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The amount a function varies on a set or interval.
The oscillation of a bounded function on a nonempty set is
Equivalently,
Oscillation is used in the oscillation criterion for Riemann integrability. A point is a discontinuity point exactly when the oscillation on its shrinking neighborhoods does not tend to .
Oscillation criterion: Let and let be bounded. For a partition , let be the oscillation of on the subinterval . Then is a Riemann integrable function on if and only if for every there exists a partition such that
This criterion is equivalent to the usual definition via upper sums and lower sums, and it is especially useful for proving integrability results like finite discontinuities imply integrability.
Let , let , and let . The point is a discontinuity point of if is not continuous at . Equivalently, there exists such that for every there is with
An indicator function of a subset is the function defined by for and for .
Indicator functions translate set operations into algebraic ones and are the basic building blocks of simple functions. In a measurable space , is measurable exactly when is a measurable set (with carrying its power-set sigma-algebra).