Need help with proof concerning chaotic behaviours of the doubling function in discrete dynamical system

Question

The doubling function is given by

$$D(x) = \begin{cases} {2x} & x < \frac{1}{2} \\ 2x -1 & x \ge \frac{1}{2} \end{cases}$$

We know that this function is chaotic on $[0,1)$ and

 A dynamical system F is chaotic if 
1. Periodic points for F are dense
2. F is transitive.
3. F depends sensitively on initial conditions

I can prove that condition 2 and 3 are true for the doubling function D(x). Can anyone help with the first condition?