In
mathematics, 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:
* 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 d ...
.
This concept first arose in
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", 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 generalizati ...
, 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 mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, a function
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 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 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
One-to-one or one to one may refer to:
Mathematics and communication
*One-to-one function, also called an injective function
*One-to-one correspondence, also called a bijective function
*One-to-one (communication), the act of an individual comm ...
(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 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 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 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 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 :
* has limits from the right and from the left at every point of its domain;
* has a limit at positive or negative infinity () of either a real number, , or .
* can only have jump discontinuities;
* can only have countably
In mathematics, a Set (mathematics), set is countable if either it is finite set, 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 fro ...
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 of positive numbers and any enumeration of the rational numbers, the monotonically increasing function 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.
Ev ...
of the discrete measure on the rational numbers, where is the weight of .
These properties are the reason why monotonic functions are useful in technical work in analysis. Other important properties of these functions include:
*if is a monotonic function defined on an interval , then is differentiable almost everywhere on ; i.e. the set of numbers in such that is not differentiable in has Lebesgue 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.
*if this set is countable, then is absolutely continuous
*if is a monotonic function defined on an interval 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 o ...
. 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 p ...
, 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.
Ev ...
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 pr ...
'' 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 on its domain, and if T is the range of f, then there is an inverse function 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 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 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 spaces 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 "On ...
/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 (t ...
/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 sets 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 comp ...
. 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 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 embeddings (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 bi ...
''f''(''x'') ≤ ''f''(''y'')) and order isomorphisms ( 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 Feasible region, search space of a problem do ...
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 whe ...
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, 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 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 ...
, 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 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 using ...
* Spearman's rank correlation coefficient - 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 nonemp ...
* Operator monotone function
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