Thanks for your reply.

The {Since "C1" is equivalent to "derivative exists"} was what I needed to know

So what do you think, is it ok to say: a function is differentiable in the neighborhood of a point A, if and only if its partial derivatives are continious in the neighborhood of A?

In other words, is it a neccesary and sufficient condition for a function to have partial derivatives continious in an interval, to be differentiable there?