I need to prove that the function $f(x)=tan(x)$ is Devaney chaotic in $\mathbb{R}$.
I know I need to prove that $f$ is transitive, has sensitive dependence on initial conditions and that the set of periodic points of $f$ is dense in $\mathbb{R}$. I saw this question being asked before here, but I don't see that the fact $|tan(x)-tan(y)|>|x-y|$ is true for every $x,y\in(\pi/2+k\pi,\pi/2+(k+1)\pi)$. I also don't see how sensitive dependence on initial conditions and dense periodic points follow from this fact.
The hints I got are:
- Use "For al $U,V$ open there is a natural number $n$ so that the n'th iterate applied to $U$ intersects $V$" for transitivity where you can ignore that $tan(x)$ is not defined everywhere.
- Show: if $J$ is an interval, there is an iterate of $f$ that maps $J$ onto the real line. Consider what happen if an iterate of $J$ contains a point of the form $\pi/2+k\pi$ for some integere $k$.
Can someone help me with proving that $f(x)=tan(x)$ is Devaney chaotic on the real line?