A definition of Conway base$13$ function
You simply require to switch over to and fro from the lexicographic definition of the base  13 development of the number (consider having ABC as opposed to.  npls) and also the crammed definition you provide to the well  created string as a base  10 number.
An instance of a number for which Conway base  13 function is 0 is
0.12  34+npls 1+2  34..11111111111 ...
where the leftmost. is the threedecimal factor (that is, it has a semantic definition), the 3 rightmost. suggest that the base  13 depiction has a boundless variety of 1 (that is, they have a metameaning), and also the various other 2. are "figures" of the number (that is, they have a syntactic definition).
An instance of a number for which Conway base  13 function is not 0 is
0.12  34+npls 1+2  34.11111111111 ...
At that number, the function has value  34.11111111111 ... (in base 10)
I recognize why the Wikipedia writeup makes use of the symbols it does, yet I locate it aggravating. Below is a transliteration, with some explanation.

Expand x ∈ (0,1) in base 13, making use of figures 0, 1, ..., 9, d , m , p    making use of the convention d = 10, m = 11, p = 12. N.B. for sensible numbers whose most lowered expression a/b is such that b is a power of 13, there are 2 such developments : an ending development, and also a non  ending one finishing in duplicated p figures. In such an instance, make use of the ending development.

Allow S ⊂ (0,1) be the set of reals whose development entails finitely several p , m , and also d figures, such that the last d figure takes place after the last p figure and also the last m figure. (We might call for that there go to the very least one figure 0   9 in between the last p or m figure and also the last d figure, yet this does not appear to be essential.) After that, every x ∈ S has a base 13 development of the kind
0. x _{1 } x _{2 }... x _{n } [ p or m ] a _{1 } a _{2 }... a _{k } [ d ] b _{1 } b _{2 }...
for some figures x _{j }∈ 0, ..., p and also where the figures a _{j } and also b _{j } are restricted to 0, ..., 9 for all j. The square brackets over are just planned for focus ; and also specifically the n+1 ^{st } base  13 figure of x is the last occurance of either p or m in the development of x . 
For x ∈ S , we specify f ( x ) by translating the string layout over. We overlook the figures x _{1 } via x _{n }, translate the p or m as a plus  indicator or minus  indicator, and also the d as a decimal factor. This generates a decimal development for an actual number, either
+a _{1 } a _{2 }... a _{k }. b _{1 } b _{2 }...
or
− a _{1 } a _{2 }... a _{k }. b _{1 } b _{2 }...
according to whether the n+1 ^{st } base  13 figure of x is a p or an m specifically. For x ∈ S , we set f ( x ) to this number ; for x ∉ S , we set f ( x ) = 0.
Keep in mind : this function is not determinable, as there is no other way that you can establish beforehand whether the base  13 development of x ∈ (0,1) has just finitely several occurances of any one of the figures p , m , or d ; also if you are given with a number which is assured to have just finitely several, as a whole you can not recognize when you have actually located the last one. Nonetheless, if you are given with a number x ∈ (0,1) for which you recognize the area of the last p , m , and also d figures, you can calculate f ( x ) really straight.
The suggestion of the Conway base  13 number is to locate a function that is not continual, yet if f(a)<x<f(b)
, after that there is some c
in between a
and also b
with f(c)=x
. This a counterexample to the reverse of the intermediate value theorem.
The function is specified by inscribing base  10 values in the tail (the figures left after missing a limited number). We make use of +
, 
, .
and also the figures to stand for an inscribed number in the tail and also call for the inscribed number to begin with a +
or 
. In base 10, every number finishing in a boundless variety of 9
s can be revised to end in a boundless variety of 0
s rather (ie. 0.999 ... = 1.0). In a similar way, we determine we will certainly revise each Conway number finishing in a boundless variety of +
, to make certain that each actual number has an oneofakind decimal depiction.
Each number can have up to one base  10 inscribed value, which is the outcome of using Conway's Base 13 function if it exists. If no such inscribing exists for x (ie. +
taking place boundless times in the development), after that we specify f (x) = 0.
We after that show that for each and every a
and also b
that we can locate a c
in between with an approximate inscribed value. We first make certain the number being created is in between a
and also b
by replicating adequate figures from a
and also incrementing a figure that will not matter. This is less complicated due to the fact that we have actually forbidden finishing in a boundless variety of +
. We after that concatenate the figures of the authorized base  10 depiction of the value we desire the function to need to those figures we have actually currently dealt with.
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