I know the formal definition of derivative. Let's consider the case $ \mathbb{R}\rightarrow \mathbb{R}$.
What i haven't understood is what principle or theorem assures us that, for a continuos function, the "instantaneous rate of change" will converge to a certain number, that is in fact the derivative.
What im looking for is a convergence proof for the derivative. If i search on google there is nothing about it.
Assuming you are looking for basic calculus stuff:
The thing is, one defines a function to be differentiable, iff the beknown sequence of difference quotients is convergent for every point in the domain. For functions that are not in this class you simply cannot form the derivative. As it is already indicated in the comments you can easily come up with counter examples to the assumption that every continuous function is differentiable (e.g the abolute value function).
Just in case : Even if the function is differentiable at evry point of its domain that does not generally imply that the derivative must be a continuous function, or even differentiable itself! There are examples of the form $f(x)=x^s \sin(x^{-s}) (f(0)=0)$ for certain values of s proving this.