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.

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.