Continuity set

In measure theory, a branch of mathematics, a continuity set of a measure μ is any Borel set B such that

[math]\displaystyle{ \mu(\partial B) = 0\,, }[/math]

where [math]\displaystyle{ \partial B }[/math] is the (topological) boundary of B. For signed measures, one asks that

[math]\displaystyle{ |\mu|(\partial B) = 0\,. }[/math]

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

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

Continuity set of a function

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