We say $X$ is continuous in probability if for any $t_0 \geqslant 0$ fixed, we have $ \mathbb{P}\left(\left\|X_{t}-X_{t_0}\right\|>\varepsilon\right) \rightarrow 0$ as $t \rightarrow t_0$ for all $\varepsilon>0$. This is also known as stochastically continuous.
When $X$ is also cadlag, it follows that $X$ cannot have fixed discontinuities. As far as I know, a martingale cannot have a fixed discontinituity and its jumps are at random times. Is there a martingale which is not continuous in probability?
I think a martingale can have a fixed discontinuity. For example, let $Z \in \mathcal F_1$ be a Bernoulli random variable independent of $\mathcal F_{1^-}$ with $P(Z=1)=P(Z=-1)=\frac 12$. Then $X_t := Z 1_{t \ge 1}$ is a martingale that is constant at $0$ before $1$, then jumps to $\pm 1$ at time $1$. This martingale is not continuous in probability.