# Continuity

One of the most important mathematical concepts, as a rule used in connection with the concept of a mapping (see Continuous function; Continuous mapping; Continuous operator). In particular, one can study the continuity of a certain algebraic operation in a given set relative to a topology on that set. When the operation in question is continuous, the term "continuity" is also applied to the set itself (for example, continuous group). The term "continuity" is also used in the sense that it is impossible to "complete by continuity" a certain mathematical object and thus (non-trivially) extend it, for example, a metric space; in this sense one speaks of the continuity of the set of real numbers.

**How to Cite This Entry:**

Continuity. L.D. Kudryavtsev (originator),

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Continuity&oldid=14554