# Continuity set

In measure theory, a continuity set of a measure μ is any Borel set B such that

${\displaystyle \mu (\partial B)=0\,,}$

where ${\displaystyle \partial B}$ is the boundary set of B. For signed measures, one asks that

${\displaystyle |\mu |(\partial B)=0\,.}$

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

${\displaystyle \Pr[X\in \partial B]=0\,,}$

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

1. ^ Cuppens, R. (1975) Decomposition of multivariate probability. Academic Press, New York.
2. ^ van der Vaart (1998) Asymptotic statistics. Cambridge University Press. ISBN 978-0-521-78450-4. Page 7