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In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. These functions first arose in calculus and were later generalized to the more abstract setting of order theory. Although the concepts generally agree, the two disciplines have developed a slightly different terminology. While in calculus, one often talks about functions being monotonically increasing and monotonically decreasing, order theory prefers the terms monotone and antitone or order-preserving and order-reversing, respectively.
- f: P → Q
be a function between two sets P and Q, where each set carries a partial order (both of which we denote by ≤). In calculus one focuses on functions between subsets of the reals and the order ≤ is just the usual ordering on real numbers, but this is not essential for this definition.
The function f is monotone if, whenever x ≤ y, then f(x) ≤ f(y). Stated differently, a monotone function is one that preserves the order.
Monotonicity in calculus and analysis
In calculus, there is often no need to call upon the abstract methods of order theory. As already noted, functions are usually mappings between (subsets of) real numbers, ordered in the natural way.
Inspired by the shape of the graph of a monotone function on the reals, such functions are also called monotonically increasing (or "nondecreasing" or, less precisely, just "increasing"). Likewise, a function is called monotonically decreasing (or "nonincreasing" or "decreasing") if, whenever x ≤ y, then f(x) ≥ f(y), i.e., if it reverses the order.
If the order ≤ in the definition of monotonicity is replaced by the strict order <, then one obtains a stronger requirement. A function with this property is called strictly increasing. Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing.
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.
- FX(x) = Prob(X ≤ x)
is a monotonically increasing function.
Monotonicity in order theory
In order theory, one does not restrict to real numbers, but one is concerned with arbitrary partially ordered sets or even with preordered sets. In these cases, the above definition of monotonicity is relevant as well. However, the terms "increasing" and "decreasing" are avoided, since they lose their appealing pictorial motivation as soon as one deals with orders that are not total. Furthermore, the strict relations < and > are of little use in many non-total orders and hence no additional terminology is introduced for them.
A monotone function is also called order-preserving. The dual notion is often called antitone, anti-monotone, or order-reversing. Hence, an antitone function f satisfies the property
- x ≤ y implies f(x) ≥ f(y),
for all x and y in its domain. It is easy to see that the composite of two monotone mappings is also monotone.
Monotone functions are central in order theory. They appear in most articles on the subject and examples from special applications are to be found in these places. Some notable special monotone functions are order embeddings (functions for which x ≤ y iff f(x) ≤ f(y)) and order isomorphisms (surjective order embeddings).
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