Monotone Functions - Revision history

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[[Image:Monotonicity example1.svg|right|thumb|Figure 1. A monotonically non-decreasing function.]]
[[Image:Monotonicity example2.svg|right|thumb|Figure 2. A monotonically non-increasing function]]
[[Image:Monotonicity example3.svg|right|thumb|Figure 3. A function that is ''not'' monotonic]]
In [[mathematics]], a '''monotonic function''' (or '''monotone function''') is a [[function (mathematics)|function]] between [[List of order structures in mathematics|ordered sets]] that preserves or reverses the given [[order relation|order]].<ref>{{Cite book|title=Oxford Concise Dictionary of Mathematics|last1=Clapham|first1=Christopher|last2=Nicholson|first2=James|publisher=Oxford University Press|year=2014|edition=5th}}</ref><ref name=":1">{{Cite web|url=http://mathworld.wolfram.com/MonotonicFunction.html|title=Monotonic Function|last=Stover|first=Christopher|website=Wolfram MathWorld|language=en|access-date=2018-01-29}}</ref><ref name=":0">{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Monotone_function|title=Monotone function|website=Encyclopedia of Mathematics|language=en|access-date=2018-01-29}}</ref> 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 <math>f</math> 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.<ref name=":1" /> 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''' (also '''increasing''' or '''non-decreasing'''),<ref name=":0" /> if for all <math>x</math> and <math>y</math> such that <math>x \leq y</math> one has <math>f\!\left(x\right) \leq f\!\left(y\right)</math>, so <math>f</math> preserves the order (see Figure 1). Likewise, a function is called '''monotonically decreasing''' (also '''decreasing''' or '''non-increasing''')<ref name=":0" /> if, whenever <math>x \leq y</math>, then <math>f\!\left(x\right) \geq f\!\left(y\right)</math>, so it ''reverses'' the order (see Figure 2).

If the order <math>\leq</math> in the definition of monotonicity is replaced by the strict order <math><</math>, then one obtains a stronger requirement. A function with this property is called '''strictly increasing''' (also '''increasing''').<ref name=":0" /><ref name=":2">{{Cite book|last=Spivak|first=Michael|title=Calculus|publisher=Publish or Perish, Inc.|year=1994|isbn=0-914098-89-6|location=1572 West Gray, #377 Houston, Texas 77019|pages=192}}</ref> Again, by inverting the order symbol, one finds a corresponding concept called '''strictly decreasing''' (also '''decreasing''').<ref name=":0" /><ref name=":2" /> 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 function|one-to-one]] (because for <math>x</math> not equal to <math>y</math>, either <math>x < y</math> or <math>x > y</math> and so, by monotonicity, either <math>f\!\left(x\right) < f\!\left(y\right)</math> or <math>f\!\left(x\right) > f\!\left(y\right)</math>, thus <math>f\!\left(x\right) \neq f\!\left(y\right)</math>.)

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 (much weaker) 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 <math>f\!\left(x\right)</math> is said to be '''absolutely monotonic''' over an interval <math>\left(a, b\right)</math> if the derivatives of all orders of <math>f</math> 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''(''x'') that is strictly monotonic, has an inverse function such that ''x'' = ''h''(''y'') 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''(''x'') is strictly monotonic on the range [''a'',''b''], then it has an inverse ''x'' = ''h''(''y'') on the range [''g''(''a''), ''g''(''b'')], but we cannot say the entire range of the function has an inverse.

Note, some textbooks {{which?|date=March 2021}} 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''' (or '''monotone 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 (see also [[monotone preferences]]).<ref>See the section on Cardinal Versus Ordinal Utility in {{harvtxt|Simon|Blume|1994}}.</ref> 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.<ref>{{cite book |last=Varian |first=Hal R. |title=Intermediate Microeconomics |edition=8th |year=2010 |publisher=W. W. Norton & Company |page=56 |isbn=9780393934243}}</ref>

=== Some basic applications and results ===
The following properties are true for a monotonic function <math>f\colon \mathbb{R} \to \mathbb{R}</math>:
*<math>f</math> has [[limit of a function|limits]] from the right and from the left at every point of its [[Domain of a function|domain]];
*<math>f</math> has a limit at positive or negative infinity (<math>\pm\infty</math>) of either a real number, <math>\infty</math>, or <math>-\infty</math>.
*<math>f</math> can only have [[jump discontinuity|jump discontinuities]];
*<math>f</math> can only have [[countable|countably]] many [[discontinuity (mathematics)|discontinuities]] in its domain. The discontinuities, however, do not necessarily consist of isolated points and may even be {{example needed span|dense in an interval (''a'', ''b'').}}

These properties are the reason why monotonic functions are useful in technical work in [[mathematical analysis|analysis]]. Some more facts about these functions are:
*if <math>f</math> is a monotonic function defined on an [[interval (mathematics)|interval]] <math>I</math>, then <math>f</math> is [[derivative|differentiable]] [[almost everywhere]] on <math>I</math>; i.e. the set of numbers <math>x</math> in <math>I</math> such that <math>f</math> is not differentiable in <math>x</math> has [[Lebesgue measure|Lebesgue]] [[measure zero]]. In addition, this result cannot be improved to countable: see [[Cantor function]].
*if this set is countable, then <math>f</math> is absolutely continuous.
*if <math>f</math> is a monotonic function defined on an interval <math>\left[a, b\right]</math>, then <math>f</math> is [[Riemann integral|Riemann integrable]].

