Existence of partial derivative and their continuity in a neighbourhood of a point where it is differentiable

Question

Let $f: \mathbb R^2 \to \mathbb R$ be differentiable at some $a \in \mathbb R^2$ , there is it necessarily true that there is a neighbourhood of $a$ in which all the partial derivatives of $f$ exists and they are continuous at $a$ ?

Answer 1

let $f(x) = \|x\|^2 1_{\mathbb{Q}^2} (x) $. Then $f$ is differentiable at $x=0$.

Since $f$ is not continuous on any line passing through any $x\neq 0$, the partials cannot exist anywhere except at $x=0$.