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

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

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    $\begingroup$ Did you already answer that question with one less derivative? $\endgroup$
    – username
    Oct 14, 2021 at 17:46
  • $\begingroup$ 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. $\endgroup$
    – G. Panel
    Oct 14, 2021 at 20:31


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