# Discrete Continuity
If everything in the material world is quantized (symbolized as
the 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
search results about them. But I think that someone might have
invented a similar concept. However, I consider this rather a
philosophical issue than a technical issue. The main point is that
"Since the material world is in fact discrete, what does continuity
mean in this discrete world?".
A possible solution is to define n-continuity. Think about it.
Since everything is discrete, every function is in fact a sequence.
Then we could call a sequence in which the difference is <=1 a
continuous 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
continuous as a whole, but if we look at (...5 4 4...) part, we
can say that the sequence is continuous at the 8th (the middle 4).
Thus we could say that a continuous sequence is a sequence of which
every point is continuous. An interesting thing is that continuity
and uniform continuity of a sequence as a whole are the same in
discrete functions.
And we could define n-continuity by setting the limit of differences
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 sequence
is n-continuous at all (the differences are bounded), the sequence
is Lipschitz continuous.
Or we could define it relatively by comparing ratios. It doesn't
makes sense that both (1 4 7) and (1111111 1111114 1111117) are
3-continuous so that those are at the same level of continuity. We
might have to compare 3/4 with 3/1111114. Like as in quantum
mechanics vs classical mechanics. By doing this, we introduce
rational numbers but I guess that's fine. Since Q is countable
anyway and a rational number is just a ratio between two integers.
But a problem is that if we use rational numbers to define continuity,
there is no absolute continuity (= 1-continuity) corresponding 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.