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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying

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. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Mathematics, the real numbers may be described informally in several different ways 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.

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)).

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 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 −∞. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined
• 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. Analysis has its beginnings in the rigorous formulation of Calculus. Two facts about these functions are:

• if f is a monotonic function defined on an interval I, then f is differentiable almost everywhere on I, i. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change 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 e. the set of numbers x in I such that f is not differentiable in x has Lebesgue measure zero. In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to In Mathematics, a null set is a set that is negligible in some sense.
• if f is a monotonic function defined on an interval [a, b], 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

An important application of monotonic functions is in probability theory. Probability theory is the branch of Mathematics concerned with analysis of random phenomena If X is a random variable, its cumulative distribution function

F_X(x) = Prob(X ≤ x)

is a monotonically increasing function. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way In Probability theory and Statistics, the cumulative distribution function (CDF, also probability distribution function or just distribution 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 In Statistics, the mode is the value that occurs the most frequently in a Data set or a Probability distribution.

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

Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives. For functional analysis as used in psychology see the Functional analysis (psychology article In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. In Mathematics, Kachurovskii's theorem is a theorem relating the convexity of a function on a Banach space to the monotonicity of its Fréchet In Mathematics, a real-valued function f defined on an interval (or on any Convex subset of some Vector space) is called convex In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) 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 Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics, especially in Order theory, preorders are Binary relations that satisfy certain conditions 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. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation 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) formalizes the intuitive concept of an ordering sequencing or arrangement

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

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. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements'

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 ↔ In the mathematical field of Order theory an order isomorphism is a special kind of Monotone function that constitutes a suitable notion of Isomorphism In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every

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, . Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s . . , a_n ≤ b_n

one has

f(a₁, . . . , a_n) ≤ f(b₁, . . . , b_n).

Conjunction, disjunction, tautology, and contradiction are monotonic boolean functions. In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield

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. Monotonicity of Entailment is a property of many Logical systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist Any true statement in a logic with this property, will continue to be true even after adding any new axioms. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Logics with this property may be called monotonic in order to differentiate them from non-monotonic logic. Logic is the study of the principles of valid demonstration and Inference. A non-monotonic logic is a Formal logic whose consequence relation is not monotonic.

Monotonicity in linguistic theory

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. Lexical functional grammar (LFG is a Grammar framework in Theoretical linguistics, a variety of Generative grammar. Head-driven phrase structure grammar (HPSG is a highly lexicalized non-derivational Generative grammar theory developed by Carl Pollard and Ivan Sag

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

References

• Bartle, Robert G. In the mathematical subfield of Numerical analysis, monotone cubic interpolation is a variant of Cubic interpolation that preserves monotonicity In Mathematics, a pseudo-monotone operator from a reflexive Banach space into its Continuous dual space is one that is in some sense almost (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.