A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right).

A monotonically decreasing function.

A function that is not monotonic.

In mathematics, a monotonic function (or monotone function) is a function which preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. Voters at the voting booths in the US in 1945 Voting systems are methods (algorithms) for groups of people to select one or more options from many, taking into account the individual preferences of the group members. ... A voting system is monotonic if it satisfies the monotonicity criterion, given below. ... Monotone refers to a sound, for example speech or music, that has a single unvaried tone. ... Image File history File links Size of this preview: 622 × 599 pixelsFull resolution (1282 × 1235 pixel, file size: 29 KB, MIME type: image/png) <?xml version=1. ... Image File history File links Size of this preview: 622 × 599 pixelsFull resolution (1282 × 1235 pixel, file size: 29 KB, MIME type: image/png) <?xml version=1. ... Image File history File links Size of this preview: 622 × 599 pixelsFull resolution (1282 × 1235 pixel, file size: 30 KB, MIME type: image/png) <?xml version=1. ... Image File history File links Size of this preview: 622 × 599 pixelsFull resolution (1282 × 1235 pixel, file size: 30 KB, MIME type: image/png) <?xml version=1. ... Image File history File links Size of this preview: 622 × 599 pixelsFull resolution (1282 × 1235 pixel, file size: 39 KB, MIME type: image/png) <?xml version=1. ... Image File history File links Size of this preview: 622 × 599 pixelsFull resolution (1282 × 1235 pixel, file size: 39 KB, MIME type: image/png) <?xml version=1. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... This article is about functions in mathematics. ... For other uses, see Calculus (disambiguation). ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...

Monotonicity in calculus and analysis

In calculus, a function f defined on a subset of the real numbers with real values is called monotonic (also monotonically increasing, increasing, or non-decreasing), if for all x and y such that x ≤ y one has f(x) ≤ f(y), so f preserves the order. Likewise, a function is called monotonically decreasing (also decreasing, or non-increasing) if, whenever x ≤ y, then f(x) ≥ f(y), so it reverses the order. For other uses, see Calculus (disambiguation). ... Superset redirects here. ... Please refer to Real vs. ...

If the order ≤ in the definition of monotonicity is replaced by the strict order <, then one obtains a stronger requirement. A function with this property is called strictly increasing. Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing. Functions that are strictly increasing or decreasing are one-to-one (because for x not equal to y, either x < y or x > y and so, by monotonicity, either f(x) < f(y) or f(x) > f(y), thus f(x) is not equal to f(y)). You may be looking for an Injective function, in which (f(a)=f(b)) -> a=b, or a Bijection function, which is both injective and surjective (ie. ...

The terms non-decreasing and non-increasing avoid any possible confusion with strictly increasing and strictly decreasing, respectively, see also strict. In mathematical writing, the adjective strict is used in to modify technical terms which have multiple meanings. ...

Some basic applications and results

The following properties are true for a monotonic function f : R → R:

• f has limits from the right and from the left at every point of its domain;
• f has a limit at infinity (either ∞ or −∞) of either a real number, ∞, or −∞.
• f can only have jump discontinuities;
• f can only have countably many discontinuities in its domain.

These properties are the reason why monotonic functions are useful in technical work in analysis. Two facts about these functions are: In mathematics, the limit of a function is a fundamental concept in analysis. ... In mathematics, the domain of a function is the set of all input values to the function. ... Continuous functions are of utmost importance in mathematics and applications. ... In mathematics the term countable set is used to describe the size of a set, e. ... Continuous functions are of utmost importance in mathematics and applications. ... Analysis has its beginnings in the rigorous formulation of calculus. ...

• if f is a monotonic function defined on an interval I, then f is differentiable almost everywhere on I, i.e. the set of numbers x in I such that f is not differentiable in x has Lebesgue measure zero.
• if f is a monotonic function defined on an interval [a, b], then f is Riemann integrable.

An important application of monotonic functions is in probability theory. If X is a random variable, its cumulative distribution function In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... For other uses, see Derivative (disambiguation). ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ... Let μ be a measure on a sigma algebra Σ of subsets of a set X. An element A in Σ is said to have measure zero if μ(A)=0. ... In the branch of mathematics known as real analysis, the Riemann integral â„›, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...

F_X(x) = Prob(X ≤ x)

is a monotonically increasing function.

A function is unimodal if it is monotonically increasing up to some point (the mode) and then monotonically decreasing. In mathematics, a function f(x) between two ordered sets is unimodal if for some value m (the mode), it is monotonically increasing for x ≤ m and monotonically decreasing for x ≥ m. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ...

Monotonicity in functional analysis

In functional analysis on a topological vector space X, a (possibly non-linear) operator T:X→X^∗ is said to be a monotone operator if Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ...

Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives. In mathematics, convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have Convex function on an interval. ... In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

A subset G of X×X^∗ is said to be a monotone set if for every pair [u₁,w₁] and [u₂,w₂] in X×X^∗,

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.

