Let be $f : \mathbb R^2 \rightarrow \mathbb R$ $$f(x,y)=\cases{2-xy,& if $ x,y \in \mathbb Q $\\ xy, & otherwise }$$ Determine the points at which f has partial derivatives. ATTEMPT: Suppose that partial derivatives exist at the point $(x_0,y_0)$.Case 1: $x_0,y_0 \in Q$. By definition $$f_x(x_0,y_0)= \lim_{x\rightarrow x_0} \frac{f(x,y_0)-f(x_0,y_0)}{x-x_0} $$ Since $\mathbb Q^2$ is dense,there exists $(x_k)\subset \mathbb Q^2$ such that $x_k \rightarrow x_0$.
$$f_x(x_0,y_0)= \lim_{k\rightarrow \infty} \frac{f(x_k,y_0)-f(x_0,y_0)}{x-x_0} $$ $$f_x(x_0,y_0)= -y_0 $$
But if I take the limit with a succession of irrational numbers, I don't conclude anything.