w_1 - w_2, u_1 - u_2) \geq 0. G 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 G(T) is a monotone set. A monotone operator is said to be ''maximal monotone'' if its graph is a ''maximal monotone set''.


In order theory

Order theory deals with arbitrary
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
s and 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 Total may refer to: Mathematics * Total, the summation of a set of numbers * Total order, a partial order without incomparable pairs * Total relation, which may also mean ** connected relation (a binary relation in which any two elements are com ...
. Furthermore, the
strict In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusive ...
relations < and > are of little use in many non-total orders and hence no additional terminology is introduced for them. Letting ≤ denote the partial order relation of any partially ordered set, a ''monotone'' function, also called ''isotone'', or ', satisfies the property : ''x'' ≤ ''y'' implies ''f''(''x'') ≤ ''f''(''y''), for all ''x'' and ''y'' in its domain. The composite of two monotone mappings is also monotone. The
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
notion is often called ''antitone'', ''anti-monotone'', or ''order-reversing''. Hence, an antitone function ''f'' satisfies the property : ''x'' ≤ ''y'' implies ''f''(''y'') ≤ ''f''(''x''), for all ''x'' and ''y'' in its domain. A
constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic properti ...
is both monotone and antitone; conversely, if ''f'' is both monotone and antitone, and if the domain of ''f'' is a lattice, then ''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 In order theory, a branch of mathematics, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order embeddings constitute a notion which is stric ...
s (functions for which ''x'' ≤ ''y''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicond ...
''f''(''x'') ≤ ''f''(''y'')) and
order isomorphism In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be con ...
s (
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
order embeddings).


In the context of search algorithms

In the context of
search algorithm In computer science, a search algorithm is an algorithm designed to solve a search problem. Search algorithms work to retrieve information stored within particular data structure, or calculated in the search space of a problem domain, with eith ...
s monotonicity (also called consistency) is a condition applied to
heuristic function In mathematical optimization and computer science, heuristic (from Greek εὑρίσκω "I find, discover") is a technique designed for solving a problem more quickly when classic methods are too slow for finding an approximate solution, or wh ...
s. A heuristic ''h(n)'' is monotonic if, for every node ''n'' and every successor ''n of ''n'' generated by any action ''a'', the estimated cost of reaching the goal from ''n'' is no greater than the step cost of getting to '' n' '' plus the estimated cost of reaching the goal from '' n' '', :h(n) \leq c\left(n, a, n'\right) + h\left(n'\right). This is a form of
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, bu ...
, with ''n'', ''n, and the goal ''Gn'' closest to ''n''. Because every monotonic heuristic is also admissible, monotonicity is a stricter requirement than admissibility. Some
heuristic algorithm In mathematical optimization and computer science, heuristic (from Greek εὑρίσκω "I find, discover") is a technique designed for solving a problem more quickly when classic methods are too slow for finding an approximate solution, or wh ...
s such as A* can be proven optimal provided that the heuristic they use is monotonic.Conditions for optimality: Admissibility and consistency pg. 94–95 .


In Boolean functions

In
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
, a monotonic function is one such that for all ''a''''i'' and ''b''''i'' in , if , , ..., (i.e. the Cartesian product ''n'' is ordered coordinatewise), then . 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 ''n''-ary Boolean function is monotonic when its representation as an ''n''-cube labelled with truth values has no upward edge from ''true'' to ''false''. (This labelled
Hasse diagram In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' one represents e ...
is the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
of the function's labelled
Venn diagram A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships ...
, which is the more common representation for .) 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 ''
and or AND may refer to: Logic, grammar, and computing * Conjunction (grammar), connecting two words, phrases, or clauses * Logical conjunction in mathematical logic, notated as "∧", "⋅", "&", or simple juxtaposition * Bitwise AND, a boolea ...
'' and '' or'' (in particular '' not'' is forbidden). For instance "at least two of ''a'', ''b'', ''c'' hold" is a monotonic function of ''a'', ''b'', ''c'', since it can be written for instance as ((''a'' and ''b'') or (''a'' and ''c'') or (''b'' and ''c'')). The number of such functions on ''n'' variables is known as the
Dedekind number File:Monotone Boolean functions 0,1,2,3.svg, 400px, The free distributive lattices of monotonic Boolean functions on 0, 1, 2, and 3 arguments, with 2, 3, 6, and 20 elements respectively (move mouse over right diagram to see description) circle 6 ...
of ''n''.


See also

*
Monotone cubic interpolation In the mathematical field of numerical analysis, monotone cubic interpolation is a variant of cubic interpolation that preserves monotonicity of the data set being interpolated. Monotonicity is preserved by linear interpolation but not guaranteed ...
*
Pseudo-monotone operator In mathematics, a pseudo-monotone operator from a reflexive Banach space into its continuous dual space is one that is, in some sense, almost as well-behaved as a monotone operator. Many problems in the calculus of variations can be expressed usi ...
*
Spearman's rank correlation coefficient In statistics, Spearman's rank correlation coefficient or Spearman's ''ρ'', named after Charles Spearman and often denoted by the Greek letter \rho (rho) or as r_s, is a nonparametric measure of rank correlation (statistical dependence between ...
- measure of monotonicity in a set of data *
Total monotonicity In real analysis, a branch of mathematics, Bernstein's theorem states that every real-valued function on the half-line that is totally monotone is a mixture of exponential functions. In one important special case the mixture is a weighted average ...
* Cyclical monotonicity *
Operator monotone function In linear algebra, the operator monotone function is an important type of real-valued function, first described by Charles Löwner in 1934. It is closely allied to the operator concave and operator concave functions, and is encountered in operator ...


Notes


Bibliography

* * * * * * * (Definition 9.31)


External links

*
Convergence of a Monotonic Sequence
by Anik Debnath and Thomas Roxlo (The Harker School),
Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
. * {{Order theory Functional analysis Order theory Real analysis Types of functions