Monotonicity in order theory

In order theory, one does not restrict to real numbers, but one is concerned with arbitrary partially ordered sets or even with preordered sets. In these cases, the above definition of monotonicity is relevant as well. However, the terms "increasing" and "decreasing" are avoided, since they lose their appealing pictorial motivation as soon as one deals with orders that are not total. Furthermore, the strict relations < and > are of little use in many non-total orders and hence no additional terminology is introduced for them. In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ... In mathematics, especially in order theory, preorders are certain kinds of binary relations that are closely related to partially ordered sets. ... In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ... A strict weak ordering is a binary relation that defines an equivalence relation and has the properties stated below. ...

A monotone function is also called isotone, or order-preserving. The dual notion is often called antitone, anti-monotone, or order-reversing. Hence, an antitone function f satisfies the property In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop. ...

x ≤ y implies f(x) ≥ f(y),

for all x and y in its domain. It is easy to see that the composite of two monotone mappings is also monotone.

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. In mathematics a constant function is a function whose values do not vary and thus are constant. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ...

Monotone functions are central in order theory. They appear in most articles on the subject and examples from special applications are to be found in these places. Some notable special monotone functions are order embeddings (functions for which x ≤ y iff f(x) ≤ f(y)) and order isomorphisms (surjective order embeddings). In mathematical order theory, an order-embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... 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. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...

Boolean functions

In Boolean algebra, a monotonic function is one such that for all a_i and b_i in {0,1} such that a₁ ≤ b₁, a₂ ≤ b₂, ... , a_n ≤ b_n Boolean algebra is the finitary algebra of two values. ...

one has

f(a₁, ... , a_n) ≤ f(b₁, ... , b_n).

Conjunction, disjunction, tautology, and contradiction are monotonic boolean functions. Logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ... Look up tautology in Wiktionary, the free dictionary. ... Broadly speaking, a contradiction is an incompatibility between two or more statements, ideas, or actions. ...

Monotonic logic

Monotonicity of entailment is a property of many logic systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions. Any true statement in a logic with this property, will continue to be true even after adding any new axioms. Logics with this property may be called monotonic in order to differentiate them from non-monotonic logic. Monotonicity of entailment - Wikipedia /**/ @import /w/skins-1. ... Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... This article is about a logical statement. ... Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... A non-monotonic logic is a formal logic whose consequence relation is not monotonic. ...

Monotonicity in linguistic theory

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Formal theories of grammar attempt to characterize the set of possible grammatical and ungrammatical sentences of any given human language, as well as the commonalities among languages. Most such theories do this by a set of rules that apply to grammatical atoms, such as the features that a given lexical item may have. So, for example, if two daughters of a node in a syntactic tree have features [E, F, G] and [F, G, H] respectively as in "John" (animate and third person and singular) and "sleeps" (third person, singular and present tense), then when their features unify at the mother node, that mother node will have the features [E, F, G, H] (animate third person singular present tense). Thus, the properties of higher nodes in a tree are simply the union of the set of features of all daughter nodes. Such questions are highly relevant in feature-logic-based grammars such as lexical-functional grammar and head-driven phrase structure grammar. Image File history File links Question_book-3. ... Lexical functional grammar (LFG) is a reaction to the direction research in the area of transformational grammar began to take in the 1970s. ... The Head-driven phrase structure grammar (HPSG) is a non-derivational generative grammar theory developed by Carl Pollard and Ivan Sag (1985). ...

Some constructions in natural languages also appear to have non monotonic properties. For example, gerund phrases like "John's singing a song was unexpected" are considered a kind of mixed category in that they have properties of both nouns and verbs. If we assume that parts of speech are not primitives but composed of features such as [±N] and [±V], and nouns are [+N, −V] and verbs [−N, +V], then the properties of gerunds appear to shift as phrases are combined in syntax, resulting in the apparent paradox that gerunds are both plus and minus in both [N] and [V] features. The properties of such mixed categories are still poorly understood.

See also

• Monotone cubic interpolation
• Pseudo-monotone operator

In the mathematical subfield of numerical analysis, monotone cubic interpolation is a variant of cubic interpolation that preserves monotonicity of the data set being interpolated. ...

References

• Bartle, Robert G. (1976). The elements of real analysis, second edition. 
• Grätzer, George (1971). Lattice theory: first concepts and distributive lattices. ISBN 0716704420. 
• Pemberton, Malcolm; Rau, Nicholas (2001). Mathematics for economists: an introductory textbook. Manchester University Press. ISBN 0719033411. 
• Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations, Second edition, Texts in Applied Mathematics 13, New York: Springer-Verlag, 356. ISBN 0-387-00444-0.  (Definition 9.31)

External links

• Convergence of a Monotonic Sequence by Anik Debnath and Thomas Roxlo (The Harker School), The Wolfram Demonstrations Project.