Monotonous stability for neutral fixed points

We give subtle, simple and precise results about the convergence or the divergence of the sequence (x(n)), where x(j) = f(x(j-1)) for every integer j, when the initial element x(0) is in the neighbourhood of a neutral fixed point, i.e. a point x* such that f(x*) = x* with \f'(x*)\ = 1 (where f is a C-infinity function defined on a subset of R).