In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a monotonic function (or monotone function) is a
function between
ordered sets that preserves or reverses the given
order
Order, ORDER or Orders may refer to:
* A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
...
.
This concept first arose in
calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
, 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 intr ...
.
In calculus and analysis
In
calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
, a function
defined on a
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 a ...
of the
real numbers
In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...
with real values is called ''monotonic'' if it is either entirely non-decreasing, or entirely non-increasing.
[ 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'')][ if for all and such that one has , so preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'')][ if, whenever , then , so it ''reverses'' the order (see Figure 2).
If the order 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 not equal to , either or and so, by monotonicity, either or , thus .)
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 is said to be ''absolutely monotonic'' over an interval if the derivatives of all orders of are ]nonnegative
In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. ...
or all nonpositive 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 is strictly increasing on the range
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
In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings.
* In a Normative economics, normative context, utility refers to a goal or ob ...
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 b ...
). 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 2009 ...
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
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a limit point (also called "accumulation point" or "cluster point") of its do ...
;
*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, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
(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 (for example,
The set of all ...
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.
*If 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 ...
at x^*\in\Bbb R and f'(x^*)>0, then there is a non-degenerate interval ''I'' such that x^*\in I and f 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 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 (38 ...
. 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 ...
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 mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has Lebesgue measure, measure zero. This can be characterized as a set that can be Cover (topology), covered by a countable union of Interval (mathematics), ...
. In addition, this result cannot be improved to countable: see Cantor function
In mathematics, the Cantor function is an example of a function (mathematics), function that is continuous function, continuous, but not absolute continuity, absolutely continuous. It is a notorious Pathological_(mathematics)#Pathological_exampl ...
.
*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 Gö ...
.
An important application of monotonic functions is in probability theory
Probability theory or probability calculus 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 expre ...
. If X is a random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
, 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'' if it is monotonically increasing up to some point (the '' mode'') 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 of its codomain; that is, implies (equivalently by contraposition, impl ...
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.
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
A map f: X \to Y is said to be ''monotone'' if each of its fibers
Fiber (spelled fibre in British English; from ) 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 inco ...
is connected; 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
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 als ...
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 function
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is conve ...
s on Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
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/math> and _2, w_2
The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/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 partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
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. 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 exclusiv ...
relations < and > are of little use in many non-total orders and hence no additional terminology is introduced for them.
Letting \leq denote the partial order relation of any partially ordered set, a ''monotone'' function, also called ''isotone'', or ', satisfies the property
x \leq y \implies f(x) \leq f(y)
for all and in its domain. The composite of two monotone mappings is also monotone.
The dual notion is often called ''antitone'', ''anti-monotone'', or ''order-reversing''. Hence, an antitone function satisfies the property
x \leq y \implies f(y) \leq f(x),
for all and in its domain.
A constant function
In mathematics, a constant function is a function whose (output) value is the same for every input value.
Basic properties
As a real-valued function of a real-valued argument, a constant function has the general form or just For example, ...
is both monotone and antitone; conversely, if is both monotone and antitone, and if the domain of is a lattice, then 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 stri ...
s (functions for which x \leq y if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
f(x) \leq 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 co ...
s (surjective
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
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 Feasible region, search space of a problem do ...
s monotonicity (also called consistency) is a condition applied to heuristic function
A heuristic or heuristic technique (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless ...
s. A heuristic h(n) is monotonic if, for every node and every successor of generated by any action , the estimated cost of reaching the goal from is no greater than the step cost of getting to plus the estimated cost of reaching the goal from ,
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 Degeneracy (mathematics)#T ...
, with , , and the goal closest to . Because every monotonic heuristic is also admissible, monotonicity is a stricter requirement than admissibility. Some heuristic algorithm
A heuristic or heuristic technique (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless ...
s such as A* can be proven optimal provided that the heuristic they use is monotonic.
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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
, a monotonic function is one such that for all and in , if , , ..., (i.e. the Cartesian product 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 -ary Boolean function is monotonic when its representation as an -cube labelled with truth values has no upward edge from ''true'' to ''false''. (This labelled Hasse diagram is the dual of the function's labelled Venn diagram
A Venn diagram is a widely used diagram style that shows the logical relation between set (mathematics), sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple ...
, 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 , , hold" is a monotonic function of , , , since it can be written for instance as (( and ) or ( and ) or ( and )).
The number of such functions on variables is known as the Dedekind number of .
SAT solving
In computer science and formal methods, a SAT solver is a computer program which aims to solve the Boolean satisfiability problem (SAT). On input a formula over Boolean data type, Boolean variables, such as "(''x'' or ''y'') and (''x'' or not ''y'' ...
, generally an NP-hard
In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assumi ...
task, can be achieved efficiently when all involved functions and predicates are monotonic and Boolean.[
]
See also
* Monotone cubic interpolation
* Pseudo-monotone operator
* Spearman's rank correlation coefficient
In statistics, Spearman's rank correlation coefficient or Spearman's ''ρ'' is a number ranging from -1 to 1 that indicates how strongly two sets of ranks are correlated. It could be used in a situation where one only has ranked data, such as a ...
- measure of monotonicity in a set of data
* Total monotonicity
* Cyclical monotonicity
In mathematics, cyclical monotonicity is a generalization of the notion of monotonicity to the case of vector-valued function.
Definition
Let \langle\cdot,\cdot\rangle denote the inner product on an inner product space X and let U be a nonempty ...
* Operator monotone function
* Monotone set function
In mathematics, a submodular set function (also known as a submodular function) is a set function that, informally, describes the relationship between a set of inputs and an output, where adding more of one input has a decreasing additional benefi ...
* Absolutely and completely monotonic functions and sequences In mathematics, the notions of an absolutely monotonic function and a completely monotonic function are two very closely related concepts. Both imply very strong monotonicity properties. Both types of functions have derivatives of all orders. In t ...
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 Open source, open-source collection of Interactive computing, interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown t ...
.
*
{{Order theory
Functional analysis
Order theory
Real analysis
Types of functions