Let $\sigma\in\mathcal{C}^0(]0,1])$ a positive function such that $\lim\limits_{t\rightarrow 0}\sigma(t)=0$, and $f\in\mathcal{C}^0(\,[0,1]\,)\cap\mathcal{C}^2(\,]0,1]\,)$ such that $\lim\limits_{t\rightarrow 0} \sigma f''(t)=0$.

**Question:** Does there exist a sequence $(f_n)\in\mathcal{C}^2(\,[0,1]\,)$ such that, in the sense of the uniform norm on $\mathcal{C}^0(\,[0,1]\,)$

$$ \lim\limits_{n\rightarrow\infty}f_n=f $$ $$\lim\limits_{n\rightarrow\infty}\sigma f''_n = \sigma f'' $$