# Monotonic function

**monotonic**or

**monotone**refer to functions between partially ordered sets. They first arose in calculus and were later generalized to the more abstract setting or order theory.

In calculus, a function *f* : *X* -> **R** (where *X* is a subset of the real numbers **R**) is **monotonically increasing** or simply **increasing** if, whenever *x* ≤ *y*, then *f*(*x*) ≤ *f*(*y*). An increasing function is also called **order-preserving** for obvious reasons.

Likewise, a function is **decreasing** if, whenever *x* ≤ *y*, then *f*(*x*) ≥ *f*(*y*). A decreasing function is also called **order-reversing**.

If the definitions hold with the inequalities (≤, ≥) replaced by strict inequalities (<, >) then the functions are called **strictly increasing** or **strictly decreasing**.

A function *f*(*x*) is **unimodal** if for some value *m* (the mode), it is monotonically increasing for *x* ≤ *m* and monotonically decreasing for *x* ≥ *m*. In that case, the maximum value of *f*(*x*) is *f*(*m*).

As was mentioned at the beginning, there is also a more **general notion of monotonicity** in case one is not concerned with the set of the real numbers (as in calculus) but with a function *f* between arbitrary partially ordered sets *A* and *B*. In this setting, a function *f* : *A* -> *B* is said to be **order-preserving** whenever *a*_{1} ≤ *a*_{2} *implies* *f*(*a*_{1}) ≤ *f*(*a*_{2}), and **order-reversing** if *a*_{1} ≤ *a*_{2} implies *f*(*a*_{1}) ≥ *f*(*a*_{2}). A function is **monotonic** if it is either order-preserving or order-reversing, and if the definitions hold when (≤, ≥) are replaced by (<, >) one adds the adverb *strictly* to the terms.

In calculus, each of the following properties of a function *f* : **R** -> **R** implies the next:

- A function
*f*is monotonic; *f*has limits from the right and from the left at every point of its domain;*f*can only have discontinuities of jump type;*f*can only have countably many discontinuities in its domain.

- if
*f*is a monotonic functions defined on an interval*I*, then*f*is differentiable almost everywhere on*I*, i.e. the set of numbers*x*in*I*such that*f*is not differentiable in*x*has Lebesgue measure zero. - if
*f*is a monotonic function defined on an interval [*a*,*b*], then*f*is Riemann integrable.

*X*is a random variable, its cumulative distribution function

*F*(_{X}*x*) = Prob(*X*≤*x*)