I know that a stochastic process is said to be predictable if it's measurable with respect to the predictable $\sigma$-field $\mathcal P$, namely the $\sigma$-field generated by all left-continuous adapted processes.
I furthermore know that if $X$ is a càdlàg process then $X(t-)$, (the left hand limit) is a predictable process.
Nevertheless I have some difficulties discerning whether a right-continuous process (the process being non-continuous) can be predictable or not.
Could you give me a hand with this? Thanks in advance.
Yes. In fact, stochastic processes of the form $Z 1_{[\tau,\infty)}$ where $Z \in \mathcal{F}_{\tau-}$ and $\tau$ is a predictable stopping time generate the predictable sigma-algebra (see here), and any such process is right-continuous.