Monotonic function

The terms 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 a₁ ≤ a₂ implies f(a₁) ≤ f(a₂), and order-reversing if a₁ ≤ a₂ implies f(a₁) ≥ f(a₂). 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.

These properties are the reason why monotonic functions are useful in technical work in analysis. Two facts about these functions are:

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.

An important application of monotonic functions is in probability theory. If X is a random variable, its cumulative distribution function

F_X(x) = Prob(X ≤ x)

is a monotonically increasing function.