# Whitney extension theorem preserving monotonicity

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.

No also in the strictly increasing case.

Let $$E = [0,1]$$ and $$f: x\mapsto x^2$$ is strictly increasing, and is the restriction of a $$C^\infty$$ function. Any $$C^2$$ extension of this function must have $$F'(0) = 0$$ and $$F''(0) = 2$$, and so for some $$\epsilon > 0$$ must have $$F'(-\epsilon) < 0$$.

On the other hand, I believe the following refined conjecture is true:

If $$f = F|_E$$ where $$F\in C^m$$, $$m\geq 1$$. Suppose $$f$$ is strictly increasing and $$F'|_E > 0$$. Then $$f$$ admits a monotone extension.

• Statement rather than conjecture, perhaps? Nov 4, 2021 at 8:24
• $F''(0)=2$, but this does, of course, not affect this nice example. Nov 4, 2021 at 8:48
• @JochenWengenroth: yes, I realized that myself as I was walking out of my department building yesterday evening. But I hope the MO folks will be more forgiving than my calculus students. :-) [Fixed.] Nov 4, 2021 at 13:34

No in the non decreasing case. Take $$E=[0,1]\cup[2,3]$$, and $$f:x\to x$$ on $$[0,1]$$ and $$f:x\to x-1$$ on $$[2,3]$$. This function is non decreasing. Its only non decreasing extension on $$[0,3]$$ is the function $$f_p = \begin{cases} x & \textrm{ for } x<1 \\ 1 & \textrm{ for } 1\leq x\leq2 \\ x-1 & \textrm{ for } x>2 \end{cases}$$ which is only $$C^{0,1}$$. However there exists many $$C^\infty$$ functions on $$\mathbb{R}$$ which is equal to $$f_p$$ on $$[0,1]\cup[2,3]$$ (you can choose any value you wish in $$\frac52$$ for example).