f(x) = 1 if x is rational, 0 otherwise.
It is defined over all real numbers but continuous nowhere. Also if you take the Dirichlet function and multiply it by x so you get
g(x) = x if x is rational, 0 otherwise
…then you have something that is continuous at exactly one place (0) and nowhere else, which also is pretty spectacular.
As well as the game of life, surreal numbers and other famous things, he also produced one of my favourite lesser-known proofs ever - the proof that 91 is the smallest number that looks prime, but isn’t https://youtu.be/S75VTAGKQpk?si=IW791RaeCsXSOrrK “This is an important theorem and a discovery that I’m very proud of”.
https://en.wikipedia.org/wiki/Thomae%27s_function
It is discontinuous at every rational number, but continuous at every irrational number.
Reminds me of the discrete metric in my Real Analysis course: distance is 0 if it is the same point, 1 otherwise.
We used to joke that it was the perfect metric for finding weird counter examples
this weierstrass function is definitely a mind bender I remember from high school.
My favorite counterexamples in analysis were based on the cantor set as it allowed counerexamples for many different scenarios while still producing functions that were “nice” in other respects
h(a/b) = b for rationals a/b (in smallest terms), 0 otherwise
This one has the properties you mention of the Dirichlet function whilst also being unbounded in every non-empty interval of the reals.
I can understand the reactions of the late 19th Century mathematicians, though I suspect much of it was exaggerated rhetoric, not real horror.
So even though there are uncountable reals, for any real number, you can find rational numbers that get arbitrarily close. You can actually see this pretty easily if you think of reals in decimal notation- sqrt(2) is irrational, but
1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414123, ... etc are all rational numbers that get closer and closer to sqrt(2).
So you should think of real numbers as uncountable, but nonetheless surrounded by rationals no matter how close you zoom in.
Take any irrational numbers x and y, and suppose for convenience that x < y.
Let d = y-x i.e. the length of the interval between x and y.
Let b be an integer so large that 1/b is less than d.
Consider fractions of the form a/b. As a increases, a/b will eventually (strictly) exceed the value of x. Suppose we fix a to be the smallest such value, i.e. (a-1)/b < x < a/b.
Adding d to both sides we get:
a/b - 1/b + d < x + d
Using the fact that 1/b < d and x + d = y, we get a/b < y.
Therefore x < a/b < y.
after a bit of thought, less surprising if we regard g(x) as a function that is defined pointwise as either i(x) = x or z(x) = 0 depending on some highly discontinuous property of x
z(x) = 0 is a pretty good approximation of i(x) = x at x=0
the reals are defined as limits of the sequences of rationals, and thus the rationals are dense in reals by that definition.
>since rational numbers are countably infinite
while the set of all the infinite convergent series of rationals happens to be strongly larger than countably infinite.
If you are referring to the observable universe being finite, then that's not relevant for the discussion: I am just putting a few more grounded terms on the theorem that computable reals (including rationals) are a countable set. The point is that "for every integer n you can get n+1" is unphysical, yet "grokkable" symbolically, so it works well within a conceptual mathematical universe (regardless of what the physical universe has to say about it). Within this math universe we build an abstract computer that can hold an arbitrary rational/computable number, but only a countable subset of the real numbers, since almost all real numbers cannot be described by any "physical" program, even if that program is larger than the entire universe.
I wish I understood the busy beaver problem / connections to Ramsey theory / etc. However for this intuitive discussion it seems like a serious digression.
In other words, it’s because while there are uncountably many irrational numbers, their representation is still only countably infinite each.
Or in yet other words, uncountable infinity is only a teensy bit larger than countable infinity, not that much larger. Consider that every prefix of an irrational number is a rational number. ;)
In fact, here is an uncountable family of them :
f(x) = x^2 * rand(0,1)
Where rand(0,1) is a uniform number between 0 and 1, sampled for every x.
f(x) = 0
f(x) = ½ x²