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Continuity set

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Source updated February 13, 2025

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.

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

01Continuity set of a function

The continuity set C(f) of a function f is the set of points where f is continuous.

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