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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.''''''<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">
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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.
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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">
  
 
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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>
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.

How to Cite This Entry:
Monotone function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monotone_function&oldid=34524
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article