I have a question. In particular it concerns the answer to the following question:
Smooth map on a "non-open" subset
Shouldn't one extend $F_x$ to $U$ ? How does one achieve this?
Note that the bump function argument only guarantees that we may get a smooth map defined on $U$ that agrees with $F_x$ on a possibly smaller neighborhood. So how does one remedy this situation?
It isn't actually necessary to extend $F_x$. This question is getting at a point which is more or less the reason that bump functions are used:
Let $U\subset M$ be an open subset, $f:U\to\mathbb{R}$ be a smooth function. If $\operatorname{supp}(f)\subset U$, $f$ can always be smoothly extended to $M$, namely by defining $$ \widetilde{f}(x):=\begin{cases} f(x) & x\in U \\ 0 & x\notin U \end{cases} $$ This means that, for instance, for a locally defined function $f:U\to\mathbb{R}$ and a bump function $\psi:M\to\mathbb{R}$ with $\operatorname{supp}(\psi)\subset U$, we may always extend $\psi f$ in this manner to a smooth function $\widetilde{\psi f}:M\to\mathbb{R}$ which vanishes outside of $U$. Frequently this step is left implicit in various constructions involving partitions of unity.