Differentiability, multivariable calculus

I have some doubts about the demonstration of the differentiability. If I'm asked to proof that an average function is differentiable on all of it domain, lets suppose its a continuous function on all of its domain, but it has not continuous partial derivatives. How should I demonstrate that its differentiable? May I use the limit with generic points $\displaystyle (x_0,y_0)$? I mean, if I use this limit (the one with the function and the tangent plane over the square root that represents a disk), and its a differentiable function, with this generic points the limit should give zero, right?

Bye there, thanks for posting.