Continuity mistake: When Marty's coming home at the beginning of the film, he's watching the discussion between his father and Biff who complains about the wrecked car and his reports. In the two shots which show Marty, there's a glass full of candies near to him. When Biff steps to the glass and lifts the lid, half of the candies are gone. (00:12:15)
Continuity mistake: Lorraine throws Joey's cake on the table and a close-up shows nothing next to the tray. When the angle changes, a Pepsi can has appeared right by the middle of it.
Continuity mistake: When Marty is trying to escape from the van he turns the time circuits on. However, the time circuits were already on - you can see them illuminated when he first puts the car in gear. (00:28:45)
Continuity mistake: 1955 Skateboard Chase Scene around Town square. Throughout the whole chase scene two visible different stunt drivers are used to drive the two different models of Biffs' Ford. One is a 1946, the other is a 1947. Note the different position and shape of the front turn sign. (01:04:25)
Continuity mistake: While being chased by the terrorists Marty passes the J C Penny's store, drives a little farther and passes it again. Just before the Libyans aim their rocket at Marty, their van changes from about 100 yards from the store to right in front of it in less than a second. (00:30:25)
Continuity mistake: When Marty is standing over his father's bed in 1955, Marty puts the headphones on George and then puts the tape in the cassette player. If you notice the hairdryer is on the left side of the belt and is touching his left arm, but when they cut to Marty holding the tape player out in front of him the hairdryer is on the right side. (01:02:10)
Continuity mistake: In one shot of Einstein strapped in the DeLorean, if you look closely in between the steering wheel, the yellow time circuits on the bottom read the month as "10" rather than "Oct." And the red and green time circuits read "11" instead of "Nov." (00:19:20)
Continuity mistake: In 1955, when Doctor Brown is watching the video of himself, he sees himself saying that the Libyans are coming. You can hear a significant pause between "They found me" and "I don't know how but they found me." But in the beginning of the film he says the entire phrase without any pauses. (00:28:45 - 01:08:10)
Continuity mistake: In the cafeteria, when Mr. Strickland is glaring at Marty and Biff, the table Marty was sitting at with George is behind him. When Biff leaves there is a close-up of the table and both the lunch tray and Marty's jacket have changed positions significantly from the previous shots (George had already left). (01:00:30)
Factual error: Marty McFly goes back in time to visit his mum-to-be and finds her watching an episode of The Honeymooners. The date is given as November 5, 1955 and yet the episode of the show - The Man From Space - was not screened until December 31, 1955. (00:44:15)
Revealing mistake: Right before the white smoke shoots out of the time machine after Einstein arrives one minute in the future in the beginning of the film, you can see the 2 fire extinguishers built in to shoot out the "smoke". (00:23:40)
Continuity mistake: After Marty has been thrown into the bookshelf after the amp explodes, he is lying on the floor with his left leg bent at the knee. In the very next shot however, he has not moved but both legs are straightened and on top of each other. (00:04:05)
Revealing mistake: When the DeLorean is struck by lightning near the end of the film, if you look closely, you can see that there are two people in the car, when it should just be Marty. You may have to look closely, but you don't need to zoom in to see it. Also, since they never re-shot this scene, you can see it in all three Back to the Future films. (01:40:40)
Continuity mistake: As Marty drives the DeLorean towards the clock tower, if you look carefully you can see that he drives past the 'Bluebird Motel' billboard at least 3 times (which is where he started from) and drives past that small building with the slanted grey roof in the square about 4 times.
Continuity mistake: When the Libyan terrorists show up in the VW bus, there is no license plate at the front, but in some of the following shots during the scene a blue license plate is affixed to the front bumper. (00:28:55)
Continuity mistake: After Lorraine drops Uncle Joey's cake on the kitchen table, Dave can be seen resting both hands on the back of his chair. When the shot changes, both of his hands are elsewhere. (00:15:00)
Continuity mistake: After Marty drops a quarter in the clock tower fund can, he briefly rests his right hand on top of its red lid. But at the next angle, his left hand is on the lid instead. (00:10:45)
Continuity mistake: The second time the Libyans kill Doc, after the DeLorean disappears, the kiosk is in flames. A frame later the flames are gone. (01:44:19)
Continuity mistake: At the beginning of the movie after Biff leaves the McFly home. There is a scene where Marty and George are talking. In one shot George has three pens in his pocket, a few seconds later he has two. (00:14:14)
Continuity mistake: When the lightning bolt hits the clock tower in 1955 and sends the DeLorean back to 1985, Doc Brown gets up and looks at the clock tower. What happened to that awful lightning storm of only a few seconds earlier? Not even a trace of wind is present in Doc's hair. (01:36:55)
Answer: The video camera was in the DeLorean. With the right kind of adapter, which was common enough in the 80s that Doc might've had it on the camera or been able to jury-rig something in the 50s, it would have been possible to connect it into the antenna screws in the back of the TV like an old Atari and play it directly from the camera.
Captain Defenestrator
TVs in the 50s had a two prong antennae connection (two screws in the back that you put a prong antennae into) TVs in the mid 80s also had this. The coax connection (the one wire that screws in) was starting to become common, but, the two prong connection would have been more likely on any given TV at the time, so, whatever wire they used to preview recordings probably had that. very convenient that Marty brought those cords with him.
An old Atari 2600 RF Adapter would be how one would link a video camera to an old-fashioned television. A simple-enough part that Doc could probably make one with 1950s technology.
Captain Defenestrator