Namespaces
Variants
Actions

Difference between revisions of "Decreasing function"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305001.png" /> defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305002.png" /> of real numbers such that the condition
+
<!--
 +
d0305001.png
 +
$#A+1 = 20 n = 0
 +
$#C+1 = 20 : ~/encyclopedia/old_files/data/D030/D.0300500 Decreasing function
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305003.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
implies
+
A function  $  f $
 +
defined on a set  $  E $
 +
of real numbers such that the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305004.png" /></td> </tr></table>
+
$$
 +
x  ^  \prime  < x  ^ {\prime\prime} ,\ \
 +
x  ^  \prime  , x  ^ {\prime\prime} \in E,
 +
$$
  
Sometimes such a function is called strictly decreasing and the term  "decreasing function"  is applied to functions satisfying for the indicated values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305005.png" /> only the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305006.png" /> (a non-increasing function). Every strictly decreasing function has an inverse function, which is again strictly decreasing. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305007.png" /> is a left-hand (respectively, right-hand) limit point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305009.png" /> is non-increasing and if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050010.png" /> is bounded from above (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050011.png" /> is bounded from below), then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050012.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050013.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050015.png" /> has a finite limit; if the given set is not bounded from above (respectively, from below), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050016.png" /> has an infinite limit, equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050017.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050018.png" />).
+
implies
  
 +
$$
 +
f ( x  ^  \prime  )  >  f ( x  ^ {\prime\prime} ).
 +
$$
  
 +
Sometimes such a function is called strictly decreasing and the term  "decreasing function"  is applied to functions satisfying for the indicated values  $  x  ^  \prime  , x  ^ {\prime\prime} $
 +
only the condition  $  f ( x  ^  \prime  ) \geq  f ( x  ^ {\prime\prime} ) $(
 +
a non-increasing function). Every strictly decreasing function has an inverse function, which is again strictly decreasing. If  $  x _ {0} $
 +
is a left-hand (respectively, right-hand) limit point of  $  E $,
 +
$  f $
 +
is non-increasing and if the set  $  \{ {y } : {y = f ( x),  x > x _ {0} ,  x \in E } \} $
 +
is bounded from above (respectively,  $  \{ {y } : {y = f ( x),  x < x _ {0} ,  x \in E } \} $
 +
is bounded from below), then for  $  x \rightarrow x _ {0} + 0 $(
 +
respectively,  $  x \rightarrow x _ {0} - 0 $),
 +
$  x \in E $,
 +
$  f ( x) $
 +
has a finite limit; if the given set is not bounded from above (respectively, from below), then  $  f ( x) $
 +
has an infinite limit, equal to  $  + \infty $(
 +
respectively,  $  - \infty $).
  
 
====Comments====
 
====Comments====
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050020.png" /> is decreasing is called increasing (cf. [[Increasing function|Increasing function]]).
+
A function $  f $
 +
such that $  - f $
 +
is decreasing is called increasing (cf. [[Increasing function|Increasing function]]).

Latest revision as of 17:32, 5 June 2020


A function $ f $ defined on a set $ E $ of real numbers such that the condition

$$ x ^ \prime < x ^ {\prime\prime} ,\ \ x ^ \prime , x ^ {\prime\prime} \in E, $$

implies

$$ f ( x ^ \prime ) > f ( x ^ {\prime\prime} ). $$

Sometimes such a function is called strictly decreasing and the term "decreasing function" is applied to functions satisfying for the indicated values $ x ^ \prime , x ^ {\prime\prime} $ only the condition $ f ( x ^ \prime ) \geq f ( x ^ {\prime\prime} ) $( a non-increasing function). Every strictly decreasing function has an inverse function, which is again strictly decreasing. If $ x _ {0} $ is a left-hand (respectively, right-hand) limit point of $ E $, $ f $ is non-increasing and if the set $ \{ {y } : {y = f ( x), x > x _ {0} , x \in E } \} $ is bounded from above (respectively, $ \{ {y } : {y = f ( x), x < x _ {0} , x \in E } \} $ is bounded from below), then for $ x \rightarrow x _ {0} + 0 $( respectively, $ x \rightarrow x _ {0} - 0 $), $ x \in E $, $ f ( x) $ has a finite limit; if the given set is not bounded from above (respectively, from below), then $ f ( x) $ has an infinite limit, equal to $ + \infty $( respectively, $ - \infty $).

Comments

A function $ f $ such that $ - f $ is decreasing is called increasing (cf. Increasing function).

How to Cite This Entry:
Decreasing function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Decreasing_function&oldid=46596
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article