Let $f$ be given as
$$ f(x,y) = \begin{cases} \dfrac{ \sin x - \sin y }{x-y}, & \text{if }\text{ $x \neq y $} \\ \cos x, & \text{if } x \text{ $=y$} \end{cases} $$
My claim is that the function is discontinuous along the diagonal. But how can I show this?