Une remarque sur la continuité et la connexité

Noncontinuous multifuctions

“…It is natural now to ask the following question which is a slight variant of that posed in [10]. Is X locally connected if each connected (C, 0) function on X, as in Corollary 2.4 or 2.6, is con- The following theorem gives an affirmative answer to a question similar to that posed above using multifunctions instead of functions.…”

“…The above corollary is a characterization of continuous real valued functions on a locally connected space similar to that in Theorem 1 of [10]. It is natural now to ask the following question which is a slight variant of that posed in [10].…”

A Modest Review of a Great Deal of Work

“…Jan has written more than ten papers 5 concerning various notions of generalized continuity. Too, this topic proved a rich source for his active collaboration with other real analysts, and his interest in notions of generalized continuity can be seen as threading a good portion of his research career, beginning with [33] in 1968 and extending through [48] in 1993. As with other sections of this paper, the setting is not always the real line R, but sometimes a general topological space; I'll try to keep the more general notions somewhat separate from those specifically related to those of the real line, R, but, of course, this is not always completely possible.…”

“…In [33], Jan began an investigation of the relationship between functions that are continuous and functions that preserve connectedness. This relationship is important for a variety of reasons not the least of which is that derivatives of functions f : R → R preserve connectedness, but need not be continuous.…”