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.
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.
Monotone function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Monotone_function&oldid=34526