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: Right when the movie begins, a machine opens a dog food can and then pours it in the dog's bowl. When pouring the food, the can isn't moved around at all, but after the food drops, the label is missing a piece. (00:02:25)
Visible crew/equipment: At the end of BTTF they all pile into the DeLorean which reverses up to the camera where the matte box and what looks like the top of the cinematographer's head are reflected in the chrome number plate. (01:46:50)
Continuity mistake: When the JVC recorder is first played, the screen on the TV is a blue colour. Later, when Doc is playing the Libyan part again and again, the screen seems to be a brownish colour. (00:50:10 - 01:05:40)
Continuity mistake: After the scene where Marty and his band are rejected from the school tryouts, Jennifer and Marty are sitting down and a lady comes by and asks them for change to save the clock tower. After they attempt a kiss, Jennifer's dad calls for her from the car. In the background, an African-American man is standing looking in the window of a building, and then walking right. In the next shot, he has moved all the way to the left and is walking right. (00:10:30)
Continuity mistake: When Marty is using the phone to call Doc while he is in the diner, his watch (which is on his left arm) starts beeping and the guy behind the counter gives him a strange look. Marty rips out the phone book page and asks the guy behind the counter where XXX Riverside Drive is. Marty is no longer wearing the watch which he just had on seconds earlier. (00:37:55)
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)
Continuity mistake: After the dance, when Marty arrives at the clock tower in Doc's car (across the street from the DeLorean), he parks precisely midway between the two stores. As the clock tolls, Marty jumps into the DeLorean, and in the overhead shot as Marty drives off, Doc's car is directly in front of the store with the blue awning. (01:31:20 - 01:36:00)
Continuity mistake: When Doc points out that Einstein's watch is in sync with his, the LED display changes from being off in a wide shot to on in the closeup. (00:20:05)
Continuity mistake: When Marty McFly is being chased around the town square by Biff and his buddies in the black convertible, an extra set of lights appear under the headlights on the car (goes from a 1946 Ford to a '47/'48). (01:06:45)
Revealing mistake: After Einstein's time travels, when both flames cross Marty and Doc's legs, they cast no shadow, giving the feeling that they are floating, thus giving away the blue screen effect. (00:22:01)
Revealing mistake: When Doc is using a remote control to drive the DeLorean, they show a close-up of the controller. The controller's power meter (the dial with a needle in the middle) shows no power - it's right over to the left. How's he driving with no battery? (00:20:15)
Continuity mistake: When young George walks to Lorraine's table to read the poetic lines, the pen in front of her books moves suddenly to being on the right side of the books. (01:04:42)
Continuity mistake: When Marty is playing back the Twin Pines Mall demonstration video for Doc, the camcorder tilts itself between shots, judging by the lens which sits higher above the TV set when Doc starts shouting, "1.21 gigawatts." (00:53:00)
Character mistake: At the beginning of the movie Marty is turning on a huge guitar amplifier. The two switches for the filaments are misspelled. They are shown as "filiment." (00:03:25)
Continuity mistake: When Marty tells George how to flirt with Lorraine, they both enter the diner and Marty sits on the back of the counter but disappears between shots, only to reappear later on. Also, the extras around him change, although they're all dressed in similarly colored clothes. Check a man in a plaid jacket next to the blonde girl in pink, who disappears when Marty trips Biff. This continuity havoc was due to the combination of Eric Stoltz scenes and new Michael J. Fox reshoots.
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. (00:21:56)
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.
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)
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)
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