Continuity set

In measure theory, a branch of mathematics, a continuity set of a measure Template:Mvar is any Borel set Template:Mvar such that $$\mu (\partial B)=0,$$ where $\partial B$ is the (topological) boundary of Template:Mvar. For signed measures, one instead asks that $$|\mu |(\partial B)=0.$$

The collection of all continuity sets for a given measure Template:Mvar forms a ring of sets.

Similarly, for a random variable Template:Mvar, a set Template:Mvar is called a continuity set of Template:Mvar if $$\mathrm{Pr}[X\in \partial B]=0.$$

Continuity set of a function

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