Continuity mistake: When Sam tells Al the bartender that he knows another Al who uses the term "caca," Al leans over toward him to say, "Common expression." The camera angle reverses, and Al is abruptly not leaning any more, but standing back from the bar in his earlier position.
Continuity mistake: In the Japanese bar, when Sam takes a drink from the beer bottle, the label is facing him. When Oswald (who is Sam's reflection in the mirror) takes the same drink, the label has rotated several degrees and is facing outward, away from him.
Revealing mistake: The background while Scott is on the raft with Brooke Shields is blatantly a studio wall. The clouds have streaks on them with dark colors and the seam where the ocean meets the wall can be spotted. The film "The Truman Show" used the same technique.
Factual error: Sam leaps into Marilyn Monroe's chauffeur in April, 1960. But the car he's driving her around in is a 1961 Lincoln Continental convertible. The 1960 had prominent tail fins, the redesigned '61 did not, so the difference is noticeable. This is a model Marilyn could not have purchased until, at the earliest, September of 1960.
Deliberate mistake: Sam, and not the man he leaps into, somehow fathers a daughter with Abigail. Al and Ziggy's answer to "How?" is "We just don't know." Since Sam has always leaped into the physical body of his subjects with no trace of his own physical presence (yes, we see Sam, but no one else does), fathering a child with his own DNA should be impossible. And the characters acknowledging the "mystery" does not excuse the breach of previously established series canon: it merely makes this, at the very least, a deliberate mistake.
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