The idea of continuity in (real) analysis—and indeed everywhere else it is used: topology, etc.—seems to me superfluous. For if, as I learnt from Stewart (Concepts of Modern Mathematics), it is based on the idea of excising jumps in so-called ‘functions’, then upon considering it carefully one sees that all functions that are called discontinuous at some point are not really functions at all; the problem, it seems, has only stuck because of the non-rigorous way analysis developed, in which Newton, Euler and the other early guys called everything that produced values a function, like the unrestricted square-root ‘function’, for example.
Now that the function concept has been placed on rigorous footing by set theory, one sees that the concept of continuous functions is just what we properly today call a mapping, and all the fuss is just linguistical. Thus, all definitions of continuity are just alternative definitions, in analytical guise, of the modern concept of mapping; the difference being just that in analysis, the sets are more specified than general—they’re usually continua, e.g.—but the idea is the same! Thus, continuity just requires that all points in the domain get mapped to exactly one point—i.e., not neither or more than one. Thus, for all x in the domain of some mapping f, there’s one and only one f(x). That, as clear as day, is the modern definition of ‘mapping’.
Or, if this is not so, please enlighten me: Does the idea of continuity capture more than what the modern set-theoretic definition of mapping does? If so, what exactly is that extra stuff it covers? Thanks.
PS. Of course I’m aware that there are theories of multi-valued functions, but that’s clearly not what I’m interested in; I’m talking about mappings as traditionally envisaged (i.e., single-valued).