Monotone function
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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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 nonlinear functional analysis, see Monotone operator. For the concept in general partially ordered sets, see Monotone mapping.
Monotone function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Monotone_function&oldid=34526