Monotonic function

Figure 1. A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right).
Figure 2. A monotonically decreasing function.
Figure 3. A function that is not monotonic.

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.

Contents

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 (see Figure 1). 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 (see Figure 2).

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" are meant to avoid confusion with "strictly increasing" respectively "strictly decreasing", but should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing"; see also strict. When functions between discrete sets are considered in combinatorics, it is not always obvious that "increasing" and "decreasing" are taken to include the possibility of repeating the same value at successive arguments, so one finds the terms weakly increasing and weakly decreasing to stress this possibility.

The term monotonic transformation can also possibly cause some confusion because it refers to a transformation by a strictly increasing function. Notably, this is the case in Economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform (see also monotone preferences).[1]

Some basic applications and results

The following properties are true for a monotonic function f : R → R:

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(X ≤ 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 × X∗ is said to be a monotone set if for every pair [u1,w1] and [u2,w2] in G,

(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 < 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

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 if and only if f(x) ≤ f(y)) and order isomorphisms (surjective order embeddings).

Boolean functions

The free distributive lattices of monotonic Boolean functions on 0, 1, 2, and 3 arguments.

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

it is true that

f(a1, ... , an) ≤ f(b1, ... , bn).

The monotonic Boolean functions are precisely those which can be defined as a composition of ands (conjunction) and ors (disjunction), but no nots (negation).

The number of such functions on n variables is known as the Dedekind number of n.

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 continues to be true, even after adding new axioms. Logics with this property may be called monotonic, to differentiate them from non-monotonic logic.

See also

Notes

  1. ↑ See the section on Cardinal Versus Ordinal Utility in (Simon and Blume, 1994).

References

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