Answer
The limit is
$$\lim_{(x,y)\to(\pi/4,2)}y\cos xy=\boxed{0}.$$
The function is continuous on its domain.
Work Step by Step
We will calculate this limit by simple substitution:
$$\lim_{(x,y)\to(\pi/4,2)}y\cos xy = 2\cos\left(\frac{\pi}{4}\cdot2\right) = 2\cos\frac{\pi}{2}= 2\cdot0=0.$$
This function is continuous in every point because at every ordered pair $(x_0,y_0)$ just by substitution we have
$$\lim_{(x,y)\to(x_0,y_0)}y\cos xy=y_0\cos x_0y_0.$$
