This question is related to Monotone version of one-dimensional Whitney extension theorem. Let $m$ be a positive integer or $m=\infty$.

Suppose that $E\subset\mathbb{R}$ is a closed set and $f:E\to\mathbb{R}$ is a non-decreasing (strictly increasing) function such that there is a function $F\in C^m(\mathbb{R})$ that coincides with $f$ on $E$. Does it follow that we can choose $F$ to be non-decreasing (strictly increasing)?

This looks like a reasonable conjecture, but I could not find it anywhere in the literature and I would like to know if this is a known fact.

**Edit.** This problem was not well thought. I have to think about it again and try to formulate it in a more reasonable way.