  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 $m^{\rm th}$ 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.
