For local martingales (as in Pro05)
$\text{[Local/general square integrability]} \overset{?}{\leftrightarrow} \text{[continuity]}$
That is, does one imply the other? I believe $ [\text{continuity}] \rightarrow [\text{local square integrability}]$
Can anyone say one way or another? Any counterexamples to help develop intuition? (I have very little here)