In measure theory, a branch of mathematics, a continuity set of a measure μ is any Borel set B such that where is the (topological) boundary of B. For signed measures, one instead asks that

The collection of all continuity sets for a given measure μ forms a ring of sets.^[1]

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

The continuity set C(f) of a function f is the set of points where f is continuous.^{[citation needed]}