# Continuity of the directional derivatives implies continuity at the point ?

This might be a trivial question.

Consider a function $f:\mathbb{R^2}\rightarrow \mathbb{R}$ and consider some point $(a,b)\in \mathbb{R^2}$.

Suppose we know that all the directional derivatives $D_{\overline{u}}f(a,b)$ for an arbitrary unit vector $\overline{u}=\langle u_1,u_2\rangle$ in $\mathbb{R^2}$ exist $...(1)$

Furthermore, we also know that $\lim_{\overline{u}\to\overline{v}}D_{\overline{u}}f(a,b) = D_{\overline{v}}f(a,b)$ for some other arbitrary arbitrary unit vector $\overline{v}$$...(2) So all the directional derivatives can smoothly transition into one another as we change the unit vector \overline{u} in \mathbb{R^2}$$...(*)$