Periodic signals in Continuous and discrete time
A periodic continuous-time signal satisfies $x(t)=x(t+T_0)$ for all $t$. The period $T_0$ doesn't need to be a rational number. A periodic discrete-time signal satisfies $x[n]=x[n+N]$ for all integers $n$. The period $N$ is an integer.
If you sample a periodic continuous-time signal, you don't necessarily get a periodic sequence. E.g., sampling the periodic signal $x(t)=\sin(\omega_0t)$ results in the sequence $x_d[n]=x(nT)=\sin(\omega_0Tn)$, which is only periodic if $\omega_0T=2\pi k/l$ with integers $k$ and $l$.
In general, sampling a $T_0$-periodic continuous-time function with sampling interval $T$ results in a periodic sequence only if
$$kT_0=lT,\qquad k,l\in\mathbb{Z}$$
is satisfied.