Dictionary Definition
increasing adj
1 becoming greater or larger; "increasing prices"
[ant: decreasing]
2 music [ant: decreasing]
User Contributed Dictionary
English
Pronunciation

 Rhymes with: iːsɪŋ
Verb
increasing present participle of increase
Extensive Definition
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 nondecreasing), 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 nonincreasing) 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 onetoone
(because for x not equal to y, either x y and so, by monotonicity,
either f(x) f(y), thus f(x) is not equal to f(y)).
The terms nondecreasing and nonincreasing 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 −∞) 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:
 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 ≤ x)
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 nonlinear) 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×X∗ is said to be a
monotone set if for every pair [u1,w1] and [u2,w2] in
X×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 nontotal orders and hence no
additional terminology is introduced for them.
A monotone function is also called isotone, or
orderpreserving. The dual
notion is often called antitone, antimonotone, or orderreversing.
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.
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 such that a1 ≤ b1, a2 ≤ b2, ... , an ≤ bn
one has
 f(a1, ... , an) ≤ f(b1, ... , bn).
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 nonmonotonic 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 featurelogicbased grammars such
as lexicalfunctional
grammar and
headdriven 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
 (Definition 9.31)
External links
 Convergence of a Monotonic Sequence by Anik Debnath and Thomas Roxlo (The Harker School), The Wolfram Demonstrations Project.
increasing in Czech: Monotónní funkce
increasing in German: Monotonie
(Mathematik)
increasing in Spanish: Función monótona
increasing in French: Fonction monotone
increasing in Icelandic: Einhalla fall
increasing in Italian: Funzione monotona
increasing in Hebrew: פונקציה מונוטונית
increasing in Japanese: 単調関数
increasing in Polish: Funkcja monotoniczna
increasing in Portuguese: Função monótona
increasing in Russian: Монотонная функция
increasing in Slovak: Monotónna funkcia
increasing in Slovenian: Monotonost
increasing in Serbian: Монотоност функције
increasing in Chinese: 单调函数