Differentiability Of Multivariable Function

Question

$$ f(x,y)= \begin{cases} x\sin(1/x) + y\sin(1/y), &xy \neq 0 \\ x \sin(1/x), &y=0, x \neq 0 \\ y \sin(1/y), &x=0, y\neq 0 \\ 0, &x=y=0. \end{cases} $$ I have to check differentiability at origin , i have seen that partial derivatives at $(0,0)$ do not exist and so it is not differentiable at $(0,0)$. Am I right ? Else how to do this. Thanks.