Monotonic function


In mathematics, a monotonic function is a function between ordered sets that preserves or reverses 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 defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease.
A function is called monotonically increasing, if for all and such that one has , so preserves the order. Likewise, a function is called monotonically decreasing if, whenever, then, 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. A function may be called strictly monotone if it is either strictly increasing or strictly decreasing. Functions that are strictly monotone are one-to-one
If it is not clear that "increasing" and "decreasing" are taken to include the possibility of repeating the same value at successive arguments, one may use the terms weakly monotone, weakly increasing and weakly decreasing to stress this possibility.
The terms "non-decreasing" and "non-increasing" should not be confused with the negative qualifications "not decreasing" and "not increasing". For example, the function of figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing.
A function is said to be absolutely monotonic over an interval if the derivatives of all orders of are nonnegative or all nonpositive at all points on the interval.

Inverse of function

A function that is monotonic, but not strictly monotonic, and thus constant on an interval, doesn't have an inverse. This is because in order for a function to have an inverse there needs to be a one-to-one mapping from the range to the domain of the function, since a monotonic function has some values that are constant in its domain, this means that there would be more than one value in the range that maps to this constant value.
However, a function y=g that is strictly monotonic, has an inverse function such that x=h because there is guaranteed to always be a one-to-one mapping from range to domain of the function. Also, a function can be said to be strictly monotonic on a range of values, and thus have an inverse on that range of value. For example, if y=g is strictly monotonic on the range , then it has an inverse x=h on the range , but we cannot say the entire range of the function has an inverse.
Note, some textbooks mistakenly state that an inverse exists for a monotonic function, when they really mean that an inverse exists for a strictly monotonic function.

Monotonic transformation

The term monotonic transformation can also possibly cause some confusion because it refers to a transformation by a strictly increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform. In this context, what we are calling a "monotonic transformation" is, more accurately, called a "positive monotonic transformation", in order to distinguish it from a “negative monotonic transformation,” which reverses the order of the numbers.

Some basic applications and results

The following properties are true for a monotonic function :
These properties are the reason why monotonic functions are useful in technical work in analysis. Some more facts about these functions are:
An important application of monotonic functions is in probability theory. If is a random variable, its cumulative distribution function is a monotonically increasing function.
A function is unimodal if it is monotonically increasing up to some point and then monotonically decreasing.
When is a strictly monotonic function, then is injective on its domain, and if is the range of, then there is an inverse function on for.
In contrast, each constant function is monotonic, but not injective, and hence cannot have an inverse.

Monotonicity in topology

A map ' is said to be monotone if each of its fibers is connected i.e. for each element ' in the set is connected.

Monotonicity in functional analysis

In functional analysis on a topological vector space, a operator is said to be a monotone operator if
Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives.
A subset of is said to be a monotone set if for every pair and in,
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 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

Order theory deals with arbitrary partially ordered sets and preordered sets as a generalization of real numbers. The above definition of monotonicity is relevant in these cases as well. However, the terms "increasing" and "decreasing" are avoided, since their conventional pictorial representation does not apply to 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.
Letting ≤ denote the partial order relation of any partially ordered set, a monotone function, also called isotone, or , satisfies the property
for all x and y in its domain. The composite of two monotone mappings is also monotone.
The dual notion is often called antitone, anti-monotone, or order-reversing. Hence, an antitone function f satisfies the property
for all x and y in its domain.
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 found in these places. Some notable special monotone functions are order embeddings ≤ f) and order isomorphisms.

Monotonicity in the context of search algorithms

In the context of search algorithms monotonicity is a condition applied to heuristic functions. A heuristic h is monotonic if, for every node n and every successor n of n generated by any action a, the estimated cost of reaching the goal from n is no greater than the step cost of getting to n' plus the estimated cost of reaching the goal from n' ,
This is a form of triangle inequality, with n, n, and the goal Gn closest to n. Because every monotonic heuristic is also admissible, monotonicity is a stricter requirement than admissibility. Some heuristic algorithms such as A* can be proven optimal provided that the heuristic they use is monotonic.

Boolean functions

In Boolean algebra, a monotonic function is one such that for all ai and bi in, if a1 ≤ b1, a2 ≤ b2,..., an ≤ bn, then f ≤ f. In other words, a Boolean function is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. Graphically, this means that an n-ary Boolean function is monotonic when its representation as an n-cube labelled with truth values has no upward edge from true to false.
The monotonic Boolean functions are precisely those that can be defined by an expression combining the inputs using only the operators and and or. For instance "at least two of a,b,c hold" is a monotonic function of a,b,c, since it can be written for instance as or or ).
The number of such functions on n variables is known as the Dedekind number of n.