Talk:Sequential Continuity is Equivalent to Continuity in the Reals
I think sequential continuity if and only if continuity extends to metric spaces, (with a virtually identical proof - we don't really use anything specific about $\size \cdot$) but not to general topological spaces. The pages on left-continuity/right-continuity may also do in some sense, but I don't know how one-sided continuity extends to metric spaces. This paragraph on Wikipedia suggests there might some room for generalisation using "filters", but it would be beyond my knowledge. I'm only putting up these pages on sequential continuity as stepping stones to results in analysis/probability, I haven't got around to tidying up this area in its own right. Anyone else is welcome to work on this in the meantime. Caliburn (talk) 14:17, 30 December 2021 (UTC)
- Yes that's what I meant, but there's 2 things here: a) we need a page with definition of sequential continuity in the context of real numbers (as in: transfer the definition as given here into a page defining it, which is the way we do it here) and b) given that we have a metric-spatial proof of this fact and the reals are a metric space, we have a second proof from that point of view.
- The first one is something that could / should have been taken care of some while ago, but I haven't had the patience to slog through a real analysis text, and so haven't encountered it in that context, so it never got done. This expand tag is the flag to indicate it's something that needs to be done.
- I may get round to it, but it may not be today. --prime mover (talk) 15:10, 30 December 2021 (UTC)
- It exists as is (probably) in the source work given -- so merits its own inclusion. It provides a proof in the context of pure real analysis, for people who have not dipped a toe in the waters of topology and metric spaces. Builds towards the synergy of all the various approaches towards the subject. The person who did create this page was working towards a rigorous underpinning of real analysis all the while keeping within the strictures of the real number axioms. He got a considerable way in before running out of steam. --prime mover (talk) 17:17, 30 December 2021 (UTC)