Continuity mistake: In 1955, when Doc Brown pulls Marty's letter out of his coat pocket, he either holds the envelope at its short side edge or at its long top edge, depending on the angle of the shots. (01:32:20)
Continuity mistake: When Doc finds Marty's letter in his pocket, the big hook fastened to the DeLorean's passenger side is seen facing the sky. Pretty soon Marty slides across the hood and into the DeLorean, but now at this point it's facing the ground instead. (01:33:35 - 01:35:50)
Continuity mistake: After the clock tower's ledge breaks and Doc's hanging off the edge, we see a shot from below, then of Doc looking straight up at the clock, which changes to 10:03. But look at the clock in the below shot: the minute hand ticks over behind him, then moves again in the close-up. (01:34:00)
Continuity mistake: At the clock tower, after the tree limb falls, Doc Brown grabs the loose, messy bundle of rope and instructs Marty, but in the next shot when Doc runs up the stairs, the rope is tied up in a rather tidy bundle. This rope bundle changes yet again as he rushes up and then reaches the top. (01:34:10)
Continuity mistake: When the clock tower bell sounds a third time, Doc backs away from the ledge and stares down below in shock. Doc's arms are hanging down at his sides, but once the shot changes he's seen cupping his ears. (01:35:35)
Continuity mistake: When the clock tower tolls at 10:00, Marty rushes to the DeLorean and as he drives off there are two huge cable spindles at the curb's red edge, near the cloth (lying on the street) that covered the DeLorean. In the next overhead shot, with Doc Brown on the ledge, the cable spindles are gone down below. (01:35:50)
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: 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: 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)
Continuity mistake: When Marty is at the "Start" line waiting for the alarm clock to ring, he moves up his return time and the engine stalls out. Next we see Marty turning an ignition key with several other keys on a chain like the ones used on ceiling fans with the little balls and a plastic fob with the paper insert that reads 555-1138 Out-A-Time on it. After Marty arrives back In 1985 in town square, he pauses and gets out for a moment. While returning the the car he puts the DeLorean in drive and again the car stalls out. Again we see Marty restarting the DeLorean, we see a different set of keys. The key mysteriously changed in the time travel to an after market type with a thin wire ring on it, no key fob and the other keys are missing. (01:38:43 - 01:42:44)
Continuity mistake: When the DeLorean sets off for the lightning strike, you can see it leaving tracks and spray on a wet road, later there is a close up of one of the wheels, which shows the road to be dry. Throughout this scene the road appears to alternate between wet and dry, during what is supposed to be a storm. (01:39:00)
Continuity mistake: When the Doc is trying to pull the cable to connect it the cable is stuck and when he pulls the cable hard the other connection breaks off. Watch when it breaks off you cn see that after Doc pulled it the cable gave him more length and would surely get Doc's arm to the other cable but when they show the Doc a second after he hasn't moved one bit. (01:39:35)
Continuity mistake: Inconsistent mileage progression on the DeLorean at the beginning of the movie has already been noted. However, mileage when Marty is going back to the future is also inconsistent, both in those scenes and with the beginning of the movie. As Marty is driving toward the cable, the mileage is shown as 33051. Just as Doc begins to slide down to reattach the cable, the odometer is visible again and shows 33044. These scenes were obviously filmed out of sequence, and were filmed before the beginning scenes (32994, 32064, etc) of the movie. (01:39:55)
Continuity mistake: In the first scuffle between Marty and Biff in the school cafeteria, they both grab a hold of each other's shirts until they were stopped by Strickland, but in the next shot, Marty is not holding on to Biff's shirt. (01:40:40)
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: Throughout most of the film, the licence plate on Doc's car reads 8N 39742, but after the DeLorean travels back to 1985, it changes to 7F 42857. (01:41:31)
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: In the scene after Marty returns to 1985 he jumps out of the DeLorean to verify he's back and then jumps back in the car to go save Doc at the mall. In the shot where he's putting the car in gear, you can read part of the yellow "Last Time Departed" display on the time circuits as "1985, 1:20AM", and in the shot where he's trying to start the car, you can read "Oct". It should read "November 12, 1955, 10:04PM" since that was the time he just left. On top of all that, the display appears to relate to when Einstein traveled a minute into the future (you can see when Doc shows Marty the time circuits after he lets Einstein out that the "Last Time Departed" is "October 26, 1985, 1:20AM"), not when Marty traveled back to 1955 (October 26, 1985, 1:35AM). You could argue that the "Last Time Departed" display is not working properly, but Doc uses Marty's last time departed (October 26, 1985, 1:35AM) to set Marty's destination time when he's returning to 1985. (01:42:40)
Continuity mistake: Lorraine Baines' hands are on her hips, a brief cut to Marty and back and she now holding a cup and saucer at chest height. You could argue she had enough time to pick it back up, but this would seem odd as she's only just put it on the table. (01:43:55)
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)
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