Continuity mistake: When Marty chases after Biff on the borrowed 'skateboard', Marty is wearing a dark grey belt and a red/blue print shirt under his red/beige jacket. However, when Marty is hanging on to the front of Biff's car as they turn a corner (and in another shot), Marty (stunt double) is wearing a light brown belt and solid tan shirt. (01:07:00)
Continuity mistake: When Biff and his goon friends are in Biff's car, as they chase Marty on his borrowed 'skateboard', the car's rearview mirror repeatedly disappears and reappears, and the side mirror changes from round to square repeatedly. (01:06:50)
Continuity mistake: When we first see the machine in action, the travel is shown from three different angles. In the second, a line of fire passes directly between Doc's legs, but in the third angle, they pass to the left of both of his legs. (00:21:00)
Continuity mistake: In 1955, when Marty drives the DeLorean to the 'start here' line that Doc painted on the street, the line and large white lettering are faded and filthy, but when Marty gets out of the car to insert the hook pole, the line and lettering are bright and immaculate. When he finally starts the DeLorean they're back to faded and filthy. (01:36:30 - 01:38:55)
Continuity mistake: When Marty is being chased by the Libyans, a few times it cuts away to show the instrumentation in the car. The first time the odometer reads 33064.2 and the trip counter says 88.8; the second time it's goes back a little to 33061.8 and 86.4. A few seconds later we see it as he turns the corner, and it goes down again from 33,061.8 to 32,994.4, the trip counter has gone from 86.4 to 19. In fact it's a completely different set of instrumentation - the bottom of the needle's angled in some shots and flat in others. (00:29:20)
Continuity mistake: When Marty is writing the letter to Doc before he goes into the future, the word "disaster" is the only word in the last line. Yet when Doc shows Marty the letter when it has been all taped up, "terrible disaster" is the last line. Also, the handwritring in each letter is different (i.e. Marty's signature). They are two different letters. (01:40:55)
Continuity mistake: In the cafe in 1955 Marty goes into the phone booth to look up Doc's name in the phone book. When he finds it the name is on the left hand page, but when he tears it out of the book he tears out the right hand page. One could say he turned the page to tear it right-handed, however when he is asking the waiter if he knew where the address was, he is reading the address on the same side he tore out - check the tears on the edge. (00:36:30)
Continuity mistake: When Doc and Marty are at the Twin Pines Mall and Doc is telling Marty about the car, he says "never mind that now, never mind that now..." and Marty has the camera facing the ground, but later when Doc in 1955 is watching the video, you can see him saying that on the video. (00:18:50 - 00:50:05)
Continuity mistake: When Marty is standing over his father's bed in 1955 persuading him to date Lorraine, there is a hair-dryer in Marty's belt that seems to disappear between different frames (note the hair dryer's cord leading to Marty's right hand, as he holds it behind his back). Really obvious. [This scene was originally longer, including a part where Marty removed the hair dryer from his belt. Still a mistake, but there's the explanation.] (00:59:45)
Continuity mistake: Marty has a brainwave about how to generate the power needed to get back to 1985, and shows Doc the flyer he was given, which Jennifer had written "I love you" on the back of, with her number. Problem is that after Doc grabs the flyer, we can see the back of it, and it's totally blank. (00:10:55 - 00:52:05)
Continuity mistake: In 1955, when Marty drives the DeLorean to the large 'start here' line that Doc painted on the street, in the first shot its front is up to the line completely covering the word 'here', with some distance to the last 't' in 'start'. Its position changes in following shots, such as when Marty tries to start the stalled car in the exterior shot, the DeLorean's front end is further back with the word 'here' perfectly visible. (01:36:30)
Continuity mistake: At the end of the movie, when Doc Brown leaves Marty at his house, he then backs up the DeLorean, turns the car around and passes in front of Marty doing 35 mph at the most, but just as he leaves the frame you can hear the swishing noises the car makes when it travels through time. No way could Doc have reached 88mph. (01:42:00)
Continuity mistake: When Marty is in 1955, he is in shock and walking on the pavement when a car comes by. In the first shot when it comes towards him it has two speakers, on the second shot when it's moving away, it only has one. In the second shot of the car with only one speaker, the A-frame sign is also turned the other way. (00:35:15)
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)
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