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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 Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
. 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.


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 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 at all points on the interval.


Inverse of function

All strictly monotonic functions are 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 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 philosoph ...
being preserved across a monotonic transform (see also monotone preferences). 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 from the right and from the left at every point of its domain; *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 (a_i) of positive numbers and any enumeration (q_i) of the rational numbers, 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 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
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 measure zero. 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. 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, 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 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 on a topological vector space 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 s ...
s (functions for which ''x'' ≤ ''y'' if and only if ''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 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 functions. 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, 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 algorithms such as A* can be proven
optimal Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
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, 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'' 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 of ''n''.


See also

* Monotone cubic interpolation *
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 betwee ...
- measure of monotonicity in a set of data * Total monotonicity * 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. * {{Order theory Functional analysis Order theory Real analysis Types of functions