A question about partial derivatives and directional derivative.

Assume $\displaystyle f$ is a real-valued function defined on an open set $\displaystyle E$ of $\displaystyle \mathbb R^n$ and all partial derivatives of $\displaystyle f$ exist on $\displaystyle E$. Is it possible that there is some $\displaystyle {\bf {a}}\in E$ and $\displaystyle {\bf u}\in\mathbb R^n$ such that the directional derivative of $\displaystyle f$ in the direction of $\displaystyle \bf u$ at $\displaystyle \bf a$ does not exist?

Thanks!