Global maximum of ?(x)-x

[Joseph Biberstine]

Gerald McGarvey wrote:
 > Another question is: Where does []x - x have a global maximum?,
 > where []x is the Conway box function (the inverse of Minkowski's
 > question mark function). Some info on []x is at
 > http://en.wikipedia.org/wiki/Minkowski_question_mark_function
 > (which answers my question about ?(1-x)+?(x)). I'm working on PARI
 > code for []x.
 >
 > - Gerry

I finally stopped lazing about and wrote Conway's box for Mathematica. 
Should work fine everywhere.

(* exact; for dyadic x *)
cbd[x_] := (y = Mod[x,1]; If[y == 0, x, d = RealDigits[y, 2]; nd = 
Join[Table[0, {i, 0, -d[[2]]}], d[[1]]]; 
FromContinuedFraction[Join[{Floor[x]}, Length /@ Split[nd]]]]);

(* approx (change magic constant where marked as desired); for all x *)
cbn[x_] := (y = Mod[x, 1]; If[y == 0, x, d = RealDigits[y, 2,(*arb 
precn*)200]; nd = Join[Table[0, {i, 0, -d[[2]]}], d[[1]]]; 
FromContinuedFraction[Join[{Floor[x]}, Length /@ Split[nd]]]]);

- Joseph Biberstine