Monotonic

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)

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