Monotonic function

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 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.

If the order ≤ in the definition of monotonicity is replaced by the strict order &lt;, 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
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 &minus;∞) of either a real number, ∞, or &minus;∞.
 * 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:
 * 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
 * FX(x) = Prob(X &le; 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.

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


 * $$(Tu - Tv, u - v) \geq 0 \quad \forall u,v \in X.$$

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 [u1,w1] and [u2,w2] in X&times;X∗,
 * $$(w_1 - w_2, u_1 - u_2) \geq 0.$$

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.

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 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


 * x &le; y implies f(x) &ge; 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 x ≤ y 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 a1 ≤ b1, a2 ≤ b2, ... , an ≤ bn

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.