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Monotonic

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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 xy, then f(x) ≤ f(y). An increasing function is also called order-preserving for obvious reasons.

Likewise, a function is decreasing if, whenever xy, 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.

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 a1a2 implies f(a1) ≤ f(a2), and order-reversing if a1a2 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:

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

FX(x) = Prob(Xx)
is a monotonically increasing function.

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