Consider the discrete-time control of a continuous-time process where control inputs are specified at regular time intervals $T_s$. This can be written as the continuous-time system $$\dot x(t) = Ax(t) + Bu(t)$$ where $u(t) = u(kT_s)$, $kT_s\le t < (k+1)T_s$.
The continuous time system can further be written as the following discrete-time equivalent system: $$x((k+1)T_s) = A_Dx(kT_s) + B_Du(kT_s)$$ where $A_D = \exp(AT_s)$, $B_D = \int_0^{T_s}\exp(Az)Bdz$, and $\exp(AT_a)$ is the matrix exponential.
Question: Can I evaluate the controllability of the original system (continuous-time system subject to discrete-time control inputs) by checking the controllability of the pair $(A,B)$?
As I understand you want to deduce the controllability of the system $$\dot{x}(t) = Ax(t) + Bu(kT), ~~ \forall t \in [kT, (k+1)T)$$ To do this first note that $x(kT) = x_k$ where $$x_{k+1} = A_D x_k + B_D u_k$$ provided that $x(0) = x_0$ and $u_k = u(kT)$. This is easy to prove with induction.
So to deduce the controllability of the first system you need to check the controllability of $(A_D,B_D)$ as they are equivalent. It is well-known that if $(A,B)$ is controllable and $T$is not pathological, then $(A_D,B_D)$ is also controllable.
$T$ is pathological if $T(\lambda - \mu) = 2i\pi k$ for some nonzero integer $k$ where $\lambda$ and $\mu$ are any pair of eigenvalues of $A$.