$f:\mathbb{R}^2\rightarrow\mathbb{R}, \ f(x,y)=x\cdot\mathbb{D}(y)$, where $\mathbb{D}$ is Dirichlet function (nowhere continuous function). Find all the limits: $\lim_{x\to 0}\lim_{y\to 0}f(x,y)$, $\lim_{y\to 0}\lim_{x\to 0}f(x,y)$ and $\lim_{(x,y)\to (0,0)}f(x,y)$.
Is there any problem with moving $x,y$ to $0$ (no matter in what order)? I think there is no problem and all limits will be equal to $0$, but then this exercise will be rather pointless, so I'm not sure..