What is "continuous" in Math?
I understand that in Math, there is a common separation between discrete and continuous.
- I understand that in Math we could say that discrete is anything (any set?) which is principally countable;
I therefore assume that in Math, "continuous" would be the opposite of discrete, hence anything (any set?) which is principally noncountable, but it might be wrong.
What is "continuous" in Math?
3 answers
The terms “discrete” and “continuous” are not related to cardinality, but to topology. In particular, you can have a discrete topology on a space of arbitrary cardinality.
Note also that “discrete” is a property of a set in a topological space, while “continuous” is a property of functions between topological spaces. Therefore they are not even really opposites of each other.
A subset of a topological space is discrete if its subspace topology is the discrete topology. This is the case if all the points of the set are isolated, that is, for any point of the set you can find a neighbourhood that contains no other point of the set.
Now the natural opposite of this would be a set that has no isolated points at all. It might make sense to call such sets continuous, but I'm not aware of that being done.
The common use of “continuous” is in respect to functions. In intuitive terms, a function is continuous if the image of points that are close together still stay close together, that is, you can stay close in the image by staying sufficiently close in the domain. The formal definition is that a function is continuous if the preimages of open sets are open.
Conversely you can call a function discrete if its image is discrete. In that case, the only way for the image of a point to stay close is to not move at all. Note that a function can be continuous and discrete at the same time. One obvious example are constant functions. But also functions on a discrete domain are always continuous.
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There are more possible cardinalities above the the Cardinality of the continuum which is $2^{\aleph_0}$ (e.g. $2^{2^{\aleph_0}}$ - that wikipedia page even mentions this, have you actually read that), sets of those sizes generally don't contain discrete things, so your distinction is simply wrong, but at lot of it lies in equalling "principally countable" with "doesn't have the Cardinality of the continuum property".
In a lot of cases it doesn't make any sense to label something as either discrete or continuous, even giving the word "continuous" meaning probably requires some metric.
My take on a criterion to differentiate would be whether every object has a next object (as a criterion for discreteness), and that leads to a countable (perhaps finite) set, whereas continuous means there's an object between (that's where the existence of a metric comes in) any two objects in the set.
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Your question was,"What is continuous in Math". So, I am just talking about continuous and discontinuous.
Continuous function : $f(x)=x$
Discontinuous function : $f(x)=\frac{1}{x-1}$
In simple word : Line of "continuous" goes along with (Continuous can be like this also). And, line of "discontinuous" is discrete.
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