How can I build (i.e. find an explicit formula) a smooth function $f_\epsilon: \mathbb R \to \mathbb R$ depending on a parameter $\epsilon >0$  which is "almost the identity" but constant around a (fixed) value $\bar x \in \mathbb R$ in the following sense:

 - $f_\epsilon \in C^2(\mathbb R)$;
 - $L_\epsilon > f'_\epsilon \ge 0$; 
 - $|f''_\epsilon| \le C_\epsilon $; 
 - $|xf''_\epsilon(x) |\le f'_\epsilon(x)$;
 - $f_\epsilon(\cdot) = \bar x$ in $(\bar x - \epsilon/2, \bar x + \epsilon/2)$.
 - $\|f_\epsilon(\cdot) - \mathrm{Id}\|_{L^p([a,b])} \lesssim \epsilon $ for any $p \in [1,\infty]$ (here $Id(x) = x$ is the identity and $[a,b]$ is any compact interval of $\mathbb R$)