Continuous Sigma-martingales and local martingales are equivalent sometimes?

Question

I was reading through this paper, and they mentioned in the beginning-most portion of it that $\sigma$-martingales and local-martingales are equivalent if they are continuous.

1. Why must they be continuous (ie: I know local implies sigma martingale in general (straightforward by one of their characterization) but what is an example of a local-martingale which is not a sigma martingale)?
2. Why are these equivalent for continuous semi-martingales?

Answer 1

As a partial answer (that is an answer to $1$ only): the process:

$M_t:=\int |Cos(t)|dB_t$, where $B_t$ is a Brownian Motion is a $\sigma$-martingale but not a local martingale.