lis[n_] := Refine[
   Table[i^4, {i, 0, 1, 1/n}], {Element[n, Reals], n > 0}];
 
$Assumptions = grad > 0 && Element[grad, Reals];
 
grad = {1, 1, 9, 9, 9, 9, 9, 9, 10, 10, 10};
 
int = Interpolation[grad];
 
$Assumptions = x > 0 && Element[x, Reals];
 
cf = Function[{x, f}, ColorData[{"AvocadoColors", {0, 10}}][int[x]]];
 
cf2 = ColorData[{"AvocadoColors", {0, 10}}] /@ grad;
 
g1 = ListLinePlot[lis, ColorFunction -> cf, 
  ColorFunctionScaling -> False]
g2 = ListLinePlot[lis, ColorFunction -> cf, 
  ColorFunctionScaling -> False]
g3 = ListLinePlot[Table[{i, i^4}, {i, 0, 1, 1/n}], 
  ColorFunction -> cf, 
     ColorFunctionScaling -> False]

Do you want g3 to be like this using cf !?g3 now varying colored