Actually, I just tried f(x,y) = x^3/(x^2 + y^2), and it is continuous along the path (x, x^1.5) (which is continuous), however, using the definition of differentiability,

lim (x, y) -> (0,0) {f(0 + x, 0 + y) - f(0, 0) - grad(0)*(x, y)}/(x^2 + y^2)^0.5

I am letting f approach along the path y = x^1.5

f(x, x^1.5) = x^4.5/(x^2 + x^3)

lim (x, x^1.5) -> (0, 0) x^4.5/[(x^2 + x^3)(x^2 + x^3)^0.5]

By using L'hopital's rule several times, I still find that this limit is undefined, so I conclude that the function is not differentiable when approached along x, x^1.5,

however, function is continuous when approached along this path because

lim (x, x^1.5) -> (0,) x^4.5/(x^2 + x^3) = 0, so it is continuous along all paths I have tried so far.

Plus, the directional derivative, I let u = (x,y), and then, Duf(0) = lim t -> 0 [f(0 + tu) - f(0)]/t = lim t -> 0 (ty)^3/((tx)2 + (ty)^2)) = 0, since t is left on numerator with no t's on the bottom.

Does this function work?