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Monotone function

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A function of one variable, defined on a subset of the real numbers, whose increment , for , does not change sign, that is, is either always negative or always positive. If is strictly greater (less) than zero when , then the function is called strictly monotone (see Increasing function; Decreasing function). The various types of monotone functions are represented in the following table.

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Increasing (non-decreasing)

Decreasing (non-increasing)

Strictly increasing

Strictly decreasing

If at each point of an interval has a derivative that does not change sign (respectively, is of constant sign), then is monotone (strictly monotone) on this interval.

The idea of a monotone function can be generalized to functions of various classes. For example, a function defined on is called monotone if the condition implies that everywhere either or everywhere. A monotone function in the algebra of logic is defined similarly.

A monotone function of many variables, increasing or decreasing at some point, is defined as follows. Let be defined on the -dimensional closed cube , let and let be a level set of . The function is called increasing (respectively, decreasing) at if for any and any not separated in by from , the relation (respectively, ) holds, and for any that is separated in by from , the relation (respectively, ) holds. A function that is increasing or decreasing at some point is called monotone at that point.

Comments

For the concept in non-linear functional analysis, see Monotone operator. For the concept in general partially ordered sets, see Monotone mapping.

How to Cite This Entry:
Monotone function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Monotone_function&oldid=34526
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article