Closure of $f\mapsto\sigma f''$ on $\mathcal{C}^2(\,[0,1]\,)$

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 $$\mathcal{C}^0(\,[0,1]\,)$$

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

• Did you already answer that question with one less derivative? Oct 14, 2021 at 17:46
• No, but I agree that treating first the same question replacing $\mathcal{C}^2$ by $\mathcal{C}^1$ and $f'',f_n''$ by $f',f_n'$ is natural. Oct 14, 2021 at 20:31