Continuity set
Encyclopedia
In measure theory, a continuity set of a measure μ is any Borel set
B such that
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
B such that
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The class of all continuity sets for given measure μ forms a ringRing (mathematics)In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
.
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.
The continuity set C(f) of a function f is the set of points where f is continuous.
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