Why do we define partitions of unity to be locally finite instead of pointwise finite? The typical definition of a partitions of unity $\{\varphi_i\}_{i\in I}$ subordinate to a cover $\{U_i\}_{i\in I}$ requires that for any $x\in X$, there is a neighborhood $N_x$ containing $x$ s.t. only a finite number of $\varphi_i$ are positive on $N_x$ (see Wikipedia). My question is why can't we define them to be only s.t. for any $x\in X$, only a finite number of $\varphi_i(x)$ are positive? Essentially, I'm asking what situations/problems require us (or at least make it extremely desirable for us) to have a local finiteness condition instead of a pointwise finiteness condition.
Henno Brandsma’s answer tells us that in fact pointwise countable partitions give rise to locally finite ones. That’s quite cool and relevant, but my question really is about asking what major theorems/proofs/concepts are easier using the locally finite definition of partitions of unity. As a counter example to what I’m looking for, the proof of the Whitney embedding theorem (what I would call a “major theorem/proof/concept”) in Section 50 of Munkres only uses that the partitions of unity are pointwise finite.
 A: It's in a sense necessary: Theorem: if $\mathcal{U}$ is an open cover of $X$ and there is a partition of unity $\{\phi_i: X \to [0,1]\}_{i \in I}$ (in the weak sense that $\sum_{i \in I} \phi(x)=1$ for all $x$ and the sum has only at most countably many non-zero terms for each fixed $x$, so that the sum is always countable) such that each $\phi_i^{-1}[(0,1]]$ is contained in some member of $\mathcal{U}$, then $\mathcal{U}$ has a locally finite open refinement and if $X$ is then moreover normal, we can find a locally finite (in the Wikipedia sense) partition of unity for $\mathcal{U}$ as well. This is Lemma 5.1.8. in Engelking's General Topology (revised and completed edition) and the normal part follows from using a closed locally finite refinement plus Urysohn's lemma.
The weak partition concept is the bare minimum of what you want (to use the $\phi_i$ to build global functions) and automatically lead to locally finite refinements of covers... So even asking for pointwise countable leads to local finiteness.
A further advantage is that when $\{f_i\mid i \in I\}$ is a locally finite family of continuous real functions then $\sum_i f_i(x)$ is continuous right away. While if we have a point-countable family of continuous functions this need not be the case (e.g. see this example of a series of countably many continuous functions from $[0,1]$ to $\Bbb R$, that is convergent but whose sum is not continuous; some Fourier series can be even worse behaved..).
