Monotonic function

“Monotonicity” redirects here. For information onmono-tonicity as it pertains to voting systems, see monotonicitycriterion.“Monotonic” redirects here. For other uses, seeMonotone (disambiguation).

In mathematics, a monotonic function (or mono-

Figure 1. A monotonically increasing function. It is strictly in-creasing on the left and right while just non-decreasing in themiddle.

Figure 2. A monotonically decreasing function

tone function) is a function between ordered sets that

Figure 3. A function that is not monotonic

preserves the given order. This concept first arose incalculus, and was later generalized to the more abstractsetting of order theory.

1 Monotonicity in calculus andanalysis

In calculus, a function f defined on a subset of the realnumbers with real values is calledmonotonic if and onlyif it is either entirely increasing or decreasing. It is calledmonotonically increasing (also increasing or nonde-creasing), if for all x and y such that x ≤ y one hasf(x) ≤ f(y) , so f preserves the order (see Figure 1).Likewise, a function is calledmonotonically decreasing(also decreasing or nonincreasing) if, whenever x ≤ y, then f(x) ≥ f(y) , so it reverses the order (see Figure2).If the order ≤ in the definition of monotonicity is re-placed by the strict order < , then one obtains a strongerrequirement. A function with this property is calledstrictly increasing. Again, by inverting the order sym-bol, one finds a corresponding concept called strictly de-creasing. Functions that are strictly increasing or de-creasing are one-to-one (because for x not equal to y ,either x < y or x > y and so, by monotonicity, eitherf(x) < f(y) or f(x) > f(y) , thus f(x) is not equal tof(y) .)

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2 4 MONOTONICITY IN ORDER THEORY

When functions between discrete sets are considered incombinatorics, it is not always obvious that “increasing”and “decreasing” are taken to include the possibility ofrepeating the same value at successive arguments, so onefinds the terms weakly increasing and weakly decreas-ing to stress this possibility.The terms “non-decreasing” and “non-increasing” shouldnot be confused with the (much weaker) negative qualifi-cations “not decreasing” and “not increasing”. For exam-ple, the function of figure 3 first falls, then rises, then fallsagain. It is therefore not decreasing and not increasing,but it is neither non-decreasing nor non-increasing.The term monotonic transformation can also possiblycause some confusion because it refers to a transforma-tion by a strictly increasing function. Notably, this is thecase in economics with respect to the ordinal propertiesof a utility function being preserved across a monotonictransform (see also monotone preferences).[1]

A function f(x) is said to be absolutely monotonic overan interval (a, b) if the derivatives of all orders of f arenonnegative at all points on the interval.

1.1 Some basic applications and results

The following properties are true for a monotonic func-tion f : R → R :

• f has limits from the right and from the left at everypoint of its domain;

• f has a limit at positive or negative infinity ( ±∞ )of either a real number,∞ , or (−∞) .

• f can only have jump discontinuities;

• f can only have countably many discontinuities inits domain.

These properties are the reason why monotonic functionsare useful in technical work in analysis. Two facts aboutthese functions are:

• if f is a monotonic function defined on an intervalI , then f is differentiable almost everywhere on I ,i.e. the set {x : x ∈ I} of numbers x in I such thatf is not differentiable in x has Lebesgue measurezero. In addition, this result cannot be improved tocountable: see Cantor function.

• if f is a monotonic function defined on an interval[a, b] , then f is Riemann integrable.

An important application of monotonic functions is inprobability theory. If X is a random variable, itscumulative distribution function FX(x) = Prob(X ≤ x)is a monotonically increasing function.

A function is unimodal if it is monotonically increasingup to some point (the mode) and then monotonically de-creasing.When f is a strictly monotonic function, then f is injectiveon its domain, and if T is the range of f , then there is aninverse function on T for f .

2 Monotonicity in topology

A map f : X → Y is said to be monotone if each ofits fibers is connected i.e. for each element y in Y the(possibly empty) set f−1(y) is connected.

3 Monotonicity in functional anal-ysis

In functional analysis on a topological vector space X, a(possibly non-linear) operator T : X → X∗ is said to be amonotone operator if

(Tu− Tv, u− v) ≥ 0 ∀u, v ∈ X.

Kachurovskii’s theorem shows that convex functions onBanach spaces have monotonic operators as their deriva-tives.A subset G of X × X∗ is said to be amonotone set if forevery pair [u1,w1] and [u2,w2] in G,

(w1 − w2, u1 − u2) ≥ 0.

