Continuity mistake: George McFly's holding a pen in his right hand, then a cut to a wide shot reveals him not holding a pen as he rests his arm away from his body on the table. Then when the camera cuts back his arm is back in front of him holding the pen. (00:14:10)
Continuity mistake: Cafe 1955, when Marty enters for the first time. The arrangement of the coats on the rack to the right-hand side of the cafe doors changes upon Marty's initial entrance and returning to the counter after using the phone booth. (00:35:50 - 00:36:40)
Continuity mistake: In the cafe, the menu in front of Marty changes after Biff leaves. It starts looking like an ice-cream menu, with swirly lines at the top, but just before the future mayor comes into shot, the design on the front changes completely. (00:38:15)
Continuity mistake: Beginning of film. When Marty arrives at Twin Pines as the Doc requested he cruises up alongside Einstein on his skateboard, saying "Where's the doc, boy?" There's a cut to the truck opening, then back to Marty, and for some strange reason the yellow wheels on his skateboard have turned pink, but the reflection on the wet car park is still yellow. It cuts back to the DeLorean coming out of the truck then back to Marty, and his wheels are yellow again. This doesn't happen in the VHS version. (00:17:40)
Revealing mistake: When the DeLorean is about to time travel for the first time, the fuel trails to achieve the following fire trails are already visible on the pavement. (00:22:00)
Continuity mistake: When Lorraine gets George's book out of the package, the bracelet she is wearing on her right arm is down by her wrist in the close-up. In the next shot as George takes the book from her, that bracelet has moved up her arm and is hidden under her sleeve, so when she puts her arm down and then places her arm around George's waist we see that the bracelet has slipped back down to her wrist. (It doesn't disappear, it changes positions between shots.) (01:49:25)
Continuity mistake: During the Johnny B. Goode song, the mike cable's position keeps changing depending on the shot, despite no-one touching it. (01:27:30)
Continuity mistake: When Biff's car slams into the manure truck, in the first shot the side of the truck is adjacent to the car's back passenger door, but in the next overhead shot the side of the truck is adjacent to the front windshield. (01:07:20)
Revealing mistake: When Biff and his gang are chasing after Marty in their car around the Courthouse Square, in the shots that takes place in the Courthouse yard, you can see that it's Biff's stunt double that's driving. (01:06:45)
Revealing mistake: Marty finds Doc's name in the phone book; when he sees the name, then runs his fingernail over to the phone number. Look closely at the phone book; it has several creases across the same place where Michael J. Fox ran his fingernail from previous takes. (00:36:15)
Continuity mistake: When Marty is being chased on the scooter turned skateboard, he is hanging onto the corner of a blue truck. Biff tries to smash him between the two, and the camera shows him pulling himself along the side of the blue truck, then hanging off the back corner again. (01:04:10)
Continuity mistake: When Biff is grousing to George about who will pay his cleaning bill (for spilling beer on his jacket when he wrecked George's car), there is a picture frame on a stand in the background. It's initially facing fairly rightwards, but in the next shot of the argument, it's facing more towards the camera. (00:11:55)
Continuity mistake: In the escape from Biff on the skateboard, Marty collides with a couple near some steps. His skateboard is left at their feet, but after a cut his skateboard's now behind the woman's head in an ideal place for Marty to grab it. The papers thrown in the air suddenly multiply and appear on the ground (under the people, which means they must have dropped the papers first before falling over!). (01:04:15)
Continuity mistake: When Marty and George are in the cafe talking about Goldie Wilson being mayor, George's head is turned facing Marty, but the next second his head is turned the other way. (00:40:35)
Continuity mistake: Near the beginning, right after Einstein time travels, Doc walks up to the time machine and says it's cold. He then opens the door and greets Einstein. In this split second you can see Einstein's stopwatch has a non-illuminated LED display, but in the next shot the face has changed to the bright red numbering. (00:23:10)
Trivia: When Marty first arrives in 1955, right when Mr. Sandman starts playing, you can see a truck directly behind him. If you look closely you can see that the logo on the truck looks like a Miller Beer logo. That's because it *is* a Miller Beer logo. The "Y" is actually a stylized twirl off the "R." (00:33:55)
Continuity mistake: During dinner, when old-Lorraine says "We all make mistakes in life, children", she swaps from having one hand on the glass to both hands, between one frame and the other. (00:15:06)
Continuity mistake: When Michael J Fox's parent kiss and the family picture (with everyone standing on the grass) gets fixed, Michael places the photo on the guitar, gets very happy and starts playing the guitar enthusiastically. Seconds later, a brief shot of the guitar shows the picture with Marty only, his brothers are missing. (01:26:18)
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)
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