$f:\mathbb{R}^{2}\to\mathbb{R}$ , we're given that for any $ v\in\mathbb{R}^{2}$ the directional derivative $ \nabla_{v}f\left(0,0\right)$ exists.
1) Is $f$ continuous at $ \left(0,0\right)?$
2) Additionally, if we assume that $f$ is continuous at $ \mathbb{R}^{2}\setminus\left\{ 0,0\right\}$ , is $f$ continuous at $ \left(0,0\right)?$