Oscillation of a function
on a set
The difference between the least upper and the greatest lower bounds of the values of on . In other words, the oscillation of on is given by
If the function is unbounded on , its oscillation on is put equal to . For constant functions on (and only for these) the oscillation on is zero. If the function is defined on a subset of , then its oscillation at any point of the closure of is defined by the formula
where the infimum is taken over all neighbourhoods of . If , then in order that be continuous at with respect to the set it is necessary and sufficient that .
The function is called the oscillation function of .
|[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|
Oscillation of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Oscillation_of_a_function&oldid=12236