Monotonic function

This article is about a mathematical concept. For the related property of voting systems, see monotonicity criterion. For the modern Greek writing system, see monotonic orthography.

In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. These functions first arose in calculus and were later generalized to the more abstract setting of order theory. Although the concepts generally agree, the two disciplines have developed a slightly different terminology. While in calculus, one often talks about functions being monotonically increasing and monotonically decreasing, order theory prefers the terms monotone and antitone or order-preserving and order-reversing, respectively.

General definition

Let

f: P → Q

be a function between two sets P and Q, where each set carries a partial order (both of which we denote by ≤). In calculus one focuses on functions between subsets of the reals and the order ≤ is just the usual ordering on real numbers, but this is not essential for this definition.

The function f is monotone if, whenever x ≤ y, then f(x) ≤ f(y). Stated differently, a monotone function is one that preserves the order.

Monotonicity in calculus and analysis

In calculus, there is often no need to call upon the abstract methods of order theory. As already noted, functions are usually mappings between (subsets of) real numbers, ordered in the natural way.

Inspired by the shape of the graph of a monotone function on the reals, such functions are also called monotonically increasing (or "non-decreasing" or, less precisely, just "increasing"). Likewise, a function is called monotonically decreasing (or "non-increasing" or "decreasing") if, whenever x ≤ y, then f(x) ≥ f(y), i.e., if 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 $a<b$ implies $a\neq b$).

The terms non-decreasing and non-increasing avoid any possible confusion with strictly increasing and strictly decreasing, respectively, see also strict.

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 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

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