Are the functions $\pm \sin(x)$ and $\pm \cos(x)$ the only functions $f:\Bbb R \to \Bbb R$ with $f^{(n)}(\Bbb R)\subset [-1,1]$ for all $n=0,1,\ldots$ and $\sup_{k\in \Bbb N} f^{(k)}(0)=1$.
If the supremum is attained at a finite $k$, then the answer follows from this post as was proved here: Functions $f$ whose values are between $-1$ and $1$, and so do its derivatives. Otherwise there muust be a strictly increasing sequence $\{k_n\}$ such that $f^{(k_n)}(0)\to 1$ as $n\to \infty$. Is this condition sufficient in order to make the conclusions as in the linked posts?
