Comments on: Sometimes all functions are continuous

By: Reference for the undefinability of modulus of continuity functional in PCF? | Q&A System

Mon, 26 Dec 2011 02:27:38 +0000

[...] Bauer has composed a quite great weblog post exploring some from the troubles in additional detail, but I’ll summarize only a little bit [...]

By: Andrej Bauer

Andrej Bauer — Tue, 20 Apr 2010 06:58:46 +0000

@Jacob: your function is of course continuous as a map from the streams of digits to digits. However, this function does not represent a function between real numbers because it does not respect equality of reals. Let me call your function `f` for easier notation. Then we have `f(0.99999...) = 0` and `f(1.00000...) = 1`, however the streams of digits `0.99999...` and `1.0000...` both represent number one. One way to get around this problem is to forbid streams which end with `99999...` If we do so your function `f` will be well-defined as a map from the real numbers to digits. However, the topology that is induced on the reals with this representation is not the usual Euclidean topology but rather something that goes under the name Fine metric. And in this topology `f` is continuous again! In general, when we compute with the points of a space `X` we do not actually use the points themselves but their representations in terms od datatypes (such as streams of digits). It is important that we take as the topology of `X` the one induced by the representation, otherwise we can make any function continuous or discontinuous at will just by changing the topology of `X`.

By: Jacob

Jacob — Mon, 19 Apr 2010 22:45:01 +0000

Isn’t it very easy to come up with functions that can be computed in finite, neigh constant time that aren’t continuous. For instance take the function in base 10:

FirstDigit(x)= 0 if the first digit of x is 0,
1 if the first digit of x is 1,
etc,

This function is obviously not continuous and should be computable for any real number, since it’s only needed to look at one digit.

I can only assume there’s something obvious that I’m missing.

By: Andrej Bauer

Andrej Bauer — Mon, 19 Apr 2010 20:17:51 +0000

@NovaDenizen: you misunderstand what it means to inspect the digits of a number, or what it means to have a real number given as a sequence of digits. Namely, the real number is not given by an infinite sequence that is actually written down anywhere, but rather as an algorithm, or a black box, which accepts a number `n` and outputs the `n`-th digit. At no point do I manipulate an infinite amount of information, yet I can look at any digit I want. Of course, I cannot look at all digits in a finite time, which is sort of the point of the post.

By: NovaDenizen

NovaDenizen — Mon, 19 Apr 2010 18:15:58 +0000

If you can write down a base-10 representation of a real number then it must follow that that number is rational and finite, because irrational numbers have an infinite numeric representation and it is impossible to write down an infinite string and all finite base-10 numerals represent finite rational numbers.

So if you can write a number down, I can implement a sgn function that will tell you its sign using effort at most proportional to what it took you to write the number down.

If you get to manipulate real numbers containing an unbounded amount of information, then I get to have functions that process an infinite amount of information in a finite time. No finite tools can work with infinite bundles of information.