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 asks that

$|\mu |(\partial B)=0\,.$

The class of all continuity sets for given measure μ forms a ring.

Similarly, for a random variable X, a set B is called continuity set 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.