Difference between revisions of "Monotone function"
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− | A function of one variable, defined on a subset of the real numbers, whose increment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064830/m0648301.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064830/m0648302.png" />, does not change sign, that is, is either always negative or always positive. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064830/m0648303.png" /> is strictly greater (less) than zero when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064830/m0648304.png" />, then the function is called strictly monotone (see [[Increasing function|Increasing function]]; [[Decreasing function|Decreasing function]]). The various types of monotone functions are represented in the following table. | + | A function of one variable, defined on a subset of the real numbers, whose increment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064830/m0648301.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064830/m0648302.png" />, does not change sign, that is, is either always negative or always positive. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064830/m0648303.png" /> is strictly greater (less) than zero when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064830/m0648304.png" />, then the function is called strictly monotone (see [[Increasing function|Increasing function]]; [[Decreasing function|Decreasing function]]). The various types of monotone functions are represented in the following table. |
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+ | <table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064830/m0648305.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Increasing (non-decreasing)</td> <td colname="3" style="background-color:white;" colspan="1"> | ||
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" /> |
Revision as of 18:17, 15 November 2014
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.
<tbody> </tbody>
|
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://encyclopediaofmath.org/index.php?title=Monotone_function&oldid=18679