For any cadlag process $\{X_t\}$ define, as it is customary, $\Delta X_t=X_t-X_{t-}$ the process of jumps of $X$.
We say that $t$ is a fixed time of discontinuity if $$ \mathsf{P}\{\Delta X_t\neq 0\}>0 $$
Following Jacod and Shiryaev's book we have that if $\{T_n\}$ is a sequence of stopping time that exhausts the jumps of $X$ the set $$ D_n =\{t\geq 0| \mathsf{P}\{T_n=t\}>0\} $$ is at most countable.
Even though this claim is quite intuitive, I miss a formal argument to convince myself.
A process is PIIS iff it has a deterministic, time-homogeneous characteristic exponent $\psi_X$ (see II.4.19. in Jacod and Shiryaev for details). By definition $\Delta X_t:=X_t-\lim_{t_n\uparrow t}X_{t_n}=X_t-X_{t^-}$. By dominated convergence and continuous mapping: $$E[e^{i\xi \Delta X_t}]=\lim_{t_n\uparrow t}E[e^{i\xi(X_t-X_{t_n})}]=\lim_{t_n\uparrow t}E[e^{i\xi X_{t-t_n}}]=\lim_{t_n\uparrow t}e^{(t-t_n)\psi_X(\xi)}=1,\,\forall \xi$$ Since $1=\int_\mathbb{R}e^{i\xi x}\delta_0(dx)$ we get $P(\Delta X_t=0)=1,\,\forall t$.