Existence of directional derivative

Question

For a two variable function, does the existence of continuous partial derivatives of order 1 with respect to $x$ and $y$ at a point $(x,y)$ imply the existence of the directional derivative in any direction at the point $(x,y)$?

Answer 1

Yes, because:

1. The continuity of the partial derivatives implies that the function is differentiable at $(x,y)$. (This is a standard multi-variable calculus theorem.)
2. When a function is differentiable, it has directional derivatives in any direction.