  This article is a review of functional $f(R)$ approximations in the
asymptotic safety approach to quantum gravity. It mostly focusses on a
formulation that uses a non-adaptive cutoff, resulting in a second order
differential equation. This formulation is used as an example to give a
detailed explanation for how asymptotic analysis and Sturm-Liouville analysis
can be used to uncover some of its most important properties. In particular, if
defined appropriately for all values $-\infty<R<\infty$, one can use these
methods to establish that there are at most a discrete number of fixed points,
that these support a finite number of relevant operators, and that the scaling
dimension of high dimension operators is universal up to parametric dependence
inherited from the single-metric approximation. Formulations using adaptive
cutoffs, are also reviewed, and the main differences are highlighted.
