Limits: continuity and differentiation

The orthodox limit formalism is a challenge-response protocol in which any positive tolerance in one thing must be shown to be attainable by imposing some (typically other) positive tolerance on another; a function is continuous at a specific input if, for any challenge tolerance on changes in the output, all inputs within a given response tolerance of the specific input give outputs within the challenge tolerance of the specific input's output. Likewise, a series converges if, for some given tolerance on variation between its entries, there is some point in the sequence beyond which entries differ by at most the given tolerance; and a function is differentiable at some given input, with a specified derivative, if every positive challenge tolerance – on how close to the specified derivative the gradients of chords of the function must be – can be met by limiting the chords to be between points within some response tolerance of the given input.

Reasoning about such challenge-response protocols typically requires use of reductio ad absurdum arguments, which I dislike; so I try to keep my use of them to a minimum. This area of my writings may be considered an implementation detail (that I dislike and wish I could replace) for things used by other areas; and shall likely grow side-discussions of how much I can get away with using more constructive methods of reasoning. In particular, for continuity and differentiation, I have a hope that one can replace the for every challenge there exists a response formulation (which leads to arguments invoking the existence of a response without necessarily showing how to construct one) with some here is a polynomial bound on the response that suffices to meet the challenge; I'm not sure how constructive the result shall be, but hope that it may at least point the way to developing some constructive mechanism that suffices for all the cases I care about.

Existing fragments:

Asymptotic truth

I think of things for which orthodoxy uses the limit machinery as being cases of what I term asymptotic truth, applicable to any continuum context. By this I mean that some condition is almost true in some quantifiable way that can be made arbitrarily close to true (for the same way of quantifying it) by narrowing attention to all sufficiently close approximations to some boundary case, in which the condition's truth might be either fatuous (so uninformative, where the asymptotic truth says something meaningful) or inapplicable (making the asymptotic truth as good a surrogate for it as we can hope for). A common boundary case is everywhere sufficiencly close to somewhere (the limit at a point, or near a line or surface); but another that does coem up is everywhere sufficiently far from somewhere (a limit at infinity). Other boundary cases are possible, but these are the two that most commonly get attention.

What I want from the limit formalism is really some way to reason about such asymptotic truth, ideally without being too firmly tied down to the details of how it is specified. It should be possible to reason about, for example, a smooth manifold with Minkovskian metric having, in a neighbourhood of any point, a family of covector fields, that is a basis at every point in the neighbourhood and asymptotically diagonalises (to the usual Minkowskian standard form) our metric near that point, with various strengths of how close to diagonal it gets as we approach the point. The coarsest case merely requires the co-ordinates to be diagonal at the point and continuous near it; but we may also be interested in the departure from diagonal form rising no faster than some power of departure from the point. Having differentiable co-ordinates at the point should hopefully correspond to (at most) linear growth; having derivative zero at the point and continuous near it should likewise correspond to quadratic growth. Reasoning about such growth should not be tied down to a particular way of reducing differences between quadratic forms to a positive real (or zero when the forms are equal), or of reducing differences of position to a positive real, in order to show that any requested upper bound on the former can be attained by imposing a suitable upper bound on the latter. Orthodoxy answers that need by using abstract topology, which abstracts away the positive real measurements behond the concept of open neighbourhoods, but gets somewhat problematic to reason about when those can be pathologically weird in form.

My approach to this thus far has been to, at least in vector spaces over the reals, effectively use voluminous (i.e. non-degenerate) simplicces. The nature of simplices gives a natural meaning to their interior for which the interior of any voluminous simplex does indeed match the interior in the orthodox topological sense; while the simplices (including boundary) are conveniently compact, in the orthodox topological sense, so this is merely a specialisation of the orthodox approach; but it's one I feel more comfortable thinking about, even though I still haven't escaped from reductio. I have at least been able to recast the challenge-response protocol in terms of taking intersctions of simplices satisfying particular conditions.

For example, for my Minkowskian local basis, the co-ordinates of the metric with respect to that basis lie in some space of lists of lists of reals, within which I can define simplices; in the smooth manifold, I have charts that map patches of the manifold to real vector spaces, within which I have simplices. So I can take any simplex within the vector space image of a chart, pull it back to the manifold and, if my point of interest lies within the result, take the collection of co-ordinates of the metric, with respect to the given local basis, and take an interest in every simplex, in my space of lists of lists of reals, that subsumes that collection of co-ordinates. The intersection of all such simplices in co-ordinate space is just the collection whose single member whose first diagonal entry is 1, the rest being −1, since the co-ordinates are continuous and that's their value at the point of interest; I can presumably concoct some similar construction to specify stronger conditions of being almost diagonal in a neighbourhood. For example, back to the simplex within a chart's vector space, in whose interior the chart's image of the point of interest lies, there is a natural scaling towards the point's image, by positive scale factors, that always has the point's image in the scaled simplex's interior; when the factor is less than 1, the scaled image lies within the original simplex. For each scaling of the simplex, we get a set of co-ordinates of the metric within the thus-scaled simplex's pull-back to the manifold; so we can limit our interest in simplices in co-ordinate space to those which, when scaled towards the standard Minkowskian diagonal matrix, by the square of any factor t < 1, contains all co-ordinates of the metric within the chart's pull-back to the manifold of its simplex scaled by factor t towards the image of the point of interest. If such simplices in co-ordinate space (exist and) all intersect in a collection whose sole member is the standard matrix, we've got a suitable sense of the local basis making the metric's co-ordinates quadratically close to diagonal near the point.

See also

This whole topic is part of the branch of mathematics called analysis, which includes topology; all of which is mighty interesting, but raddled with reductio. So I haven't written much about it, but others have. One example is (the later portion of) David Morgan-Marr's annotation for one of Bilbert's puns.


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