An important application of monotonic functions is in [[probability theory]]. If <math>X</math> is a [[random variable]], its [[cumulative distribution function]] <math>F_X\!\left(x\right) = \text{Prob}\!\left(X \leq x\right)</math> is a monotonically increasing function.

A function is ''[[unimodal function|unimodal]]'' if it is monotonically increasing up to some point (the ''[[Mode (statistics)|mode]]'') and then monotonically decreasing.

When <math>f</math> is a ''strictly monotonic'' function, then <math>f</math> is [[injective function|injective]] on its domain, and if <math>T</math> is the [[range of a function|range]] of <math>f</math>, then there is an [[inverse function]] on <math>T</math> for <math>f</math>. In contrast, each constant function is monotonic, but not injective,<ref>if its domain has more than one element</ref> and hence cannot have an inverse.

== In topology ==

A map ''<math>f: X \rightarrow Y</math>'' is said to be '''monotone''' if each of its fibers is connected; i.e. for each element ''<math>y</math>'' in <math>Y</math> the (possibly empty) set <math>f^{-1}(y)</math> is connected.

== In functional analysis ==

In [[functional analysis]] on a [[topological vector space]] <math>X</math>, a (possibly non-linear) operator <math>T: X \rightarrow X^*</math> is said to be a '''monotone operator''' if

:<math>(Tu - Tv, u - v) \geq 0 \quad \forall u,v \in X.</math>

[[Kachurovskii's theorem]] shows that [[convex function]]s on [[Banach space]]s have monotonic operators as their derivatives.

A subset <math>G</math> of <math>X \times X^*</math> is said to be a '''monotone set''' if for every pair <math>[u_1, w_1]</math> and <math>[u_2, w_2]</math> in <math>G</math>,

:<math>(w_1 - w_2, u_1 - u_2) \geq 0.</math>

<math>G</math> 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 <math>G(T)</math> is a monotone set. A monotone operator is said to be '''maximal monotone''' if its graph is a '''maximal monotone set'''.

@@ Line 65: / Line 65: @@
 <math>G</math> 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 <math>G(T)</math> is a monotone set. A monotone operator is said to be '''maximal monotone''' if its graph is a '''maximal monotone set'''.
-==Resources==
+== Licensing ==
-* [https://en.wikipedia.org/wiki/Monotonic_function Monotonic function], Wikipedia
+Content obtained and/or adapted from:

← Older revision		Revision as of 18:58, 20 October 2021
Line 64:		Line 64:

	<math>G</math> 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 <math>G(T)</math> is a monotone set. A monotone operator is said to be '''maximal monotone''' if its graph is a '''maximal monotone set'''.		<math>G</math> 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 <math>G(T)</math> is a monotone set. A monotone operator is said to be '''maximal monotone''' if its graph is a '''maximal monotone set'''.
		+
		+	==Resources==
		+	* [https://en.wikipedia.org/wiki/Monotonic_function Monotonic function], Wikipedia

