Monotonic function
[1][2][3] This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
in the definition of monotonicity is replaced by the strict order
[3][4] A function with either property is called strictly monotone.
The terms "non-decreasing" and "non-increasing" should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing".
All strictly monotonic functions are invertible because they are guaranteed to have a one-to-one mapping from their range to their domain.
However, functions that are only weakly monotone are not invertible because they are constant on some interval (and therefore are not one-to-one).
A function may be strictly monotonic over a limited a range of values and thus have an inverse on that range even though it is not strictly monotonic everywhere.
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).
[5] In this context, the term "monotonic transformation" refers to a positive monotonic transformation and is intended to distinguish it from a "negative monotonic transformation," which reverses the order of the numbers.
: These properties are the reason why monotonic functions are useful in technical work in analysis.
is a random variable, its cumulative distribution function
In contrast, each constant function is monotonic, but not injective,[7] and hence cannot have an inverse.
In functional analysis on a topological vector space
Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives.
Order theory deals with arbitrary partially ordered sets and preordered sets as a generalization of real numbers.
However, the terms "increasing" and "decreasing" are avoided, since their conventional pictorial representation does not apply to orders that are not total.
are of little use in many non-total orders and hence no additional terminology is introduced for them.
denote the partial order relation of any partially ordered set, a monotone function, also called isotone, or order-preserving, satisfies the property
The dual notion is often called antitone, anti-monotone, or order-reversing.
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.
In the context of search algorithms monotonicity (also called consistency) is a condition applied to heuristic functions.
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.
[8] In Boolean algebra, a monotonic function is one such that for all ai and bi in {0,1}, if a1 ≤ b1, a2 ≤ b2, ..., an ≤ bn (i.e. the Cartesian product {0, 1}n is ordered coordinatewise), then f(a1, ..., an) ≤ f(b1, ..., bn).
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 (which may appear more than once) using only the operators and and or (in particular not is forbidden).
The number of such functions on n variables is known as the Dedekind number of n. SAT solving, generally an NP-hard task, can be achieved efficiently when all involved functions and predicates are monotonic and Boolean.
