Here is what I am trying to do:
Let $X$ be a paracompact smooth manifold.
Let $C$ be closed, $U$ open and $C\subset U \subset X$ and $f$ is a smooth map on $U$. I want to show that then there exists an open $V$ and smooth $F$ on $X$ with $F\mid_V = f\mid_V$ and $C\subset V \subset U$.
To this end I would like to smoothly extend $f$ to $X$.
Please could someone show me how one can extend a smooth map smoothly?
Because $X$ is paracompact a partition of unity subordinate to $U, C^c$ exists. Let's call the maps in this partition of unity $\varphi_U$ and $\psi$.
I suspect the answer will involve convolution of $f$ with $\varphi_U$ but I have been unable to prove that this is smooth or that it extends $f$ (in fact, it may not extend $f$).