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'' $  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====
<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>