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Continuity: There is a scene after Tony came up with the idea for the fake movie and presented it to the CIA, where he is looking at papers under a magnifying glass. Tony is holding his cigarette in his left hand under the magnifying glass. The very next shot shows his face, and he suddenly has the cigarette in his mouth. Which he then removes with his right hand. The next shot shows it in his left hand again.
Factual error: The embassy takeover was November 3rd. A scene in Washington DC described as 69 days later showed Tony Mendez driving around town with yellow/fall like leaves on the trees, something that would not have been true in the middle of January. A similar scene at the end of the movie has Tony Mendez approaching his house in Virginia and a very fall like scene is shown.
Continuity: When Mendez and the other two are reviewing possible scripts to choose, Mendez walks into the home, shuffles through scripts and finds Argo. When he goes to give the script to the other guys the script is missing its black cover and back cover. It is a plain white script, obviously not the same as the original one he picked up off the desk.
Factual error: In the beginning of the film, when Agent Mendez is called to CIA Headquarters, as he passes the secretary's desk, the outgoing letter propped against her word processor has two postage stamps affixed (2-cent Navajo necklace & 37-cent US Flag) which were first issued and then reissued by the US Postal Service beginning 2003-04, 23+ years after the hostage event in 1980.
Factual error: CIA agent Mendez takes-off from Washington to Tehran, apparently via London because he boarded a British Airways flight, on a twin engine aircraft. Twin engine aircraft flights across the Atlantic were just being introduced by TWA around 1980 and weren't flown by BA until years later, if at all.