Is the following condition enough for differentiability?

Question

If the partial derivatives of a field $\psi(x,y,z)$ always exists at point $(x_0,y_0,z_0)$ even if we rotate the Cartesian coordinate system in any angle, then can we say $\psi(x,y,z)$ is differentiable at $(x_0, y_0,z_0)$?

Answer 1

No, for example: $$ \psi(x,y,z)=\left(\frac{xy^2}{x^2+y^2},\frac{xz^2}{x^2+z^2},\frac{yx^2}{x^2+z^2}\right) $$ Has a partial derivative in every direction, but is not differentiable on (0,0,0).

Answer 2

No. You also need the partials to be continuous. Then the function is differentiable.