spawn[{a_Rational, b_Rational}] := 
 Module[{w = (b - a)/3}, {{a - 2 w, a - w}, {b + w, b + 2 w}}]
spawn[intervals_List] := spawn /@ intervals

f[0] = 0;
f[x_] := f[3 x]/2 /; x <= 1/3
f[x_] := 1 - f[1 - x] /; x >= 2/3
CantorFunctionIntervals[n_] := 
 CantorFunctionIntervals[n] = 
  Module[{intervals = 
     Flatten[Union[{0, 1}, 
       Cases[NestList[spawn, {1/3, 2/3}, 
         n], _Rational, \[Infinity]]]]}, intervals]

CantorFunctionPlot[n_, opts___] := 
 Module[{intervals = CantorFunctionIntervals[n]}, 
  ListPlot[Transpose[{intervals, f /@ intervals}], opts, 
   PlotJoined -> True, Axes -> True]]

CantorFunctionValues[n_] := 
 Module[{intervals = 
    CantorFunctionIntervals[n]}, {Transpose[{intervals, 
     f /@ intervals}]}]

CantorFunction[x_?NumericQ, n_: 5] := 
 f[Select[CantorFunctionIntervals[n], # >= x &, 1][[1]]]
Plot[CantorFunction[x, 10], {x, 0, 1}]