# monotonicity

## Information about monotonicity

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.

## 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 xy 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 xy, 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.

### Some basic applications and results

In calculus, each of the following properties of a function f : R → R implies the next:
• A function f is monotonic;
• 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:
An important application of monotonic functions is in probability theory. If X is a random variable, its cumulative distribution function
FX(x) = Prob(Xx)
is a monotonically increasing function.

A function is unimodal if it is monotonically increasing up to some point (the mode) and then monotonically decreasing.

## Monotonicity in functional analysis

In functional analysis, a (possibly non-linear) operator A from a topological vector space V into its dual space V is said to be a monotone operator if the dual pairing

has constant sign for all x and y in V. If

then A is said to be an increasing operator; if the "≥" inequality is replaced by "≤", then A is said to be a decreasing operator. As a special case, a vector field A on n-dimensional Euclidean space Rn is said to be monotone if the dot product

has constant sign for all x and y in Rn. An increasing (respectively decreasing) vector field may also be said to be outward-pointing (respectively inward-pointing). Kachurovskii's theorem shows that convex functions on Banach spaces have monotonically increasing operators as their derivatives.

## 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.

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

xy 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.

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 xy iff f(x) ≤ f(y)) and order isomorphisms (surjective order embeddings).

## Boolean functions

In Boolean algebra, a monotonic function is one such that for all ai and bi in {0,1} such that a1b1, a2b2, ... , anbn

one has

f(a1, ... , an) ≤ f(b1, ... , bn).

Conjunction, disjunction, tautology, and contradiction are monotonic boolean functions.

## 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 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.

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.

## References

• 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)
Monotone refers to a sound, for example speech or music, that has a single unvaried tone.

Monotone or monotonicity may also refer to:
• Monotone (software), an open source revision control system
• Monotone class theorem, in measure theory

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education.
Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. This article gives a detailed introduction to the field and includes some of the most basic definitions.
Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education.
subset of a set B if A is "contained" inside B. Notice that A and B may coincide. The relationship of one set being a subset of another is called inclusion or containment.
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
non-injective function.]] In mathematics, an injective function is a function which associates distinct arguments to distinct values. More precisely, a function f is said to be injective if it maps distinct x in the domain to distinct y
In mathematical writing, the adjective strict is used to modify technical terms which have multiple meanings. It indicates that the exclusive meaning of the term is to be understood. (More formally, one could say that this is the meaning which implies the other meanings.
In mathematics, the limit of a function is a fundamental concept in analysis. Informally, a function f(x) has a limit L at a point p if the value of f(x) can be made as close to L as desired, by making x close enough to
domain is most often defined as the set of values, D for which a function is defined.[1] A function that has a domain N is said to be a function over N, where N is an arbitrary set.
Jump point redirects here. For the book by Tom Hayes, see Jump Point.
Continuous functions are of utmost importance in mathematics and applications. However, not all functions are continuous.
countable set is a set with the same cardinality (i.e., number of elements) as some subset of the set of natural numbers. The term was originated by Georg Cantor; it stems from the fact that the natural numbers are often called counting numbers.
Jump point redirects here. For the book by Tom Hayes, see Jump Point.
Continuous functions are of utmost importance in mathematics and applications. However, not all functions are continuous.
Analysis has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function.
In algebra, an interval is a set that contains every real number between two indicated numbers and may contain the two numbers themselves. Interval notation is the notation in which permitted values for a variable are expressed as ranging over a certain interval; "" is an
derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
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.e. is a set with measure zero.
In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration.
In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In set theory, there is only one null set, and it is the empty set.
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.
Probability theory is the branch of mathematics concerned with analysis of random phenomena.[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities
A random variable is an abstraction of the intuitive concept of chance into the theoretical domains of mathematics, forming the foundations of probability theory and mathematical statistics.
In probability theory, the cumulative distribution function (CDF), also called probability distribution function or just distribution function,[1] completely describes the probability distribution of a real-valued random variable X.
unimodal if for some value m (the mode), it is monotonically increasing for xm and monotonically decreasing for xm.