This question is a follow up question to this question that I recently asked.
A $C^{\infty}$ function $f:(c,d)\rightarrow\mathbb{R}$ shall be called a local polynomial if whenever $f:(c,d)\rightarrow\mathbb{R}$, then there is some collection of intervals $(a_{n},b_{n})$ such that $\bigcup_{n}(a_{n},b_{n})$ is dense in $(c,d)$ and where $f|_{(a_{n},b_{n})}$ coincides with some polynomial for some $n$. In this question, it is shown that not every local polynomial is a polynomial.
Let's call a function $f\in C^{\infty}(c,d)$ Lebesgue supersmooth if whenever $a_{n}\in\mathbb{R}$ for all $n$, then $\lim_{n\rightarrow\infty}a_{n}f^{(n)}(x)\rightarrow 0$ for almost every $x\in(c,d)$. We shall call a function $f\in C^{\infty}$ 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 for a meager set of real numbers $x\in (c,d)$.
Is every Lebesgue supersmooth function a local polynomial?
Is every Baire supersmooth function a local polynomial?
I am also interested in generalizations of this problem such as when we take higher dimensions or if we replace measure zero or meager with some other notion of smallness.
