The notion of continuity characterises an absence of discrete chaanges; a
function is continuous if it maps inputs near
each other (in relevant
senses) to outputs (in corresponding senses) near
to each other. Its
formalisation may be approached in various ways: a fairly canonical orthodox
specification of it says that, for any open set of outputs the collection of
inputs the function maps into that open set is itself open. That, however,
requires topology; and is an example of a challenge-response protocol; these are
things I'd rather use less of. Another way to characterise continuity is that,
if a sequence of inputs converges then so does the
sequence of outputs and, if the former has a limit, so does the latter and the
function maps the former's limit to the latter's. This, of course, depends on
the notion of convergence, which I've yet to articulate in a way that avoids a
need for topology and challenge-response; but at least it factorises them via
convergence and will eliminate them if I find a way to describe convergence
without them.
Fortunately, I'm not interested in quite such a general case as orthodox topologists; I only really care about continuity in linear spaces over {positives} and spaces that can be described entirely in terms of these (e.g. smooth manifolds, described via charts). So, as for differentiation, I can use voluminous simplices (whose edges, out of a single vertex, form a basis) to tacitly generate a topology, without directly invoking the topology itself. As in the case of differentiation, I can handle this in various ways.
For any simplex in the inputs space, I can ask which simplices in the output space enclose the function's image of that input simplex; for a given input, I can ask for all such (closed) output simplices for input simplices with the given input in their interior; and I can take the intersection of these. The function is then continuous at the given input precisely if the intersection contains exactly the value that the function maps the input to.
Alternatively, given any simplex S about the input x and simplex T about the output f(x) for which y in S implies f(y) in T, I can scale S down towards x and T down towards f(x), asking whether there is some positive power of the input scaling that provides an upper bound on the output scaling required to contain all f's values within the scaled input simplex.
I intuit that all of these approaches produce near-enough the same notion of continuity for the purposes of the mathematics I need in order to talk about physics.
I consequently aim to discuss topological questions in terms of some notion
of continuity, provided by context, for relations
– albeit possibly mostly used for mappings – between relevant
collections. I anticipate that, from such a notion of continuity of relations
among some given collections, one may induce notions of continuity among
relations between them and collections derived from them in various ways,
notably including Cartesian products of the given collections. This tacitly
presumes a categoric characterisation of how continuous relations compose with
one another. The continuous mappings are
then homomorphisms between mathematical
structures, for which the sense of continuity is part of the specification of
the structure type (along with any other properties that may be given, e.g. if
the continua between which we're mapping are groups and we're restricting
attention to continuous group homomorphisms). I shall reter to an object of
such a category as a continuum
(plural: continua) and use
the term continuum isomorhpism
for a continuous
isomorphism with continuous inverse. (Orthodoxy calls
these homeomorphisms
;
however, I always have trouble remembering what that word means, so I chose a
phrase for it that makes clear it's the relevant isomorphism for a category in
which continuity is a defined property of all morphisms.)
Written by Eddy.