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Suppose I have a bounded open set $\Omega$ in $\mathbb{R}^n$; suppose that I have a map $f:\bar{\Omega}\to \mathbb{R}^m$ which is of smoothness class $C^1$ on $\Omega$ and the derivative $D(f|_{\Omega}):\Omega\to \mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)$ can be extended to a continuous map on $\bar{\Omega}$. Is it true that $f$ is then of class $C^1$ on $\bar{\Omega}$? (Meaning that there is an extension of $f$ to a $C^1$ function defined on an open subset of $\mathbb{R}^n$ containing $\bar{\Omega}$.) I have proven the converse holds, but cannot prove this direction. I would like to avoid using the Whitney Extension Theorem, as I believe this is overkill; my "intuition" tells me that there should be some kind of partition of unity argument.

The reason I would like the above to be true is so that I can finish proving $$ C^k(\bar{\Omega};\mathbb{R}^m) = \{ f\in C^k(\Omega;\mathbb{R}^m) : f\text{ can be extended to a }C^1 \text{ map defined on an open set containing }\bar{\Omega}\} $$ is a Banach space with the $C^k$ norm.

Edit: This question is answered here.

The extension of smooth function

Milk
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  • (1) Are you assuming $\bar\Omega$ is a (piecewise)-$C^1$ manifold with boundary? Without some sort of regularity condition, you won't be able to do anything easy even locally, so a partition of unity argument is moot. (2) If you are assuming regularity, then you can see easily how to get a local $C^1$ extension from an open subset of the closed half-space, and then proceed. – Ted Shifrin Nov 13 '20 at 22:28
  • Sure, let us assume that $\bar{\Omega}$ is a $C^1$ submanifold of $\mathbb{R}^n$ with boundary. Does the problem now reduce to the case $\bar{\Omega}=\mathbb{H}^n$, i.e. if we can do it on the upper half-space, then we can do it on $\bar{\Omega}$? – Milk Nov 13 '20 at 22:43
  • Yes, locally, and then use a partition of unity as you suggested. – Ted Shifrin Nov 13 '20 at 22:46
  • I'm not sure how to prove this special case. If $f:\mathbb{H}^n\to \mathbb{R}^m$ is a map which is $C^1$ smooth on the open upper half space, and the derivative has a continuous extension to $\mathbb{H}^n$, then how can I extend $f$ to a $C^1$ function on an open subset of $\mathbb{R}^n$ containing $\mathbb{H}^n$? – Milk Nov 13 '20 at 23:10
  • This is easier than Whitney. See Borel's Lemma. – Ted Shifrin Nov 13 '20 at 23:14

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