$f$ is differentiable at $(0,0)$ because all directional derivatives exist and are finite.

$f$ is not differentiable at $(0,0)$ because the directional derivative in the direction of the unit vector $\mathbf{u} = \langle a, b \rangle$ is not a linear function of $a$ and $b$.

The partial derivatives $f_x(0,0)$ and $f_y(0,0)$ do not exist, meaning the gradient cannot be defined.

$f$ is discontinuous at $(0,0)$, which immediately rules out differentiability.