Continuity set

In measure theory, a branch of mathematics, 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 (topological) boundary of B. For signed measures, one instead asks that $${\displaystyle |\mu |(\partial B)=0.}$$

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

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