Hi AKTilted,

I have assumed in your question that $\displaystyle \mathbf{u}$ can be any unit vector. If this isn't true then I don't think it can be shown.

If this is true for any unit vector $\displaystyle \mathbf{u}$, then it is true for each $\displaystyle \mathbf{u}_{i} = (0,\ldots,1,\ldots,0),$ where 1 is in the $\displaystyle i^{\text{th}}$ position. Then we have that

$\displaystyle D_{u}g(0)=\nabla g(0) \cdot \mathbf{u}_{i} = \frac{\partial g}{\partial x_{i}}(0) = 0.$

Now since $\displaystyle g$ does not change in any direction from 0, then it must be a constant function near 0, which is continuous.