# Can a discrete sinusoidal signal represent only limited number of frequencies?

If $\omega$ is the frequency of a continuous time sinusoidal signal, $-\infty < \omega < \infty$.

But when it comes to discrete time sinusoidal signals, the range becomes limited : $-\pi \le \omega_{discrete} \le \pi$.

Does that mean discrete sinusoidal signals cannot represent all frequencies possible ? What is the true picture here ?

It is important to realize that in the discrete-time case, $\omega$ is normalized by the sampling frequency $f_s$:
$$\omega=2\pi f/f_s$$
The inequality in your question is basically the sampling theorem, which says that a discrete-time signal can only represent frequencies lower than half the sampling frequency. All higher frequencies will be folded back into the frequency band $[0,f_s/2]$ (this is called aliasing). If you need higher frequencies then you need to increase the sampling rate.