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I gave continuous map a little bit of substance by giving it an actual Idea-paragraph and by writing out the epsilontic definition for the case of metric spaces, together with its equivalence to the “abstract” definition in terms of opens.
I added the nonstandard definition, largely because it makes precise an intuitive version that can go in the Idea section.
Something should be said about continuity at a point, but I don't have time to do that right now.
Another way to make precise the “preservation of closeness” is in terms of closeness to sets: if $x\approx A$ means that $x$ belongs to the closure of $A$ (a relation in terms of which the notion of topological space can equivalently be defined, at least in classical mathematics), then $f$ is continuous iff $x\approx A$ implies $f(x)\approx f(A)$.
True, and of course that's directly related to the stuff you've been doing with proximity and apartness spaces.
I added to continuous map Frank Waaldijk’s very nice negative result that in constructive mathematics (by which I mean the kind of mathematics possible in a topos with natural numbers object) it cannot be shown that there is a notion of continuity of set-theoretic functions such that certain natural desiderata hold.
That result is already listed at fan theorem (since the existence of such a notion is equivalent to the fan theorem). Which is not to say that it should not also be listed at continuous map, of course.
I added some remarks beforehand about the different notions of continuity in constructive analysis.
Also, the problem is not so much that a kontinuous function on $[0,1]$ might not be bounded (since a uniformly continuous function must be bounded, even without the Fan Theorem, and it is specified that kontinuous functions on $[0,1]$ are uniformly continuous) but that a kontinuous, positive-valued function on $[0,1]$ might not be bounded below by a positive number (so that its reciprocal could not be bounded above, hence could not itself be kontinuous). So I rephrased that bit.
Toby, is there a significance to spelling it ’kontinuous’?
I presume that Waaldijk used a nonstandard spelling because he was speaking about an arbitrary set (with given properties) of partial functions while wanting to invoke a sense that these were more or less the continuous functions.
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