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