@@ Line 2: / Line 2: @@
 [[Image:Monotonicity example2.svg|right|thumb|Figure 2. A monotonically non-increasing function]]
 [[Image:Monotonicity example3.svg|right|thumb|Figure 3. A function that is ''not'' monotonic]]
-In [[mathematics]], a '''monotonic function''' (or '''monotone function''') is a [[function (mathematics)|function]] between [[List of order structures in mathematics|ordered sets]] that preserves or reverses the given [[order relation|order]].<ref>{{Cite book|title=Oxford Concise Dictionary of Mathematics|last1=Clapham|first1=Christopher|last2=Nicholson|first2=James|publisher=Oxford University Press|year=2014|edition=5th}}</ref><ref name=":1">{{Cite web|url=http://mathworld.wolfram.com/MonotonicFunction.html|title=Monotonic Function|last=Stover|first=Christopher|website=Wolfram MathWorld|language=en|access-date=2018-01-29}}</ref><ref name=":0">{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Monotone_function|title=Monotone function|website=Encyclopedia of Mathematics|language=en|access-date=2018-01-29}}</ref> This concept first arose in [[calculus]], and was later generalized to the more abstract setting of [[order theory]].
+In mathematics, a '''monotonic function''' (or '''monotone 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.
 == In calculus and analysis ==
-In [[calculus]], a function <math>f</math> 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.<ref name=":1" /> That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease.
+In calculus, a function <math>f</math> 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''' (also '''increasing''' or '''non-decreasing'''),<ref name=":0" /> if for all <math>x</math> and <math>y</math> such that <math>x \leq y</math> one has  <math>f\!\left(x\right) \leq f\!\left(y\right)</math>, so <math>f</math> preserves the order (see Figure 1). Likewise, a function is called '''monotonically decreasing''' (also '''decreasing''' or '''non-increasing''')<ref name=":0" /> if, whenever <math>x \leq y</math>, then <math>f\!\left(x\right) \geq f\!\left(y\right)</math>, so it ''reverses'' the order (see Figure 2).
+A function is called '''monotonically increasing''' (also '''increasing''' or '''non-decreasing'''), if for all <math>x</math> and <math>y</math> such that <math>x \leq y</math> one has  <math>f\!\left(x\right) \leq f\!\left(y\right)</math>, so <math>f</math> preserves the order (see Figure 1). Likewise, a function is called '''monotonically decreasing''' (also '''decreasing''' or '''non-increasing''') if, whenever <math>x \leq y</math>, then <math>f\!\left(x\right) \geq f\!\left(y\right)</math>, so it ''reverses'' the order (see Figure 2).
-If the order <math>\leq</math> in the definition of monotonicity is replaced by the strict order <math><</math>, then one obtains a stronger requirement. A function with this property is called '''strictly increasing''' (also '''increasing''').<ref name=":0" /><ref name=":2">{{Cite book|last=Spivak|first=Michael|title=Calculus|publisher=Publish or Perish, Inc.|year=1994|isbn=0-914098-89-6|location=1572 West Gray, #377 Houston, Texas 77019|pages=192}}</ref> Again, by inverting the order symbol, one finds a corresponding concept called '''strictly decreasing''' (also '''decreasing''').<ref name=":0" /><ref name=":2" /> 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 function|one-to-one]] (because for <math>x</math> not equal to <math>y</math>, either <math>x < y</math> or <math>x > y</math> and so, by monotonicity, either <math>f\!\left(x\right) < f\!\left(y\right)</math> or <math>f\!\left(x\right) > f\!\left(y\right)</math>, thus <math>f\!\left(x\right) \neq f\!\left(y\right)</math>.)
+If the order <math>\leq</math> in the definition of monotonicity is replaced by the strict order <math><</math>, then one obtains a stronger requirement. A function with this property is called '''strictly increasing''' (also '''increasing'''). Again, by inverting the order symbol, one finds a corresponding concept called '''strictly decreasing''' (also '''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 (because for <math>x</math> not equal to <math>y</math>, either <math>x < y</math> or <math>x > y</math> and so, by monotonicity, either <math>f\!\left(x\right) < f\!\left(y\right)</math> or <math>f\!\left(x\right) > f\!\left(y\right)</math>, thus <math>f\!\left(x\right) \neq f\!\left(y\right)</math>.)
 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.
@@ Line 15: / Line 15: @@
 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 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 <math>f\!\left(x\right)</math> is said to be '''absolutely monotonic''' over an interval <math>\left(a, b\right)</math> if the derivatives of all orders of <math>f</math> are [[nonnegative]] or all [[nonpositive]] at all points on the interval.
+A function <math>f\!\left(x\right)</math> is said to be '''absolutely monotonic''' over an interval <math>\left(a, b\right)</math> if the derivatives of all orders of <math>f</math> are nonnegative or all nonpositive at all points on the interval.
 === Inverse of function ===
@@ Line 23: / Line 23: @@
 However, a function ''y'' = ''g''(''x'') that is strictly monotonic, has an inverse function such that ''x'' = ''h''(''y'') 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''(''x'') is strictly monotonic on the range [''a'',''b''], then it has an inverse ''x'' = ''h''(''y'') on the range [''g''(''a''), ''g''(''b'')], but we cannot say the entire range of the function has an inverse.
-Note, some textbooks {{which?|date=March 2021}} mistakenly state that an inverse exists for a monotonic function, when they really mean that an inverse exists for a strictly monotonic function.
