  The gradient flow transformation can be interpreted as continuous real-space
renormalization group transformation if a coarse-graining step is incorporated
as part of calculating expectation values. The method allows to predict
critical properties of strongly coupled systems including the renormalization
group $\beta$ function and anomalous dimensions at nonperturbative fixed
points. In this contribution we discuss a new analysis of the continuous
renormalization group $\beta$ function for $N_f=2$ and $N_f=12$ fundamental
flavors in SU(3) gauge theories based on this method. We follow the approach
developed and tested for the $N_f=2$ system in arXiv:1910.06408. Here we
present further information on the analysis, emphasizing the robustness and
intuitive features of the continuous $\beta$ function calculation. We also
discuss the applicability of the continuous $\beta$ function calculation in
conformal systems, extending the possible phase diagram to include a 4-fermion
interaction. The numerical analysis for $N_f=12$ uses the same set of ensembles
that was generated and analyzed for the step scaling function in
arXiv:1909.05842. The new analysis uses volumes with $L \ge 20$ and determines
the $\beta$ function in the $c=0$ gradient flow renormalization scheme. The
continuous $\beta$ function predicts the existence of a conformal fixed point
and is consistent between different operators. Although determinations of the
step scaling and continuous $\beta$ function use different renormalization
schemes, they both predict the existence of a conformal fixed point around
$g^2\sim 6$.
