# Discrete Continuity

If everything in the material world is quantized (symbolized as th
e Planck constant), then what does continuity mean? I mean, as an
analytical sense and not a topological sense. I don't want to talk
about open sets and sheaves and blah blah blah.


These n-continuity, &c. are what I invented. So there will be no s
earch results about them. But I think that someone might have inve
nted a similar concept. However, I consider this as rather a philo
sophical issue than a technical issue. The main point is that "Sin
ce the material world is in fact discrete, what does continuity me
an in this discrete world?".


A possible solution is to define n-continuity. Think about it. Sin
ce everything is discrete, every function is in fact a sequence. T
hen we could call a sequence in which the difference is <=1 a cont
inuous sequence.

For example, a sequence (1 2 3 3 2 3 4) is continuous but (1 4 2 7
3 2 5 4 4) is not.

We could define this locally. Like the latter sequence is not cont
inuous as a whole, but if we look at (...5 4 4...) part, we can sa
y that the sequence is continuous at the 8th (the middle 4). Thus
we could say that a continuous sequence is a sequence of which eve
ry point is continuous. An interesting thing is that continuity an
d uniform continuity of a sequence as a whole are the same in disc
rete functions.

And we could define n-continuity by setting the limit of differenc
es n instead of 1. Then the latter sequence is 5-continuous since
the maximum difference is 7-2 = 5. And we could say that if a sequ
ence is n-continuous at all (the differences are bounded), the seq
uence is Lipschitz continuous.


Or we could define it relatively by comparing ratios. It doesn't m
akes sense that both (1 4 7) and (1111111 1111114 1111117) are 3-c
ontinuous so that those are at the same level of continuity. We mi
ght have to compare 3/4 with 3/1111114. Like as in quantum mechani
cs vs classical mechanics. By doing this, we introduce rational nu
mbers but I guess that's fine. Since Q is countable anyway and a r
ational number is just a ratio between two integers.

But a problem is that if we use rational numbers to define continu
ity, there is no absolute continuity (= 1-continuity) correspondin
g to a quantum. Well, there is still 0-continuity i.e. a constant
function though. Hence I think the original (integers-only) way is
better. But maybe we should use both methods.