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.