Continuity set
In measure theory, a continuity set of a measure μ is any Borel set B such that
where is the boundary set of B. For signed measures, one asks that
The class of all continuity sets for given measure μ forms a ring.[1]
Similarly, for a random variable X a set B is called continuity set if
otherwise B is called the discontinuity set. The collection of all discontinuity sets is sparse. In particular, given any collection of sets {Bα} with pairwise disjoint boundaries, all but at most countably many of them will be the continuity sets.[2]
The continuity set C(f) of a function f is the set of points where f is continuous.
References
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