If yes, should both lower and upper semicontinuity have separate tags or should they be under the same tag.
If not, should we expand the scope of the continuity tag to include this topic. (After all, some users have been using the tag in this way. But it would be nice to know the consensus of the community about this. And if the consensus is that the questions about semicontinuity belong here, this should be mentioned explicitly in the tag info.)
Should the tag include also semicontinuity of multifunctions?
I think there should be tag semicontinuous-functions that covers both upper-semicontinuous and lower-semicontinuous functions but excludes multi-valued functions. Upper- and semicontinuous functions are dual to each other and many duplicates of one version will be in terms of the dual concept.
I prefer semicontinuous-functions to semicontinuity because the latter concept seems to include multi-valued functions. But if we treat a function as a special case of a multi-valued functions, all semi-continuity notions for multi-valued-functions boil down to ordinary continuity (but not upper-or semi-continuity). For multi-valued functions, a tag hemicontinuity, an often used synonym in this context, could be used, but we don't have that many questions on it under multivalued-functions now.
I think that it might be useful to have a tag for semicontinuoity of functions and multifunctions. (It seems to be a rather common topic in real analysis and general topology. This class of semicontinuous functions is useful in some contexts.)
I think that it would be better to have single tag rather than two separate tags. (The two classes of functions are rather close to each other. Moreover, having too specific tags is not ideal since for a question we can have at most 5 tags.)
The new tag could serve for questions about semicontinuity of both functions and multifunctions. (Notice that some people use the name hemicontinuity for the latter, and we have some questions using this name. As far as I can tell the term semicontinuity is also used for multifuctions. For example Engelking's book, which I consider a classical reference, uses the name semicontinuity for set-valued mappings; see Problem 1.7.17.)