Let f(x) = 2x (mod 1) = 2x - Floor(2x).
For any (Subscript[x, 0]) ∈ [0, 1) and any eps > 0, there is a positive integer k and an x with | x - (Subscript[x, 0]) | < eps such that 
| (f^k)(x) - (f^k)((Subscript[x, 0]))| = 1/2.
In other works, every (Subscript[x, 0]) is a sensitive point for f.

x0 = Random[]; (* put this line second for a random x0 *)
x0 = .0001;  (*put this second to choose x0 *)
eps = 10^-4;  (* This positive number can be chosen arbitrarily *)
(*  Make changes only above this line *)
f[x_] := 2x - Floor[2x];
xMin = 0;
xMax =1;
k = Max[1, Floor[-Log[2,eps]] ]; 
initialDist = (2^(-(k+1)));
Print["The initial condition is (Subscript[x, 0]) = ", x0];
Print["For eps = ", eps, ", the smallest positive integer k with (2^(-(k + 1))) < eps is k = ", k];
Print["The initial distance between (Subscript[x, 0]) and x is (2^(-(k + 1))) =  ",(2^(-(k+1)))];
Print["Check: This initial distance is smaller than eps, since eps - (2^(-(k + 1))) = ", N[eps - initialDist], " is positive."  ];
If[k > 1,
Print["Check: Choosing k smaller would not give a distance smaller than eps, since eps - (2^(-k)) = ", N[eps - 2^-k], " is not positive."] ];
nIterations =k;
(* Make changes only above this line *)
path0 = {};
iterates0 = {x0};  (* this will give the seqence of iterates *)
p = x0;  (* p is always the current p_n, p0 is unchanged *)
Do[
newp = f[p];
iterates0 = Join[iterates0, {newp}];
path0 = Join[path0, {{p,p},{p, newp}, {newp, newp}}];
p = newp,
{nIterations}  (*Do it this many times *)
];
If[iterates0[[-1]] > 1/2, x = x0 -(2^(-(k+1))), (*else *) x = x0 +(2^(-(k+1)))];
path = {};
iterates = {x};  (* this will give the seqence of iterates *)
p = x;  (* p is always the current p_n, p0 is unchanged *)
Do[
newp = f[p];
iterates = Join[iterates, {newp}];
path = Join[path, {{p,p},{p, newp}, {newp, newp}}];
p = newp,
{nIterations}  (*Do it this many times *)
];
Print["The itinerary of (Subscript[x, 0]) = ", x0, " is ", Floor[2* iterates0], " etc." ];
Print["The itinerary of x = ", x, " is ", Floor[2* iterates] , " etc."];
Print["The blue dot is at (Subscript[x, 0]), the black dot is at (f^k)((Subscript[x, 0])) = ", iterates0[[-1]], ", and the red dot is at (f^k)(x) = ", iterates[[-1]]];
Print["Check: | (f^k)((Subscript[x, 0])) - (f^k)(x) | = ", Abs[iterates0[[-1]] - iterates[[-1]] ] ];
Plot[{f[x], x}, {x, xMin, xMax}, AspectRatio -> Automatic (* y = x is 45° degree line *), PlotStyle ->{Automatic, Dotted}, Epilog -> {PointSize[Large], Blue, Point[path0[[1]] ], Red, Line[path], Point[path[[-1]] ], Black,Point[path0[[-1]] ],Line[path0]}
]

x0

x