This survey paper contains a brief account of the literature on preservation
of continuity under various convergences from the beginning to current research. We
include stronger forms of continuity such as uniform continuity, proximal continuity,
strong uniform continuity and weaker forms of continuity such as quasi-continuity, near
continuity and almost continuity. A few proofs of the results are given and references
are given for others. A standard reference for this research area is [28]