I have been reading a lot, trying to find this. But all I have been able to find is the construction of partition of unity, but explicit detail of the utility seems to be difficult to find. Te be precise, my doubt is the following:
Given a manifold $M$ and a cover $\mathcal{O}$ to which a partition of unity $\{\phi_U \}_{U\in \mathcal{O}}$ is subordinate. Suppose that $\mathcal{O}$ is constructed by local charts $(x_U,U)$. If $f_U:U\rightarrow \mathbb{R}$ is a smooth function for each member of the cover, then we can ''glue'' them together by defining $f:M\rightarrow \mathbb{R}$ as $$f=\sum_{U\in \mathcal{O}} \phi_U \cdot f_U $$ My question is, why is $f$ smooth? But really, why? Because, given the partition of unity, we have that this sum will be finite for any $p\in M$, thus it is a finite sum of the product of smooth functions, thus it is smooth. It seems as if all that we needed was that for each $U\in \mathcal{O}$: the set $\{p\in M: \phi_U(p)\neq 0 \}$ to be locally finite. Why do we demand supp $\phi_U\subseteq U $ and $\sum_{U\in \mathcal{O}} \phi_U(p)=1 $ for every $p\in M$. Are these conditions really necessary?