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 $;
• $f_\epsilon(\cdot) = \bar x$ 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 $\mathbb R$)