Closing lemma on the Interval

Question

Let $f:$ $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ a $C^1$ diffeomorphism, $x\in\Omega(f)$.$\space$

How do I prove that $\forall\space\epsilon\gt0$, $\exists\space$ $g:$$\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ with $d(f,g)\lt\epsilon$ $\space$ such that $x\in Per(g)$?

$\Omega(f)=$ { $y\in\mathbb{R}\mid\space\forall\space U, y\in U, $ there is $n\ge1 $ such that $f^n(U) \cap U\neq\emptyset $ }

The distance is given by the uniform convergence of the functions and their first derivative.

Answer 1

Since your $f$ is a diffeo, it is monotone, and a wandering point of a monotone map on $\mathbb{R}$ is already either a fixed point or a point of period 2 (the latter can only happen if $f$ is decreasing).