# Continuity set

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In [measure theory](/source/measure_theory), a branch of [mathematics](/source/mathematics), a '''continuity set''' of a [measure](/source/measure_(mathematics)) {{mvar|μ}} is any [Borel set](/source/Borel_set) {{mvar|B}} such that
<math display=block>\mu(\partial B) = 0,</math>
where <math>\partial B</math> is the (topological) [boundary](/source/boundary_(topology)) of {{mvar|B}}.  For [signed measure](/source/signed_measure)s, one instead asks that
<math display=block>|\mu|(\partial B) = 0.</math>

The collection of all continuity sets for a given measure {{mvar|μ}} forms a [ring of sets](/source/ring_of_sets).<ref>Cuppens, R. (1975) Decomposition of multivariate probability. Academic Press, New York.</ref>

Similarly, for a [random variable](/source/random_variable) {{mvar|X}}, a set {{mvar|B}} is called a '''continuity set''' of {{mvar|X}} if
<math display=block>\Pr[X \in \partial B] = 0.</math>

==Continuity set of a function==
The '''continuity set''' {{math|''C''(''f'')}} of a [function](/source/function_(mathematics)) {{mvar|f}} is the set of points where {{mvar|f}} is [continuous](/source/continuous_function).{{citation needed|date=February 2025}}

== References ==
{{reflist}}

Category:Measure theory

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Adapted from the Wikipedia article [Continuity set](https://en.wikipedia.org/wiki/Continuity_set) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Continuity_set?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.