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.