Editing Monotonic function (section)

== 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 it is either entirely non-decreasing, or entirely non-increasing.<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 termed ''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>, 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=Houston, Texas |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 with either property is called ''strictly monotone''. 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>.)

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 <math>f</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 ===

All strictly monotonic functions are [[Inverse function|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 <math>y = g(x)</math> is strictly increasing on the range <math>[a, b]</math>, then it has an inverse <math>x = h(y)</math> on the range <math>[g(a), g(b)]</math>.

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.{{cn|reason=Give an example for such a source.|date=August 2022}}

=== 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]]).<ref>See the section on Cardinal Versus Ordinal Utility in {{harvtxt|Simon|Blume|1994}}.</ref> 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.<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 ===
[[File:Monotonic dense jumps svg.svg|thumb|550px|Monotonic function with a dense set of jump discontinuities (several sections shown)]]
[[File:Growth equations.png|550px|thumb|Plots of 6 monotonic growth functions]]
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 discontinuities]];
*<math>f</math> can only have [[countably]] many [[Discontinuities of monotone functions|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]] <math display>(a_i)</math> of positive numbers and any enumeration <math>(q_i)</math> of the [[rational number]]s, the monotonically increasing function <math display=block>f(x)=\sum_{q_i\leq x} a_i</math> is continuous exactly at every irrational number (cf. picture). It is the [[cumulative distribution function]] of the [[discrete measure]] on the rational numbers, where <math>a_i</math> is the weight of <math>q_i</math>.
*If <math>f</math> is [[differentiable]] at <math>x^*\in\Bbb R</math> and <math>f'(x^*)>0</math>, then there is a non-degenerate [[interval (mathematics)| interval]] ''I'' such that <math>x^*\in I</math> and <math>f</math> 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 [[mathematical analysis|analysis]]. Other important properties of these functions include:
*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]] 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.

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.