Continuity of Functions Made Literal
In function analysis and topology, two closely tied fields of mathematics with very broad reach, a central concept is that of the continuous function: a formal property of functions that ensures there are no ‘gaps’ in their graphs. This concept is given a formal definition that can be confusing, and comes with significant technical limitations anyway, but neither of these things are necessary! In this article we’ll look at a first-principles approach to deriving an alternative continuity property one might call ‘incidence continuity’, which can be shown to be equivalent to Cauchy continuity.
First let’s discuss the common definition of continuity, taking as a motivating example the following function:
This function is visibly discontinuous, in that it has a large jump at x=0. In function analysis continuity is given 3 necessary conditions, none of which are satisfied by this function:
- the function is defined when x is any number c
- the limit of the function is as x approaches any number c
- the two values defined above are equal, i.e. f(x) → f(c) as x → c
Clearly this function fails to satisfy the second condition, which means regardless of what we might define f(0) to be, the third condition will never have a formal meaning, let alone be satisfied. This is promising, as there really is no value for f(0) that would make f continuous, but our definition depends now on a concept of limits in order to be useful, which is a major problem when dealing with number systems that can’t always have a well defined concept of limits, such as the system of rational numbers.
In order to overcome this, the field of abstract (or point-set) topology defines continuity in terms of sets that contain all of their boundary points, called closed sets; a function is continuous if the pre-image of any closed set is closed. In number systems or topological spaces where limits are always defined, this definition becomes equivalent to the third property of continuous functions.
Looking at our discontinuous function again, with the closed set S = {y | 1 ≤ y}, and its pre-image {x | 0 < x}, we see that the function being undefined at 0 is exactly the reason why it is not included in this pre-image, despite being a boundary point of it. Modifying this function to take on different values at x = 0, and choosing different closed sets S, can demonstrate how the other necessary conditions of continuity relate to pre-images of open sets, but despite this rigorous match-up to the formal definition we started with, this still won’t do for the rational number system.
Consider the function g(x) = (x-π)/|x-π|, defined only for rational values x, this function will be -1 when x < π, and +1 when x > π, clearly jumping when x is π, but x is never exactly π, since x is always rational. As a real-valued function it would have an undefined limit as x → π, and so the closed set {1} has pre-image {x | x > π }, which is not closed, but in the system of rational numbers this set (or rather the set of rational numbers in it) is a closed set, since it has no rational boundary at all!
In some sense point-set topology treats all incomplete systems as themselves filled with ‘holes’, making many functions continuous simply by ‘jumping’ at locations that don’t actually exist within the space. To overcome this we need a more rigorous concept of what a jump is, which is where it becomes much more useful to consider the number line as a metric space rather than a point-set topology. Metric spaces formalize the concept of distance, and on the number line the most obvious metric is the absolute difference:
d(x, y) = |x−y|
With a metric in place we can define a concept of incidence, saying that two sets are incident if the distance between their elements can become arbitrarily small. If two sets have an element in common, we shall say that they overlap, which is a stronger property than incidence, in that the common element will have ‘arbitrarily small’ distance from itself simply by having no distance at all.
If we understand continuity to be related to ‘gaps’ or ‘jumps’, i.e. non-incident sets, it becomes much more intuitive and versatile to define formally. Taking our example function, we see that the set of positive numbers is incident with the set of negative numbers, but they map to sets that are not incident:
It should be clear that any ‘gap’ or ‘jump’ in a function should correspond to some pair of sets with this property, and that any continuous function on the real number line should map bounded, incident intervals to incident sets, but neither of these will quite do as a defining property of continuity, since there are continuous functions that map for example the incident sets {n+1/n} and {n−1/n} to non-incident images, and an example contradicting the other way, the discontinuous function mapping terminating decimal numbers to 0 and all other numbers to 1 will map any incident intervals to images overlapping at both 0 and 1.
With these in mind we propose to define functions between two metric spaces to be ‘incidence’ continuous if and only if they map any pair of (totally) bounded, incident sets to incident images, a property that is clearly true of functions that do not ‘jump’, and should be false of functions that do. [counterexamples given by readers will be discussed at the end]
For the (much more technically oriented) remainder of this article we take this to be the definition of incidence continuity, but to avoid confusion we will mostly avoid talking about the standard/topological continuity, preferring to simply describe functions as being defined everywhere and/or as approaching values as limits depending on which of those properties hold.
Before continuing, it is worth noting that although it is typical to characterize a definition with a theorem in order to confirm our intuitions about what the definition refers to, this definition was instead constructed by carefully analyzing our own intuitions and directly formalizing that. This means that when we show how the more mainstream properties of being defined and approached by limits relate to this [almost] literal concept of continuity, we do so not to confirm our definition, but rather to demonstrate where the existing usage of the term is and is not accurate at an intuitive level.
The first thing we can show about incidence continuity is that continuous functions always approach their values as limits (in mainstream language they are ‘continuous’ on their domain) and that in complete spaces a function that is defined everywhere and approaches its values as limits is always incidence continuous. In other words when defined in terms of incidence, continuity is simultaneously weaker than the usual meaning, in that a continuous function can now have singularities, and stronger than the usual meaning in the relative topology of the function’s domain, in that a continuous function can no longer have jumps even if the relative topology of the domain would allow it.
