I have seen the proof of it, but I can't quite intuitively understand why continuous partial derivatives imply that a function is differentiable.
Thanks!
EDIT: The only reason I can think of is that if the partials exist at a point and are continuous at a neighborhood around that point, then it is only logical that the tangent plane to that point exists because the equation for the tangent plane at that point would be a good approximation of the function around that point. If the partials exist at the point but are not continuous at a neighborhood around that point, then I can't see how a tangent plane at the point can exist because would not be a good approximation of the function near the point since the function would have discontinuous slope along certain direction(s)(by the definition of a partial derivative). Is my intuition right?