Continuity mistake: When Shepherd Book tells everyone about the electrical nets, Jayne says, "One day you're gonna tell us all how a preacher knows so damn much about crime." Shepherd then turns to leave. But in the immediate close up of him and Mal, he turns to leave again.
Continuity mistake: When Zoë and Wash are on the bridge talking about Saffron, Zoë's necklace moves position between shots.
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Continuity mistake: When Saffron first hits Wash, he falls with his head in the centre of the doorway (from our point of view), but a second later, when she goes to lift his legs through the door so she can shut it, his head is noticeably more to the left (again, from the viewer's point of view). (00:27:05)
Continuity mistake: Just before Serenity disembarks from Triumph near the beginning of the episode, there are some small areas of shrubbery and foliage surrounding the exit ramp. But once Serenity takes off, the shrubbery and foliage is gone.
Continuity mistake: When Inara finds Mal unconscious in his quarters, his head is resting up against the wall. When the angle changes to show her running over to him, he has moved away from the wall and his head is on the floor. After she has called for help, his head is once again up against the wall. (00:31:00)
Factual error: Jayne doesn't need the suit to fire his gun, because ammunition contains an oxidizer that would allow it to be fired even without oxygen present.
Suggested correction: From a different perspective it's not a factual error. Because Jane shot multiple bullets. It means the gun still can be shot after the space suit case got pierced from the first bullet. So it's a possibility that the characters thought wrong (even though it's a bit silly for space travellers to not know this fact but doesn't mean it won't happen).
Visible crew/equipment: In the celebration at the beginning, the shot where Jayne receives the rain stick, a camera crew is briefly visible just to the right of Jayne's head right before he says "I will treasure this."