We show both relations by contra-positive, allowing us to reason about the existence of discontinuities rather than their absence.
First suppose that a function f does not approach all of its values as limits, which formally means we have some value c and some positive real number ε so that d(f(x), f(c)) > ε, for some family of values x with d(x, c) as small as we like. This directly allows us to construct the two sets {c} and {x | d(f(x), f(c)) > ε}, to be incident with non-incident image, and more specifically we can construct the convergent sequence x(n) to further satisfy d(x(n), c) < 1/n, so that it can be totally bounded, while still being incident with {c} and having non-incident image. By definition we therefore have an incidence discontinuity.
Next suppose that a function is defined everywhere on a complete metric space, but is not incidence continuous, i.e. there are totally bounded sets R and S that are incident but have non-incident images. We can construct sequences r(n) and s(n) to have d(r(n), s(n)) < 1/n, and since both sets are totally bounded, we can choose a sub-sequence of r (or s) that is Cauchy and thus converges to some value c in the closure of R. This sub-sequence gets arbitrarily close to both c and to the corresponding sub-sequence of s, so the sub-sequence of s must also converge to c by the triangle inequality, putting c in the closure of S as well.
Now that we have c common to the closures of R and S, we can show that f(x) does not approach f(c), by first observing that f(c) cannot be incident with both the image of R and of S, since this would make these images incident with each-other, so we can assume that f(c) is at least distance ε from all elements of the image of one of R or S, meaning the respective sub-sequence we chose to converge to c will give us arbitrarily small d(x, c) where x always satisfies d(f(x), f(c)) > ε, i.e. it gives us x approaching c with f(x) not approaching f(c). □
The above discussion and a little thought should make it clear that this interpretation of continuity has a lot of power for reasoning about functions defined on metric spaces that are incomplete, which should prove invaluable for rigorous symbolic/constructive proofs that ought not be weighed down by the complexity of uncountable and undecidable spaces such as the real number line, or any foundational or elementary discussion of Cauchy sequences or Dedekind cuts as concrete instances of this real number line, in which any continuous function can be extended to the closure of its domain, allowing for very natural definitions of square roots and other inverses as removable singularities of partial functions.
Additionally any abstract field of study currently dependent on the topological sense of continuity could be strengthened and/or weakened to depend on this metrical sense of continuity instead, giving a more powerful alternative meaning to objects such as paths, homotopy groups, homeomorphisms, and perhaps derivatives and integrals as well.
[Final thoughts, written 2020/10/15]
Since writing this article, a number of astute readers have pointed out functions that appear continuous while violating the above formulation, while others have connected the above formulation to both uniform and Cauchy continuity. First a known result was recalled by one reader, that uniform continuity is equivalent to mapping all incident sets to incident images, then later a connection to Cauchy continuity was suggested by another.
The above argument for how topological continuity and incidence continuity relate, can be very easily modified to show that incidence continuity is always equivalent to Cauchy continuity, helpful in showing this is the observation that if two Cauchy sequences approach each other then the spliced sequence q(2n) = s(n); q(2n+1) = r(n) will be Cauchy, and must have a Cauchy image by Cauchy continuity.
Multiple readers pointed out a function that is topologically continuous when restricted to the (incomplete) domain of the positive reals, but nonetheless incidence (i.e. Cauchy) discontinuous, the example given was sin(1/x) but 1/x does the same so we discuss this. We can pick sequences like r(n) = 1/(2n) and s(n) = 1/(2n+1), both Cauchy, convergent to 0, and hence incident with each other, but with images 1/r(n) = 2n and 1/s(n) = 2n+1, which are neither incident with each other nor Cauchy on their own. This is fine in the broader context of Cauchy continuity, but since our stated goal was to characterize an intuitive notion of ‘gaps’, this counterexample is a significant contradiction.
The error in our formulation of incidence continuity was in thinking that “all bounded incident intervals have incident images” was a stronger statement than “all bounded incident sets have incident images”, as seen in the 1/x counterexample, where all bounded incident intervals do in fact have incident images. The appropriate amendment would have been to forget about general metric spaces, and simply require that all incident, dense subsets of intervals should have incident images.
If incident sets characterize uniform continuity, and totally bounded incident sets characterize Cauchy continuity, then it seems that incident, dense subsets of intervals characterize a kind of ‘Dedekind’ continuity, that whenever sup A = inf B, f(A) is incident with f(B). This one might be a genuinely new formulation of continuity, and one that actually characterizes ‘gaps’ in the most literal way possible.
In a previous version of this article I did not call my formulation ‘incidence continuity’ but instead referred to it simply as continuity, since it was supposed to be the most literal possible formulation of continuity. Now that I have found something that is actually a literal formulation of continuity, I have edited this article to refer to incidence continuity as such. I have also made other small edits including a change to the opening paragraph to better indicate what the article was about, based on small criticisms from some readers.
While I turned out to be wrong in some of my philosophical claims in the original version of this article, gaining insight from others was a major goal of writing it, (in addition to having a time stamp on my semi-original thoughts) so ultimately the criticisms have made it more worthwhile to have written rather than less. Overall this is an encouraging anecdote for the power of the contemporary web to connect people and ideas, so I hope to continue!
