  The asymptotic safety scenario of gravity conjectures that (i) the quantum
field theory of gravity exists thanks to the presence of a non-trivial
ultraviolet fixed point of the renormalization group, and that (ii) the fixed
point has only a finite number of relevant perturbations, i.e. a finite number
of UV-stable directions (or in other words, a finite number of free parameters
to be fixed experimentally). Within the f(R) approximation of the functional
renormalization group equation of gravity, we show that assuming the first half
of the conjecture to be true, the remaining half follows from general
arguments, that is, we show that assuming the existence of a non-trivial fixed
point, the fact that the number of relevant directions is finite is a general
consequence of the structure of the equations.
