Given a sequence $(g_n)$ of continuous functions $g_n:\mathbb R \to [0,\infty)$ with the properties $\operatorname{supp} g_n^{(1)} \subset (n,n+1)$ and $\int g_n \, d\lambda=1$ for all $n$, how can I show that $$f:\mathbb R^2 \to \mathbb R, (x,y) \mapsto \sum_{n=1}^\infty(g_n(x)-g_{n+1}(x))g_n(y)$$is continuous but still $$\int\int f(x,y)\,dx\,dy \neq\int\int f(x,y)\,dy\,dx$$
(1) $\operatorname{supp} g_n =\overline{\{ x\in \mathbb R \mid g_n(x)\neq 0\}}$
No idea how to start, a hint would be nice.