Let's call a $C^{\infty}$-function $f:\mathbb{R}\rightarrow\mathbb{R}$ Lebesgue supersmooth if whenever $a_{n}\in\mathbb{R}$ for all $n$, then $\lim_{n\rightarrow\infty}a_{n}f^{(n)}(x)\rightarrow 0$ almost everywhere. Similarly, let's call $f:\mathbb{R}\rightarrow\mathbb{R}$ Baire supersmooth if whenever $a_{n}\in\mathbb{R}$ for all $n$, then $\lim_{n\rightarrow\infty}a_{n}f^{(n)}(x)\rightarrow 0$ except on a meager set.

Is every Lebesgue supersmooth function a polynomial? $\mathbf{No}$. Answered by Pietro Majer in the comments by referring to this question.

Is every Baire supersmooth function a polynomial? $\mathbf{No}$. Answered by Pietro Majer in the comments by referring to this question.

Do we get the same solutions if we replace "meager" and "measure zero" with "countable"?

What happens in higher dimensions?

I am asking this question mainly out of curiosity and because it seems like such a function would be a good example to have in mind if one exists.