Editing Monotonic function (section)

=== 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.