Continuity mistake: In 1955, when George walks into his house, Marty thinks of Mystery Science Theatre and looks at the vanishing photograph, and at the dance, after George punches Biff and rescues Lorraine, again Marty looks at the vanishing photo. In these two close-ups the thumb holding the photo looks nothing like Marty's thumb/nail in other close-ups, such as when he powers up the amp at Doc Brown's at the start. (00:03:15 - 01:23:45)
Super Grover
Revealing mistake: Right before Einstein is sent to the future, when a very bright light surrounds the DeLorean, not only is the dog gone, but a black bulky bag has appeared on the passenger's seat. Seemingly a hidden stunt driver or a driving mechanism.
Sacha
Continuity mistake: After the DeLorean disappears with the dog inside, flames appear underneath Marty and Doc. In the close-up of their legs, a shot later, the road is spotless and the flames reappear a second later.
Sacha
Continuity mistake: After Goldie Wilson arrives at the manure truck scene, we can see onlookers approaching at the street corner in front of Texaco - one of which is the kid who Marty borrowed a skateboard from. But once the shot changes, that same kid is instantly standing on the sidewalk near Marty holding a crate instead. (01:07:35)
ryguy_1983
Continuity mistake: Whilst George McFly is pegging out the laundry, Marty is explaining his plan as to how he can make Lorraine fall in love with him. At the start of the scene, both of Marty's breast pocket flaps are untucked. After putting the bag down, one tucks itself in, then they return to normal when Marty says "let's go over the plan again". (01:09:55)
Continuity mistake: As the Delorean accelerates towards Doc and Marty in the car park, in a shot from the car's perspective they're a few feet apart, then we cut to a close up, and they're right next to each other. (00:21:45)
Jon Sandys
Continuity mistake: When George is in the school cafeteria talking to Marty about science fiction stories, he holds the notepad differently with each changing shot. (00:58:55)
ryguy_1983
Continuity mistake: When Doc and Marty are watching the DeLorean burn out from across the vacant Twin Pines Mall parking lot, the radio control unit's antenna is seen lowered one shot then fully raised in the next, despite Doc not having touched it. (00:21:20)
ryguy_1983
Continuity mistake: While Marty is talking with Jennifer in the square he is holding a tape case in his left hand which disappears in the next shots.
oswal13
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 and play it directly from the camera.
Captain DefenestratorTVs 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