Continuity set

In measure theory, a branch of mathematics, a continuity set of a measure $μ$ is any Borel set $B$ such that $\mu (\partial B)=0,$ where $\partial B$ is the (topological) boundary of $B$ . For signed measures, one instead asks that $|\mu |(\partial B)=0.$

The collection of all continuity sets for a given measure $μ$ forms a ring of sets.^[1]

Similarly, for a random variable $X$ , a set $B$ is called a continuity set of $X$ if $\Pr[X\in \partial B]=0.$

Continuity set of a function

The continuity set $C (f)$ of a function $f$ is the set of points where $f$ is continuous.^{[citation needed]}

References

↑ Cuppens, R. (1975) Decomposition of multivariate probability. Academic Press, New York.

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[1] Cuppens, R. (1975) Decomposition of multivariate probability. Academic Press, New York.

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Continuity set of a function

References