Revealing mistake: When Doc flies the DeLorean up and the wheels get stuck to a banner string, the close-ups of Doc show a lit cockpit, but the shots from the outside show it's dark and one can't really see who's at the wheel. However, for a couple of seconds you get to see it's not Doc but a stunt double or a stiff dummy. (01:23:25)
Continuity mistake: At the start, when Marty and Jennifer kiss, Marty turns toward his parents standing at the door. The screen door has three lower sections, and the house numbers are black. When Doc says, "Where we're going we don't need roads," it then cuts to Biff rushing out the door. Now, in this shot the changes include a screen door with four lower sections, stone wall differs, the house numbers are bright white, and a light fixture directly over the house numbers. (00:01:05 - 00:02:20)
Continuity mistake: When Doc tells Marty, "Don't do anything, don't interact with anyone, and try not to look at anything," note Doc's red shirt design. Then it cuts to a shot of Marty as he says, "I don't get it, I thought you said this had something to do with my kids." The next shot facing Doc, as he says, "Look what happens to your son," his red shirt's design has changed, though it's the same style shirt. The same thing occurs with Doc's yellow shirt with the train design, but it's more subtle. (00:09:25)
Continuity mistake: When Old Biff drives into the garage with Young Biff in the passenger seat, Young Biff (body double) is not wearing a belt, but the belt reappears when they exit the car and Old Biff places the Sports Almanac in Young Biff's back pocket. Also, the Levi's small red tab is now missing from that pocket. (01:07:55 - 01:10:05)
Continuity mistake: When Marty and Doc find Biff's car, Marty is not present in the DeLorean, but he appears in the next shot. (01:29:54)
Visible crew/equipment: When Marty realises that the almanac is not inside the dust jacket, a set light can be seen on the far right side of the screen. It can also be seen when he realises where Biff is. (01:21:08 - 01:21:45)
Continuity mistake: When Marty, Jennifer, and Doc head to the 2015 Hill Valley, part of the text on the dashboard is faded, but when they travel back to 1985, the text is completely intact. Additionally, in the second shot, the 'Last Time Departed' is mentioned as Nov. 12, 1955, despite it being Oct. 26, 1985. (00:38:22)
Continuity mistake: When the police car arrives in Hilldale, there are no door handles on the car, but in the next shot, some appear. (00:27:16)
Continuity mistake: When Marty hides in Biff's car as he walks over to it after leaving the auto shop, the left visor on the car is raised, but in the next shot, it is lowered. (01:05:53)
Factual error: When Marty enters the 1955 Hill Valley, one of the cars has a white licence plate with blue text and a sunset, which was not used until 1982. (01:04:54)
Continuity mistake: When Doc turns the time circuits on as he is about to head over to the high school, the 'Destination Time' is under 'Last Time Departed', but in the next shot, it is back to being over it like normal. It should also be noted that the month is represented by a number in the first shot, but with letters in the second. Additionally, one of the LED's is burnt out in the first, but is back to being lit in the second. (01:23:30)
Revealing mistake: When Doc holds the newspaper with the 'Gang jailed' headline, the same paragraph is repeated several times. (00:24:03)
Revealing mistake: When old Biff turns to face Griff, his elbow passes through Marty's Pepsi bottle. (00:14:41)
Answer: At the end of the first film, Doc says, "what the hell" in response to the letter Marty gave him. So this shows it is not that strong a conviction, especially in the face of definite knowledge about something bad happening. It is also a way for Doc to repay Marty.
MasterOfAll