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1 General definition

Let

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.

2 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. Likewise, a function is called monotonically decreasing (or just "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.

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).

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 differentiableCalculus In mathematics, the derivative of a function is one of the two central concepts of calculus. The inverse of a derivative is called the antiderivative, or indefinite integral. The derivative of a function at a certain point is a measure of the rat almost everywhereIn measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. is a set with measure zero. If used for properties of the real numbers, t on I, i.e. the set of numbers x in I such that f is not differentiable in x has LebesgueMeasure theory The Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration. Sets which can be assigned a volume are called Lebesgue measura measure zeroA set of numbers has measure zero if its Lebesgue measure is zero. A classic example is the set of rational numbers in the real line under standard (Euclidean) Lebesgue measure. As another example, a monotonic function on the real line is continuous every.
• if f is a monotonic function defined on an interval [a, b], then f is Riemann integrableIf you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. In a branch of mathematics known as real analysis, the Riemann integral created by Bernhard Riemann, was the first rig.

An important application of monotonic functions is in probability theoryProbability theory Discrete mathematics Mathematical analysis Probability theory is the mathematical study of probability. Mathematicians think of probabilities as numbers in the interval from 0 to 1 assigned to "events" whose occurrence or failure to occ. If X is a random variableA random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. For example, rolling a dice and recording the outcome yields a random variable w, its cumulative distribution functionIn probability theory, the cumulative distribution function (abbreviated cdf completely describes the probability distribution of a real-valued random variable, X''. For every real number x the cdf is given by : where the right-hand side represents the pr

F_X(x) = Prob(X ≤ x)

is a monotonically increasing function.