+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''' (or '''monotone 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 (see also [[monotone preferences]]).<ref>See the section on Cardinal Versus Ordinal Utility in {{harvtxt|Simon|Blume|1994}}.</ref> 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.<ref>{{cite book |last=Varian |first=Hal R. |title=Intermediate Microeconomics |edition=8th |year=2010 |publisher=W. W. Norton & Company |page=56 |isbn=9780393934243}}</ref>
+The term '''monotonic transformation''' (or '''monotone 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 (see also monotone preferences). 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 <math>f\colon \mathbb{R} \to \mathbb{R}</math>:
-*<math>f</math> has [[limit of a function|limits]] from the right and from the left at every point of its [[Domain of a function|domain]];
+*<math>f</math> has limits from the right and from the left at every point of its domain;
 *<math>f</math> has a limit at positive or negative infinity (<math>\pm\infty</math>) of either a real number, <math>\infty</math>, or <math>-\infty</math>.
-*<math>f</math> can only have [[jump discontinuity|jump discontinuities]];
+*<math>f</math> can only have jump discontinuities;
-*<math>f</math> can only have [[countable|countably]] many [[discontinuity (mathematics)|discontinuities]] in its domain.  The discontinuities, however, do not necessarily consist of isolated points and may even be {{example needed span|dense in an interval (''a'', ''b'').}}
+*<math>f</math> 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'').}}
-These properties are the reason why monotonic functions are useful in technical work in [[mathematical analysis|analysis]]. Some more facts about these functions are:
+These properties are the reason why monotonic functions are useful in technical work in analysis. Some more facts about these functions are:
-*if <math>f</math> is a monotonic function defined on an [[interval (mathematics)|interval]] <math>I</math>, then <math>f</math> is [[derivative|differentiable]] [[almost everywhere]] on <math>I</math>; i.e. the set of numbers <math>x</math> in <math>I</math> such that <math>f</math> is not differentiable in <math>x</math> has [[Lebesgue measure|Lebesgue]] [[measure zero]]. In addition, this result cannot be improved to countable: see [[Cantor function]].
+*if <math>f</math> is a monotonic function defined on an interval <math>I</math>, then <math>f</math> is differentiable almost everywhere on <math>I</math>; i.e. the set of numbers <math>x</math> in <math>I</math> such that <math>f</math> is not differentiable in <math>x</math> has Lebesgue measure zero. In addition, this result cannot be improved to countable: see Cantor function.
 *if this set is countable, then <math>f</math> is absolutely continuous.
-*if <math>f</math> is a monotonic function defined on an interval <math>\left[a, b\right]</math>, then <math>f</math> is [[Riemann integral|Riemann integrable]].
+*if <math>f</math> is a monotonic function defined on an interval <math>\left[a, b\right]</math>, then <math>f</math> is Riemann integrable.
-An important application of monotonic functions is in [[probability theory]]. If <math>X</math> is a [[random variable]], its [[cumulative distribution function]] <math>F_X\!\left(x\right) = \text{Prob}\!\left(X \leq x\right)</math> is a monotonically increasing function.
+An important application of monotonic functions is in probability theory. If <math>X</math> is a random variable, its cumulative distribution function <math>F_X\!\left(x\right) = \text{Prob}\!\left(X \leq x\right)</math> is a monotonically increasing function.
-A function is ''[[unimodal function|unimodal]]'' if it is monotonically increasing up to some point (the ''[[Mode (statistics)|mode]]'') and then monotonically decreasing.
+A function is ''unimodal'' if it is monotonically increasing up to some point (the ''mode'') and then monotonically decreasing.
-When <math>f</math> is a ''strictly monotonic'' function, then <math>f</math> is [[injective function|injective]] on its domain, and if <math>T</math> is the [[range of a function|range]] of <math>f</math>, then there is an [[inverse function]] on <math>T</math> for <math>f</math>. In contrast, each constant function is monotonic, but not injective,<ref>if its domain has more than one element</ref> and hence cannot have an inverse.
+When <math>f</math> is a ''strictly monotonic'' function, then <math>f</math> is injective on its domain, and if <math>T</math> is the range of <math>f</math>, then there is an inverse function on <math>T</math> for <math>f</math>. In contrast, each constant function is monotonic, but not injective, and hence cannot have an inverse.
 == In topology ==
@@ Line 53: / Line 53: @@
 == In functional analysis ==
-In [[functional analysis]] on a [[topological vector space]] <math>X</math>, a (possibly non-linear) operator <math>T: X \rightarrow X^*</math> is said to be a '''monotone operator''' if
+In functional analysis on a topological vector space <math>X</math>, a (possibly non-linear) operator <math>T: X \rightarrow X^*</math> is said to be a '''monotone operator''' if
 :<math>(Tu - Tv, u - v) \geq 0 \quad \forall u,v \in X.</math>
-[[Kachurovskii's theorem]] shows that [[convex function]]s on [[Banach space]]s have monotonic operators as their derivatives.
+Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives.
 A subset <math>G</math> of <math>X \times X^*</math> is said to be a '''monotone set''' if for every pair <math>[u_1, w_1]</math> and <math>[u_2, w_2]</math> in <math>G</math>,