Let $X$ and $Y$ be locally compact Hausdorff spaces. Consider the following fragment from Folland's text "Real analysis", p226:
I checked the errata and the $\overline{U}\times \overline{V}$ in the definition of $F$ should be replaced by $U \times V$ (otherwise we have a problem because $\phi$ is not defined on the closure of $U$ and similarly for $\psi$).
Why is the function $F$ continuous and has support contained in $U \times V?$ I tried to apply the pasting lemma but I'm guessing I'm in desparate need of some coffee because I couldn't show it.
It is clear that $F \mid_{U \times V}$ and $F \mid_{X \times Y \setminus U \times V} = 0$ are continuous.
We know that $K = \text{supp}(\phi) \subset U$ and $L = \text{supp}(\psi) \subset V$ are compact. We claim that $\text{supp}(\phi \otimes \psi) \subset K \times L \subset U \times V$. In fact, $\{ (x,y) \in U \times V \mid (\phi \otimes \psi)(x,y) = \phi(x)\psi(y) \ne 0 \} = \{ (x,y) \in U \times V \mid \phi(x) \ne 0, \psi(y) \ne 0 \}$ is contained in $K \times L$ and the claim follows because $K \times L$ is closed in $ U \times V$.
Since $F(x,y) \ne 0$ is possible only when $(x,y) \in U \times V$ and $(\phi \otimes \psi)(x,y) \ne 0$, we conclude $\text{supp}(F) \subset \text{supp}(\phi \otimes \psi) \subset K \times L \subset U \times V$. In particular $F$ has compact support which is contained in $U \times V$.
The sets $R = U \times V$ and $S = X \times Y\setminus K \times L$ form an open cover of $X \times Y$ and $F$ is continuous on both parts (in fact, $F = 0$ on $S$). Therefore $F$ is continuous.