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