multivariable calculus - Show that $frac{sin(xy)}{y}$ is differentiable at $(0,0)$.

How do I show that the following function is differentiable at $(0,0)$?

$$\begin{cases} \dfrac{\sin(xy)}{y}, & \text{if }y \neq 0 \\ \\ 0, & \text{if }y = 0 \end{cases}$$
I calculated the partial derivatives and

• $f(x) = \cos(xy)$ exists near $(0,0)$ and is continuous


• $f(y) = \dfrac{xy \cos(xy) - \sin(xy)}{y^2}$ exists, but how do I show that it is continuous?