<P> In several contexts, the topology of a space is conveniently specified in terms of limit points . In many instances, this is accomplished by specifying when a point is the limit of a sequence, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a directed set, known as nets . A function is (Heine -) continuous only if it takes limits of sequences to limits of sequences . In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition . </P> <P> In detail, a function f: X → Y is sequentially continuous if whenever a sequence (x) in X converges to a limit x, the sequence (f (x)) converges to f (x). Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous . If X is a first - countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous . In particular, if X is a metric space, sequential continuity and continuity are equivalent . For non first - countable spaces, sequential continuity might be strictly weaker than continuity . (The spaces for which the two properties are equivalent are called sequential spaces .) This motivates the consideration of nets instead of sequences in general topological spaces . Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions . </P> <P> Instead of specifying the open subsets of a topological space, the topology can also be determined by a closure operator (denoted cl) which assigns to any subset A ⊆ X its closure, or an interior operator (denoted int), which assigns to any subset A of X its interior . In these terms, a function </P> <Dl> <Dd> f: (X, c l) → (X ′, c l ′) (\ displaystyle f \ colon (X, \ mathrm (cl)) \ to (X', \ mathrm (cl)')) </Dd> </Dl>

Every sequence of real numbers is a continuous function