A $2$-variable non-differentiable function whose partial derivatives exist If a $2$-variable function is not differentiable at some single point, but has finite partial derivatives for both variables at that point, can it also have a derivative in any direction, I mean is it valid to say that $dF/dl = \text{grad}F\cdot(l_1\cdot i + l_2\cdot j)$?
For example $f(x,y) = (x^3 - xy^2)/(x^2 + y^2)$ and $f(0,0) = 0$,
We need to determine the set of points $(x,y)$ from $R^2$ such that the function  has a derivative in every direction at any point.

Your $f$ has directional derivatives everywhere on $\mathbb {R}^2.$ The only problem point is $(0,0).$ Check that $\partial f/\partial x (0,0) = 1, \partial f/\partial y (0,0)=0.$ Look along the lines $y = mx$ and see what happens in other directions. One interesting thing is that $f(x,x)\equiv 0.$ Thus the directional derivative in the direction $(1/\sqrt 2,1/\sqrt 2)$ is $0.$ However, $\nabla f (0,0) \cdot (1/\sqrt 2,1/\sqrt 2)  =1/\sqrt 2.$
