Let $1 \leq m \leq n$, $g$ and $h$ continuous function of compact support, $g: \mathbb R^m \to \mathbb R$ and $h:\mathbb R^{n-m} \to \mathbb R$.
We define $g \star h: \mathbb R^n \to \mathbb R$ such that $$ g \star h(x_1, \ldots , x_n) = g(x_1, \ldots, x_m)h(x_{m+1}, \ldots, x_n) $$
Prove that $g \star h$ is continuous of compact support and that $\int g \circ h = (\int g)(\int h)$
Any ideas? I'm trying to prove that it's continuous using the definition, but I can't get anywhere useful. I know how to do the last part (integrals) I just need to see that it is a continuous function and that its support is compact.
Note the projections $\pi_1 : \mathbb R^n \to \mathbb R^m$ and $\pi_2 : \mathbb R^n \to \mathbb R^{n-m}$ defined by $$ \pi_1(x_1, \ldots, x_n) = (x_1, \ldots, x_m), \quad \text{and} \quad \pi_2(x_1, \ldots, x_n) = (x_{m+1}, \ldots, x_n)$$ are continuous. Then $g \star h = (g \circ \pi_1) (h \circ \pi_2)$ is continuous as compositions and products of continuous functions. If the support of $g$ and $h$ are contained in the compact sets $K_1$ and $K_2$ resepectively then it shouldn't be difficult to show the support of $g \star h$ is contained in $K_1 \times K_2$ which will still be compact.