Revealing mistake: When De Niro gets shot in the neck, his bald skin cap wrinkles considerably when he presses the wound with his hand. Confirmed by make-up artist Dick Smith in the making-of documentary. (01:40:00)
Revealing mistake: Travis kills Iris' Time Keeper and fails to commit suicide. He sits by the corpse; you can steel see Murray Moston's belly move, breathing. (01:42:20)
Revealing mistake: Sport initially is cautious enough to inquire Travis about a gun, and suspect him of being a policeman. Forgetting the fact that he's suddenly dumb enough to not recognize just a few days later the weirdo with a huge mole on the cheek he talked to, just because he shaved his hair (maybe he was stoned), he still is approached by a guy with a hand in his pocket, in a dark street, making weird questions, who asks him if he has a GUN, and that then says "Suck on this." That already should make him at least quite jumpy, but then, at this point Robert De Niro pulls the gun out of his pocket...but flubs it, it does not happen smoothly. For a good couple of seconds Harvey Keitel stands in front of him motionless, unfazed like no human being would, just because there is no cue for the actor to do anything but stand there like an idiot and get shot. (01:39:10)
Revealing mistake: When the Senator finishes his speech and steps off the stage, Travis begins to move making his way through the crowd. Several extras are recycled appearing in inconsistent positions - look no further than the black woman with a bright flower patterned pink shirt; she is simultaneously in front of Travis as he begins to move in the outer section, and right between Charles Palantine and the stage in the shot immediately after. And then she keeps popping up where she's not supposed to be, continuously. (01:36:20)
Revealing mistake: After taking Iris out for breakfast, Travis is in his car and looks at the building when Iris lives and/or works. And in that establishing shot, that lasts about a second fading almost immediately, from his car he is watching...himself, because it is re-used footage from the scene of one day earlier, when he was entering the building to pay for her service. The person you see go through the door, is De Niro, same shirt and jeans of that scene. Not only that, most people at the window are the same and in the exact same position, and the passersby in the street are the same. (01:18:30 - 01:29:30)
Revealing mistake: When Travis starts spelling his phony name out to the security guy, a ginger kid walks in, stares at the camera, and hides away as quick as he came. In similar fashion, another kid with a red baseball cap looks at the camera behind De Niro once this walks by evading the photographer. (01:04:20)
Revealing mistake: During De Niro's voiceover in the "Dear father and mother" letter, the camera pans towards the stage where Senator Charles Palantine is giving his speech. As De Niro mentions the anniversary, a dude in the crowd spots the camera and waves hi to it, says something like "Wonder what's up?" to someone next to him in the crowd and leans to his left trying to stay in frame longer. (01:12:05)
Answer: This is just my opinion. Remember Iris's line "Have you ever tried looking at your own eyeballs in the mirror?" or something to that effect. Well, Travis sees the madness reflected in his own eyes, doesn't like what he sees (as it reminds him of what he has done, what he might yet do).