Monotonic function explained
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.[1] [2] [3] This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
In calculus and analysis
In calculus, a function
defined on a
subset of the real numbers with real values is called
monotonic if it is either entirely non-decreasing, or entirely non-increasing. 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 termed monotonically increasing (also increasing or non-decreasing) if for all
and
such that
one has
f\left(x\right)\leqf\left(y\right)
, so
preserves the order (see Figure 1). Likewise, a function is called
monotonically decreasing (also
decreasing or
non-increasing) if, whenever
, then
f\left(x\right)\geqf\left(y\right)
, 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).
[4] Again, by inverting the order symbol, one finds a corresponding concept called
strictly decreasing (also
decreasing). A function with either property is called
strictly monotone. Functions that are strictly monotone are
one-to-one (because for
not equal to
, either
or
and so, by monotonicity, either
f\left(x\right)<f\left(y\right)
or
f\left(x\right)>f\left(y\right)
, thus
f\left(x\right) ≠ f\left(y\right)
.)
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 nonnegative or all nonpositive at all points on the interval.
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.
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 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.[6]
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
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 of 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
.
is
differentiable at
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:
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
Lebesgue measure zero. In addition, this result cannot be improved to countable: see
Cantor function.
- if this set is countable, then
is absolutely continuous
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 FX\left(x\right)=Prob\left(X\leqx\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
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,
[7] 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.
In topology
A map
is said to be
monotone if each of its fibers is
connected; that is, for each element
the (possibly empty) set
is a connected
subspace of
In functional analysis
, 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 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 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
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 order embeddings (functions for which
if and only if
and
order isomorphisms (
surjective order embeddings).
In the context of search algorithms
In the context of search algorithms monotonicity (also called consistency) is a condition applied to heuristic functions. A heuristic
is monotonic if, for every node and every successor of generated by any action, the estimated cost of reaching the goal from is no greater than the step cost of getting to plus the estimated cost of reaching the goal from,
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.[8]
In Boolean functions
In
Boolean algebra, a monotonic function is one such that for all and in, if,, ..., (i.e. the Cartesian product is ordered coordinatewise), then . 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 -ary Boolean function is monotonic when its representation as an
-cube labelled with truth values has no upward edge from
true to
false. (This labelled
Hasse diagram is the dual of the function's labelled
Venn diagram, which is the more common representation for .)
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,, hold" 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.[9]
See also
Bibliography
- Book: Bartle
, Robert G.
. The elements of real analysis . second . 1976.
- Book: Grätzer
, George
. Lattice theory: first concepts and distributive lattices . 1971 . 0-7167-0442-0.
- Book: Pemberton
, Malcolm
. Rau, Nicholas . Mathematics for economists: an introductory textbook . Manchester University Press . 2001 . 0-7190-3341-1.
- Book: Renardy, Michael . Rogers, Robert C. . amp . An introduction to partial differential equations. Texts in Applied Mathematics 13. Second. Springer-Verlag. New York. 2004. 356. 0-387-00444-0.
- Book: Riesz, Frigyes . Béla Szőkefalvi-Nagy . amp . Functional Analysis . 1990. Courier Dover Publications. 978-0-486-66289-3.
- Book: Russell . Stuart J. . Norvig . Peter . Artificial Intelligence: A Modern Approach . 2010 . 3rd . Prentice Hall . Upper Saddle River, New Jersey . 978-0-13-604259-4.
- Book: Simon . Carl P. . Lawrence . Blume . Mathematics for Economists . first . April 1994 . 978-0-393-95733-4. (Definition 9.31)
External links
Notes and References
- Book: Oxford Concise Dictionary of Mathematics. Clapham. Christopher. Nicholson. James. Oxford University Press. 2014. 5th.
- Web site: Monotonic Function. Stover. Christopher. Wolfram MathWorld. en. 2018-01-29.
- Web site: Monotone function. Encyclopedia of Mathematics. en. 2018-01-29.
- Book: Spivak, Michael. Calculus. Publish or Perish, Inc.. 1994. 0-914098-89-6. Houston, Texas . 192.
- See the section on Cardinal Versus Ordinal Utility in .
- Book: Varian, Hal R. . Intermediate Microeconomics . 8th . 2010 . W. W. Norton & Company . 56 . 9780393934243.
- if its domain has more than one element
- Conditions for optimality: Admissibility and consistency pg. 94–95 .
- 10.1609/aaai.v29i1.9755 . SAT Modulo Monotonic Theories . Sam . Bayless . Noah . Bayless . Holger H. . Hoos . Alan J. . Hu . Proc. 29th AAAI Conf. on Artificial Intelligence . AAAI Press . 3702 - 3709 . 2015 . free . 1406.0043 . live . https://web.archive.org/web/20231211023402/https://ojs.aaai.org/index.php/AAAI/article/view/9755 . Dec 11, 2023 .