Example Double Integral: exists or not

Example Double Integral: exists or not

Let $\Omega = [0,1] \times [0,1]$. Let $f \colon \Omega \to \mathbb{R}$ be
$f(x,y)=x$ if $x=y$, and $f(x,y)=0$ otherwise. I would like to show the
integral exists or not using the criterion of Riemann $$
\overline{\int_\Omega} f = \underline{\int_\Omega} f \text{ iff } \forall
\;\epsilon > 0 \; \exists \; P_\epsilon \text{ partition of }\Omega :
U(f,P_\epsilon)&#8722;L(f,P_\epsilon)< \epsilon, $$ some help?