What does "discrete" really mean, in plain English?

Question

Can someone explain what a "discrete" function really means, in a philosophical sense, in plain English?

As a guess, does discrete mean there are only points with known values, and nothing in between? And if that's the case, is it possible to truly know what's in between the points somehow?

I mean, linear interpolation would be "fudging it" of course, simplifying a curve to a series of lines. Polynomial interpolation maybe?

Is there a way to 100% accurately represent what would go in between the dots?

Come to mention it, aren't all computed values "discrete"? I.e., when the graphing calculator, or desmos.com or whatever, draws out a graph, isn't it actually plotting a series of output values of an equation, only at small enough increments that you can't see the gaps?

So what I'm asking is, is there actually a deeper, fundamental difference between a discrete function like 

y_0 = 10
y_(i+1) = C/2 + y_i

vs a "regular" function like

y = x

or is it just a matter of similar patterns being represented differently by the computer? Bc both functions can go on forever. And even though the first progresses in discrete "steps," the pattern that it represents must exist at a smaller scale, just maybe not "captured" by the "lens" of this equation? Idk.

Answer 1

A discrete set in a metric space or other topological space, such as the line or the plane or $3$-dimensional Euclidean space, is a space in which every points is (topologically) isolated, and that means each point in the set has an open neighborhood that contains no other points in the set.

For example, the set of integers $\{0,\pm1,\pm2,\pm3,\ldots\}$ is discrete because about every integer, say for example $5,$ you can find an open interval, say $(5-0.1,5+0.1),$ which contains no other integer.

And the set $\left\{ \tfrac 1 n : n=1,2,3,\ldots\right\}$ is discrete, but but if you add the limit point $0,$ getting $\left\{ \tfrac 1 n : n=1,2,3,\ldots\right\} \cup\{0\},$ that is not discrete because $0$ is a limit point rather than an isolated point. In other words, no matter how small an open interval you consider that contains $0,$ that interval also contains other members of the set.

A discrete probability distribution is one consisting entirely of point masses. Thus if a random variable (capital) $X$ has the property that $\sum_x \Pr(X=x)=1,$ where the sum is over all values (lower-case) $x$ that (capital) $X$ could be equal to.

Answer 2

There are many different kinds of infinity. The smallest infinity is the natural numbers, $\mathbb{N} = \{1, 2, 3, \dots \}$. We call its size $\omega$.

Many sets have the same size as $\mathbb{N}$. For example, the set of even numbers also has size $\omega$, or the set of prime numbers, or that of pairs of numbers.

Intuitively, you might think that this doesn't make sense. Aren't there half as many even numbers as natural numbers? Aren't there much more pairs of two numbers?

Mathematically, two sets are the same size, if you can pair up their elements so that no element of either set is left alone. In other words, if you take an infinite set and can count its elements, like $1, 2, 3, \dots$, and make sure every element appears at some point in this counting, then the size of this set is $\omega$.

We call discrete (also, countable) a set that is no bigger than $\mathbb{N}$ - its elements can be counted.

There are, however, bigger infinities than $\omega$. Such an example is the set of real numbers. It's proven that, however you try to match real numbers with naturals, there will always be real numbers left out. As such, the real line is not discrete.

The examples you have with the set of points you have is tricky. There are many different discrete sets.

Suppose we work with integers. Then, we have only integers, and there's nothing in between them.

However, there's rational numbers - fractions. Fractions are also discrete, since they're essentially pairs of integers - numerator and denominator. However, between any two fractions there are more fractions - countably infinite number of them, in fact. Still, the irrational numbers are simply not there.

If you're interested in what happens on the whole real line, even at irrational points - then for all intents and purposes, you're no longer working on a discrete set.

Answer 3

When we collect data, the data is discrete. We have a finite number of observations. We then might fit that data to a continuous curve to model that data. It is worth keeping in mind that ALL data has error to it. Is it possible to know where any of points really are? Not just the interpolated points -- any of the points.

Mathematics works in an idealized world that is not the real world. We are working with purely mathematical objects. Curves are continuous because of how we have defined the curve. For every x in an interval there exist a y. Whether we choose to explicitly calculate those y values or not, they are there. That the graphing software is only representing finitely many points, the software is a simulation of a deeper mathematical world.