A devilish stair case

We have seen in class an example of a discrete random variable, where the values are dense in the unit interval. The distribution function is a devilish stair: when climbing it up, one has to take in any time interval infinitely many steps. Here is my Mathematica program, which draws the stair. The Deepneslevel measures, how devilish you want to make the stair.


DevilSkin := {Line[{{0,0},{1,0}}]};
Hair[n_]:=Table[Line[{{(2k-1)/2^n,0},{(2k-1)/2^n,2/4^n}}],{k,2^(n-1)}];
DevilHair:=Show[Graphics[{DevilSkin,Table[Hair[n],{n,DeepnesLevel}]}],
  PlotRange->{{0,1},{0,2}},DisplayFunction->Identity];
LevelValues[n_]:=Table[{(2k-1)/2^n,2/4^n},{k,2^(n-1)}];
UpToLevel[nn_]:=Sort[Module[{s={}}, Do[s=Union[s,LevelValues[n]],{n,nn}];s]];
DevilStair[x_]:=Module[{v=.0},Do[If[u[[k,1]]<=x,v=v+u[[k,2]]],{k,Length[u]}];v];
DevilStairCase :=Plot[DevilStair[x],{x,0,1},DisplayFunction->Identity];
HellsBells[DL_]:=Module[{}, DeepnesLevel=DL; u=UpToLevel[DeepnesLevel];
  Show[GraphicsArray[{DevilHair, DevilStairCase}],
  DisplayFunction->$DisplayFunction]];
Display["!psfix -land>devilstair.ps",HellsBells[6]];