Why is this function continuous in the proof of Folland's proposition 7.21

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.