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 orderpreserving for obvious reasons.
Likewise, a function is decreasing if, whenever x ≤ y, then f(x) ≥ f(y). A decreasing function is also called orderreversing.
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 orderpreserving whenever a_{1} ≤ a_{2} implies f(a_{1}) ≤ f(a_{2}), and orderreversing if a_{1} ≤ a_{2} implies f(a_{1}) ≥ f(a_{2}). A function is monotonic if it is either orderpreserving or orderreversing, 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:
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
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