In Robert L. Devaney's "An Introduction to Chaotic Dynamical Systems" it defines a function $f:J\longrightarrow J$ to be expansive if there exists $\nu>0$ such that, for any $x,y\in J$, there exists $n$ such that $\vert f^n(x)-f^n(y)\vert >\nu$, where $f^n$ is the $n$-th iteration of $f$.
One of the exercises in the section asks to prove that $T(x)=\tan(x)$ is chaotic in the entire real line, despite the fact that there are a dense set of points at which an iterate of $T$ fails to be defined.
As I see it, the fact that $\vert\tan(x)-\tan(y)\vert>\vert x-y\vert$ for every $x,y$ in the same interval $(\pi /2+k\pi , \pi /2+(k+1)\pi)$ proves that every interval $U$, however small, is eventually mapped (by some $T^n$) to the entire real line.
Does this prove topological transitivity as $T^k(U)=\mathbb{R}$ will intersect every other interval $V$? Also, if $$A_0=\{\pi/2+k\pi:k\in\mathbb{Z}\}$$ $$A_1=\{x\in\mathbb{R}:T(x)\in A_0\}$$ $$\vdots$$ $$A_n=\{x\in\mathbb{R}:T^n(x)\in A_0\}$$ Then $\bigcup_{n=0}^{\infty}A_n$ is the set where some iterate of $T$ is not defined, does the fact above prove it is dense? For every $x\in\mathbb{R}-\bigcup_{n=0}^{\infty}A_n$ and every interval $I$ centered at $x$, the interval will be eventually mapped to $\mathbb{R}$ so at least one point of $I$ lands in $\pi/2$ so there is a sequence in $\bigcup_{n=0}^{\infty}A_n$ converging to $x$.
Does the same fact prove sensitive dependence on initial conditions for similar reasons? Also how do I prove that periodic points (if there are any) are dense?
Thanks