  Realizing a quantum theory for gravity based on Asymptotic Safety hinges on
the existence of a non-Gaussian fixed point of the theory's renormalization
group flow. In this work, we use the functional renormalization group equation
for the effective average action to study the fixed point underlying Quantum
Einstein Gravity at the functional level including an infinite number of
scale-dependent coupling constants. We formulate a list of guiding principles
underlying the construction of a partial differential equation encoding the
scale-dependence of $f(R)$-gravity. We show that this equation admits a unique,
globally well-defined fixed functional describing the non-Gaussian fixed point
at the level of functions of the scalar curvature. This solution is constructed
explicitly via a numerical double-shooting method. In the UV, this solution is
in good agreement with results from polynomial expansions including a finite
number of coupling constants, while it scales proportional to $R^2$, dressed up
with non-analytic terms, in the IR. We demonstrate that its structure is mainly
governed by the conformal sector of the flow equation. The relation of our work
to previous, partial constructions of similar scaling solutions is discussed.
