calculus - If all directional derivatives of $f$ in point $p$ exist, is $f$ differentiable?

I am a little bit confused by the various theorems concerning the differentiability of a multivariable function.
Let $f : D \subseteq R^n \to R$ have all directional derivatives in point $p$. Does it directly imply that $f$ is differentiable in $p$? I know that the opposite is true: If $f$ were differentiable, it would imply that it has directional derivatives.

Answer

No, this is not true. Take, for instance

$$\begin{array}{rccc}f\colon&\mathbb{R}^2&\longrightarrow&\mathbb{R}\\&(x,y)&\mapsto&\begin{cases}\frac{x^2y}{x^4+y^2}&\text{ if }(x,y)\neq(0,0)\\0&\text{ otherwise.}\end{cases}\end{array}$$ You can check that, at $(0,0)$, every directional derivative is $0$. However, $f$ is not differentiable there.