G is said to be maximal monotone if it is maximalamong all monotone sets in the sense of set inclusion. Thegraph of a monotone operator G(T) is a monotone set. Amonotone operator is said to be maximal monotone ifits graph is a maximal monotone set.

4 Monotonicity in order theory

Order theory deals with arbitrary partially ordered setsand preordered sets in addition to real numbers. Theabove definition of monotonicity is relevant in these casesas well. However, the terms “increasing” and “decreas-ing” are avoided, since their conventional pictorial repre-sentation does not apply to orders that are not total. Fur-thermore, the strict relations < and > are of little use inmany non-total orders and hence no additional terminol-ogy 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 antitonefunction f satisfies the property

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x ≤ y implies f(x) ≥ f(y),

for all x and y in its domain. The composite of two mono-tone mappings is also monotone.A constant function is both monotone and antitone; con-versely, if f is both monotone and antitone, and if thedomain of f is a lattice, then f must be constant.Monotone functions are central in order theory. They ap-pear in most articles on the subject and examples fromspecial applications are found in these places. Some no-table special monotone functions are order embeddings(functions for which x ≤ y if and only if f(x) ≤ f(y)) andorder isomorphisms (surjective order embeddings).

5 Monotonicity in the context ofsearch algorithms

In the context of search algorithms monotonicity (alsocalled consistency) is a condition applied to heuristicfunctions. A heuristic h(n) is monotonic if, for everynode n and every successor n' of n generated by any ac-tion a, the estimated cost of reaching the goal from n isno greater than the step cost of getting to n' plus theestimated cost of reaching the goal from n' ,

h(n) ≤ c(n, a, n′) + h(n′).

This is a form of triangle inequality, with n, n', and thegoal Gn closest to n. Because every monotonic heuris-tic is also admissible, monotonicity is a stricter require-ment than admissibility. In some heuristic algorithms,such as A*, the algorithm can be considered optimal ifit is monotonic.[2]

6 Boolean functions

In Boolean algebra, a monotonic function is one suchthat for all ai and bi in {0,1}, if a1 ≤ b1, a2 ≤ b2, ...,an ≤ bn (i.e. the Cartesian product {0, 1}n is orderedcoordinatewise), then f(a1, ..., an) ≤ f(b1, ..., bn). Inother words, a Boolean function is monotonic if, for everycombination of inputs, switching one of the inputs fromfalse to true can only cause the output to switch from falseto true and not from true to false. Graphically, this meansthat a Boolean function is monotonic when in its Hassediagram (dual of its Venn diagram), there is no 1 (redvertex) connected to a higher 0 (white vertex).The monotonic Boolean functions are precisely those thatcan be defined by an expression combining the inputs(which may appear more than once) using only the op-erators and and or (in particular not is forbidden). For

instance “at least two of a,b,c hold” is a monotonic func-tion of a,b,c, since it can be written for instance as ((a andb) or (a and c) or (b and c)).The number of such functions on n variables is known asthe Dedekind number of n.

7 See also

• Monotone cubic interpolation

• Pseudo-monotone operator

• Total monotonicity

8 Notes

[1] See the section on Cardinal Versus Ordinal Utility inSimon & Blume (1994).

[2] Conditions for optimality: Admissibility and consistencypg. 94-95 (Russell & Norvig 2010).

9 Bibliography

• Bartle, Robert G. (1976). The elements of real anal-ysis (second ed.).

• Grätzer, George (1971). Lattice theory: first con-cepts and distributive lattices. ISBN 0-7167-0442-0.

• Pemberton, Malcolm; Rau, Nicholas (2001). Math-ematics for economists: an introductory textbook.Manchester University Press. ISBN 0-7190-3341-1.

• Renardy, Michael and Rogers, Robert C. (2004).An introduction to partial differential equations.Texts in Applied Mathematics 13 (Second ed.).New York: Springer-Verlag. p. 356. ISBN 0-387-00444-0.

• Riesz, Frigyes and Béla Szőkefalvi-Nagy (1990).Functional Analysis. Courier Dover Publications.ISBN 978-0-486-66289-3.

• Russell, Stuart J.; Norvig, Peter (2010). ArtificialIntelligence: A Modern Approach (3rd ed.). Up-per Saddle River, New Jersey: Prentice Hall. ISBN978-0-13-604259-4.

• Simon, Carl P.; Blume, Lawrence (April 1994).Mathematics for Economists (first ed.). ISBN 978-0-393-95733-4. (Definition 9.31)

4 10 EXTERNAL LINKS

10 External links• Hazewinkel, Michiel, ed. (2001), “Monotone func-tion”, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

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

• Weisstein, Eric W., “Monotonic Function”,MathWorld.

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