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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o0705101.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o0705102.png" />''
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$#C+1 = 27 : ~/encyclopedia/old_files/data/O070/O.0700510 Oscillation of a function
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The difference between the least upper and the greatest lower bounds of the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o0705103.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o0705104.png" />. In other words, the oscillation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o0705105.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o0705106.png" /> is given by
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{{TEX|auto}}
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{{TEX|done}}
  
<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/o/o070/o070510/o0705107.png" /></td> </tr></table>
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'' $  f $
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on a set  $  E $''
  
If the function is unbounded on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o0705108.png" />, its oscillation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o0705109.png" /> is put equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051010.png" />. For constant functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051011.png" /> (and only for these) the oscillation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051012.png" /> is zero. If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051013.png" /> is defined on a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051015.png" />, then its oscillation at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051016.png" /> of the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051017.png" /> is defined by the formula
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The difference between the least upper and the greatest lower bounds of the values of  $  f $
 +
on $  E $.  
 +
In other words, the oscillation of $  f $
 +
on  $  E $
 +
is given by
  
<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/o/o070/o070510/o07051018.png" /></td> </tr></table>
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$$
 +
\omega _ {E} ( f  )  = \
 +
\sup _ {P  ^  \prime  , P  ^ {\prime\prime} \in E }
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\{ | f ( P  ^  \prime  ) - f ( P  ^ {\prime\prime} ) | \} .
 +
$$
  
where the infimum is taken over all neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051021.png" />, then in order that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051022.png" /> be continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051023.png" /> with respect to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051024.png" /> it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051025.png" />.
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If the function is unbounded on  $  E $,
 +
its oscillation on  $  E $
 +
is put equal to  $  \infty $.  
 +
For constant functions on  $  E $(
 +
and only for these) the oscillation on  $  E $
 +
is zero. If the function  $  f $
 +
is defined on a subset  $  E $
 +
of $  \mathbf R  ^ {n} $,  
 +
then its oscillation at any point  $  Q $
 +
of the closure of  $  E $
 +
is defined by the formula
  
 +
$$
 +
\omega _ {Q , E }  ( f  )  = \
 +
\inf _ {\begin{array}{c}
 +
U \\
 +
Q \in U
 +
\end{array}
 +
}  \omega _ {U \cap E }  ( f  ) ,
 +
$$
  
 +
where the infimum is taken over all neighbourhoods  $  U $
 +
of  $  Q $.
 +
If  $  Q \in E $,
 +
then in order that  $  f $
 +
be continuous at  $  Q $
 +
with respect to the set  $  E $
 +
it is necessary and sufficient that  $  \omega _ {Q,E }  ( f  ) = 0 $.
  
 
====Comments====
 
====Comments====
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051026.png" /> is called the oscillation function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070510/o07051027.png" />.
+
The function $  Q \rightarrow \omega _ {Q,E }  ( f  ) $
 +
is called the oscillation function of $  f $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)  pp. 120</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.R. Goldberg,  "Methods of real analysis" , Blaisdell  (1964)  pp. 129</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)  pp. 120</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.R. Goldberg,  "Methods of real analysis" , Blaisdell  (1964)  pp. 129</TD></TR></table>

Latest revision as of 08:04, 6 June 2020


$ f $ on a set $ E $

The difference between the least upper and the greatest lower bounds of the values of $ f $ on $ E $. In other words, the oscillation of $ f $ on $ E $ is given by

$$ \omega _ {E} ( f ) = \ \sup _ {P ^ \prime , P ^ {\prime\prime} \in E } \{ | f ( P ^ \prime ) - f ( P ^ {\prime\prime} ) | \} . $$

If the function is unbounded on $ E $, its oscillation on $ E $ is put equal to $ \infty $. For constant functions on $ E $( and only for these) the oscillation on $ E $ is zero. If the function $ f $ is defined on a subset $ E $ of $ \mathbf R ^ {n} $, then its oscillation at any point $ Q $ of the closure of $ E $ is defined by the formula

$$ \omega _ {Q , E } ( f ) = \ \inf _ {\begin{array}{c} U \\ Q \in U \end{array} } \omega _ {U \cap E } ( f ) , $$

where the infimum is taken over all neighbourhoods $ U $ of $ Q $. If $ Q \in E $, then in order that $ f $ be continuous at $ Q $ with respect to the set $ E $ it is necessary and sufficient that $ \omega _ {Q,E } ( f ) = 0 $.

Comments

The function $ Q \rightarrow \omega _ {Q,E } ( f ) $ is called the oscillation function of $ f $.

References

[a1] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) pp. 120
[a2] R.R. Goldberg, "Methods of real analysis" , Blaisdell (1964) pp. 129
How to Cite This Entry:
Oscillation of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Oscillation_of_a_function&oldid=12236
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article