Monotonic function



In
In calculus and analysis
In calculus, a function defined on a
A function is termed monotonically increasing (also increasing or non-decreasing)[3] if for all and such that one has , so preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing or non-increasing)[3] if, whenever , then , so it reverses the order (see Figure 2).
If the order in the definition of monotonicity is replaced by the strict order , one obtains a stronger requirement. A function with this property is called strictly increasing (also increasing).
To avoid ambiguity, the terms weakly monotone, weakly increasing and weakly decreasing are often used to refer to non-strict monotonicity.
The terms "non-decreasing" and "non-increasing" should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing". For example, the non-monotonic function shown in 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
Inverse of function
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. For example, if is strictly increasing on the range , then it has an inverse on the range .
The term monotonic is sometimes used in place of strictly monotonic, so a source may state that all monotonic functions are invertible when they really mean that all strictly monotonic functions are invertible.[citation needed]
Monotonic transformation
The term monotonic transformation (or monotone transformation) may also cause 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
Some basic applications and results


The following properties are true for a monotonic function :
- has limits from the right and from the left at every point of its domain;
- has a limit at positive or negative infinity () of either a real number, , or .
- can only have jump discontinuities;
- can only have summable sequenceof positive numbers and any enumeration of the rational numbers, the monotonically increasing function is continuous exactly at every irrational number (cf. picture). It is the cumulative distribution function of the discrete measure on the rational numbers, where is the weight of .
- If is differentiableat and , then there is a non-degenerate interval I such that and is increasing on I. As a partial converse, if f is differentiable and increasing on an interval, I, then its derivative is positive at every point in I.
These properties are the reason why monotonic functions are useful in technical work in analysis. Other important properties of these functions include:
- if is a monotonic function defined on an interval , then is differentiable almost everywhere on ; i.e. the set of numbers in such that is not differentiable in has measure zero. In addition, this result cannot be improved to countable: see Cantor function.
- if this set is countable, then is absolutely continuous
- if is a monotonic function defined on an interval , then is Riemann integrable.
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
When is a strictly monotonic function, then is
The graphic shows six monotonic functions. Their simplest forms are shown in the plot area and the expressions used to create them are shown on the y-axis.
In topology
A map is said to be monotone if each of its
In functional analysis
In functional analysis on a topological vector space , a (possibly non-linear) 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.
In order theory
Order theory deals with arbitrary
Letting denote the partial order relation of any partially ordered set, a monotone function, also called isotone, or order-preserving, 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 (functions for which if and only if and
In the context of search algorithms
In the context of
This is a form of
In Boolean functions
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In
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). 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 ((a and b) or (a and c) or (b and c)).
The number of such functions on n variables is known as the Dedekind number of n.
See also
- Monotone cubic interpolation
- Pseudo-monotone operator
- Spearman's rank correlation coefficient - measure of monotonicity in a set of data
- Total monotonicity
- Cyclical monotonicity
- Operator monotone function
- Monotone set function
- Absolutely and completely monotonic functions and sequences
Notes
- ^ Clapham, Christopher; Nicholson, James (2014). Oxford Concise Dictionary of Mathematics (5th ed.). Oxford University Press.
- ^ a b Stover, Christopher. "Monotonic Function". Wolfram MathWorld. Retrieved 2018-01-29.
- ^ a b c d e "Monotone function". Encyclopedia of Mathematics. Retrieved 2018-01-29.
- ^ ISBN 0-914098-89-6.
- ^ See the section on Cardinal Versus Ordinal Utility in Simon & Blume (1994).
- ISBN 9780393934243.
- ^ if its domain has more than one element
- ^ Conditions for optimality: Admissibility and consistency pg. 94–95 (Russell & Norvig 2010).
- from the original on Dec 11, 2023.
Bibliography
- Bartle, Robert G. (1976). The elements of real analysis (second ed.).
- Grätzer, George (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman. ISBN 0-7167-0442-0.
- Pemberton, Malcolm; Rau, Nicholas (2001). Mathematics for economists: an introductory textbook. Manchester University Press. ISBN 0-7190-3341-1.
- Renardy, Michael & 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 & Béla Szőkefalvi-Nagy (1990). Functional Analysis. Courier Dover Publications. ISBN 978-0-486-66289-3.
- Russell, Stuart J.; Norvig, Peter (2010). Artificial Intelligence: A Modern Approach (3rd ed.). Upper Saddle River, New Jersey: Prentice Hall. ISBN 978-0-13-604259-4.
- Simon, Carl P.; Blume, Lawrence (April 1994). Mathematics for Economists (first ed.). Norton. ISBN 978-0-393-95733-4. (Definition 9.31)
External links
- "Monotone function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Convergence of a Monotonic Sequence by Anik Debnath and Thomas Roxlo (The Harker School), Wolfram Demonstrations Project.
- Weisstein, Eric W. "Monotonic Function". MathWorld.