directional derivatives implication

Question

Supposing we have a function $f(x,y)$ that has directional derivatives at every direction at $(0,0)$ then can we say that $f$ necessarily has continuous partial derivatives at $(0,0) $ ?

Answer 1

Let $f(x,y) = 1$ if $y\geq x^2$ or if $y\leq -x^2$ or if $y=0,$ and $f(x,y) = 0$ otherwise. Then every directional derivative is $0$, but the function is not even continuous at $(0,0)$.