Can you give a definition of the Conway base-13 function better than the one actually present on wikipedia (here), which isn't clear? Maybe with some examples?
I understand why the Wikipedia article uses the notation it does, but I find it annoying. Here is a transliteration, with some elaboration.
Note: this function is not computable, as there is no way that you can determine in advance whether the base-13 expansion of x ∈ (0,1) has only finitely many occurances of any of the digits p, m, or d; even if you are provided with a number which is promised to have only finitely many, in general you cannot know when you have found the last one. However, if you are provided with a number x ∈ (0,1) for which you know the location of the final p, m, and d digits, you can compute f(x) very straightforwardly.
You just need to switch back and forth from the lexicographic meaning of the base-13 expansion of the number (think of having ABC instead of .-+) and the loaded meaning you give to the well-formed string as a base-10 number.
An example of a number for which Conway base-13 function is 0 is
where the leftmost . is the threedecimal point (that is, it has a semantic meaning), the three rightmost . mean that the base-13 representation has an infinite number of 1 (that is, they have a metameaning), and the other two . are "digits" of the number (that is, they have a syntactic meaning).
At that number, the function has value -34.11111111111... (in base 10)
The idea of the Conway base-13 number is to find a function that is not continuous, yet if
The function is defined by encoding base-10 values in the tail (the digits left after skipping a finite number). We use
Each number can have up to one base-10 encoded value, which is the result of applying Conway's Base 13 function if it exists. If no such encoding exists for x(ie.
We then show that for each