Summary: We study the properties of a conformal field theory (CFT) driven periodically with a continuous protocol characterized by a frequency \(\omega_D\). Such a drive, in contrast to its discrete counterparts (such as square pulses or periodic kicks), does not admit exact analytical solution for the evolution operator \(U\). In this work, we develop a Floquet perturbation theory which provides an analytic, albeit perturbative, result for \(U\) that matches exact numerics in the large drive amplitude limit. We find that the drive yields the well-known heating (hyperbolic) and non-heating (elliptic) phases separated by transition lines (parabolic phase boundary). Using this and starting from a primary state of the CFT, we compute the return probability \((P_n)\), equal \((C_n)\) and unequal \((G_n)\) time two-point primary correlators, energy density \((E_n)\), and the \(m^{th}\) Renyi entropy \(( {S}_n^m )\) after \(n\) drive cycles. Our results show that below a crossover stroboscopic time scale \(n_c, P_n, E_n\) and \(G_n\) exhibits universal power law behavior as the transition is approached either from the heating or the non-heating phase; this crossover scale diverges at the transition. We also study the emergent spatial structure of \(C_n, G_n\) and \(E_n\) for the continuous protocol and find emergence of spatial divergences of \(C_n\) and \(G_n\) in both the heating and non-heating phases. We express our results for \({S}_n^m\) and \(C_n\) in terms of conformal blocks and provide analytic expressions for these quantities in several limiting cases. Finally we relate our results to those obtained from exact numerics of a driven lattice model.