Sequential continuity
is sequentially continuous at when every sequence in with satisfies .
Definition 2.11 (Sequential Continuity)
Let . We say that is sequentially continuous at if for every sequence in such that , we have .
Characterisation of continuity
is continuous at if and only if is sequentially continuous at .
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If is continuous at , then for on with and , pick from continuity and then with for all ; hence for all , so .
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Conversely, if is not continuous at , then some admits, for every , a point with but . Taking builds with , so , while stays at least away from and does not converge to it — contradicting sequential continuity.
Related
Stated in
- Definition 2.11 (Sequential Continuity)§2.2 Continuity of Functions