Monotonic functioncalculus 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.
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.
- 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.
- FX(x) = Prob(X ≤ x)