Monotonic function

 ( Functional Analysis ) 

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 $f$ is said to be absolutely monotonic over an interval $\backslash left(a,\; b\backslash right)$ if the derivatives of all orders of $f$ are nonnegative or all nonpositive at all points on the interval.

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 $y\; =\; g(x)$ is strictly increasing on the range $a,$, then it has an inverse $x\; =\; h(y)$ on the range $g(a),$.

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.

• $f$ has limits from the right and from the left at every point of its domain;
• $f$ has a limit at positive or negative infinity ($\backslash pm\backslash infty$) of either a real number, $\backslash infty$, or $-\backslash infty$.
• $f$ can only have jump discontinuities;
• $f$ can only have countably many discontinuities in its domain. The discontinuities, however, do not necessarily consist of isolated points and may even be dense in an interval ( a, b). For example, for any summable sequence (a\_i) of positive numbers and any enumeration $(q\_i)$ of the , the monotonically increasing function $$f(x)=\backslash sum\_\{q\_i\backslash leq\; x\}\; a\_i$$ is continuous exactly at every irrational number (cf. picture). It is the cumulative distribution function of the discrete measure on the rational numbers, where $a\_i$ is the weight of $q\_i$.
• If $f$ is differentiable at $x^*\backslash in\backslash Bbb\; R$ and $f\text{'}(x^*)>0$, then there is a non-degenerate interval I such that $x^*\backslash in\; I$ and $f$ 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 $f$ is a monotonic function defined on an interval $I$, then $f$ is derivative almost everywhere on $I$; i.e. the set of numbers $x$ in $I$ such that $f$ is not differentiable in $x$ has Lebesgue measure measure zero. In addition, this result cannot be improved to countable: see Cantor function.
• if this set is countable, then $f$ is absolutely continuous
• if $f$ is a monotonic function defined on an interval $\backslash lefta,$, then $f$ is Riemann integral.

An important application of monotonic functions is in probability theory. If $X$ is a random variable, its cumulative distribution function $F\_X\backslash !\backslash left(x\backslash right)\; =\; \backslash text\{Prob\}\backslash !\backslash left(X\; \backslash leq\; x\backslash right)$ is a monotonically increasing function.

A function is unimodal if it is monotonically increasing up to some point (the mode) and then monotonically decreasing.

When $f$ is a strictly monotonic function, then $f$ is injective on its domain, and if $T$ is the range of $f$, then there is an inverse function on $T$ for $f$. In contrast, each constant function is monotonic, but not injective,if its domain has more than one element and hence cannot have an inverse.

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.

$$(Tu\; -\; Tv,\; u\; -\; v)\; \backslash geq\; 0\; \backslash quad\; \backslash forall\; u,v\; \backslash in\; X.$$ Kachurovskii's theorem shows that on have monotonic operators as their derivatives.

A subset $G$ of $X\; \backslash times\; X^*$ is said to be a monotone set if for every pair $u\_1,$ and $u\_2,$ in $G$,

$$(w\_1\; -\; w\_2,\; u\_1\; -\; u\_2)\; \backslash geq\; 0.$$ $G$ 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.

Letting $\backslash leq$ denote the partial order relation of any partially ordered set, a monotone function, also called isotone, or , satisfies the property

$$x\; \backslash leq\; y\; \backslash implies\; f(x)\; \backslash leq\; f(y)$$

for all and 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 satisfies the property

$$x\; \backslash leq\; y\; \backslash implies\; f(y)\; \backslash leq\; f(x),$$

for all and in its domain.

A constant function is both monotone and antitone; conversely, if is both monotone and antitone, and if the domain of is a lattice, then 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 (functions for which $x\; \backslash leq\; y$ if and only if $f(x)\; \backslash leq\; f(y))$ and order isomorphisms (surjective order embeddings).

$$h(n)\; \backslash leq\; c\backslash left(n,\; a,\; n\text{'}\backslash right)\; +\; h\backslash left(n\text{'}\backslash right)\; .$$

This is a form of triangle inequality, with , , and the goal closest to . 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.Conditions for optimality: Admissibility and consistency pg. 94–95 .

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 negation is forbidden). For instance "at least two of , , hold" (the ternary majority function) is a monotonic function of , , , since it can be written for instance as (( and ) or ( and ) or ( and )).

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

SAT solving, generally an NP-hard task, can be achieved efficiently when all involved functions and predicates are monotonic and Boolean.

• 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

• George Grätzer (1971). 9780716704423, W. H. Freeman. ISBN 9780716704423
• Malcolm Pemberton (2025). 9780719033414, Manchester University Press. ISBN 9780719033414
• (2025). 9780387004440, Springer-Verlag. ISBN 9780387004440
• (1990). 9780486662893, Courier Dover Publications. ISBN 9780486662893
• (2025). 9780136042594, Prentice Hall. ISBN 9780136042594
• (1994). 9780393957334, Norton. ISBN 9780393957334
  (Definition 9.31)

• Convergence of a Monotonic Sequence by Anik Debnath and Thomas Roxlo (The Harker School), Wolfram Demonstrations Project.