It's easy to find a function $f(x,y)$ whose directional derivatives are all zero at a point but which is not differentiable at that point. But my question is, is it true that a function $f(x,y)$ is differentiable at a point if and only if the tangent line of every differentiable curve lying on the surface and passing through the point lies on the same plane?
If not, does anyone know of a counterexample? For instance, is it possible for $$ \frac{d}{dt} f\bigl( x(t), y(t) \bigr)\Big|_{t=t_0} $$ to equal $0$ for every differentiable curve $\langle x, y \rangle = \langle x(t), y(t) \rangle$ passing through $\langle x(t_0), y(t_0) \rangle$ at $t=t_0$ without $f$ being differentiable at that point?