real analysis - differentiability at a point (0,0) based on partial derivatives

For $$f(x,y)=\begin{cases} y^2 sin\left(\frac{x}{y}\right) & \text{if } y\neq0 \\ 0 & \text{if } y=0 \end{cases}$$
i've shown that it is continuous and that the partial derivatives exist in $\mathbb{R}^2$. However, it appears that the partials are not continuous at $(0,0)$; is this sufficient to show that $f$ is not differentiable at $(0,0)$?